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Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations

Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations FUZZY INFORMATION AND ENGINEERING https://doi.org/10.1080/16168658.2022.2119041 RESEARCH ARTICLE Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations a b b c Zeyad Al-Zhour , Ahmad El-Ajou , Moa’ath N. Oqielat , Osama N. Al-Oqily ,Shadi a d Salem and Mousa Imran Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia; Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan; Department of Basic Sciences, Faculty of Arts and Sciences, Al-Ahliyyah Amman University, Amman, Jordan; Department of Physics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan ABSTRACT ARTICLE HISTORY Received 10 February 2021 Purpose: We construct the analytical approximate resiual power Revised 16 May 2021 fuzzy series solutions of fuzzy conformable fractional differential Accepted 23 August 2022 equations in an r-level depiction in the sense of strongly generalized α-fuzzy conformable derivative in which of the all initial conditions KEYWORDS are taken to be fuzzy numbers. Fuzzy fractional operator; Methodology: The certain fuzzy conformable fractional differential fuzzy fractional power series; equation under strongly generalized α-fuzzy derivative is converted strongly generalised fuzzy derivative to a crisp one as a family of differential inclusions and solved via resiual power method. The main drawback concerning the use of dif- ferential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a fuzzy valued function. Findings: (i) To show the efficiency of our proposed method: Several important and attractive test examples, which included the fractional conformable fuzzy integro-differential equation are discussed and solved in detail. (ii) To show the stability of approximate solutions to specific prob- lems: some graphical results, numerical comparisons and tabulate data are created and discussed at different values of Value: Using the residual power series analysis methos is a pow- erful and easy-to-use analytic tool to solve initial problems on fuzzy conformable fractional differential equations and it suc- cessfully applied to solve real life problems such as the induc- tance–resistance–capacitance, RLC-series circuit. 1. Introduction Most real-world optimisation problems often involve uncertainty or inaccuracy in the data due to measurement errors or some unexpected things. Fuzzy logic is a type of uncertainty that is very common in solving those kinds of problems and it is also used as an important mathematical tool to express the ambiguity and inaccuracy of human thinking [1–4]. Zadeh CONTACT Zeyad Al-Zhour zeyad1968@yahoo.com; zalzhour@iau.edu.sa © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 Z. AL-ZHOUR ET AL. established the fuzzy set theory [1], fuzzy derivative [2] and also introduced the concept of the linguistic approach and its applications in artificial intelligence, decision-making pro- cess information retrieval, and economics [3]. However, Dubois and Prade [4] proposed the concepts of fuzzy differential and differential integration for fuzzy valued functions (F-VFs) and developed by many researchers such as Stefanini [5], Dubois, Prade [6–8], and Puri, and Ralescu [9–10]. Fuzzy fractional differential equations (F-FDEs) appeared as a generalisation of fuzzy differential equations (F-DEs) and they are playing a fundamental role in modelling many real-life fuzzy difficulties and various phenomena under uncertainty that arise many applications such as in quantum field theory and optics, doffing oscillator and population models, electronic, dynamical and control systems, artificial intelligence, industrial engi- neering, financial and management of banking, nature studies and technological processes [11–32]. For example, Guo et al. [11] introduced the fuzzy population models; Agarwal et al. [12] presented the concept of F-FDE; Fard and Salehi [13]; Soolaki et al. [14] discussed the fuzzy fractional variational problems using Caputo and combined Caputo differentiability; Zhang et al. [15] proved the generalised necessary and sufficient optimality conditions for fuzzy fractional problems based on Atangana-Baleanu fractional derivative (FD) and gen- eralised Hukuhara difference; Das and Roy [16] presented a new numerical method for solving F-FDEs using Adomian decomposition method in terms Riemann-Livoullie sense; Salahshour et al. [17] solved F-FDEs by fuzzy Laplace transforms; Ghaemi et al. [18] discussed the numerical solution (NS) of a fuzzy fractional kinetic equation in the term of fuzzy Caputo FD; Marc et al. [19] discussed the potential applications of fuzzy logic in financial and man- agement of banking; Venkat et al. [20] used fuzzy logic in financial markets for decision making; Bede and Gal [21] introduced the strongly-generalised differentiability (SGD) for a F-VF; Hasan et el. [22] investigated the analytical and NSs of fractional fuzzy hybrid system in Hilbert space are devoted to model control via Atangana-Baleanu Caputo FD; Ahma- dian et al. [23–25] used tau method for finding the NSs of a fuzzy fractional kinetic model, uncertain fractional viscoelastic model and linear F-FDEs, respectively; and Sh. Behzadi et al. [26] applied the Fuzzy Picard method for solving fuzzy quadratic Riccati and Fuzzy Painlev equations. Generally, it is not straightforward to obtain exact solutions (ESs) for these kinds of prob- lems because of the difficulties involved, so reliable numerical techniques are needed to deal with these types of problems. Recently, some methods were suggested for creating analytical NSs for uncertain F-FDEs in terms of SGD sense. These methods are the finite ele- ment, reproducing kernel, fuzzy Picard method, homotopy, and residual power series (R-PS) methods [21,26–33]. The R-PS method developed by some authors [30–31] which is considered as effec- tive optimisation technique to construct power series solutions (PSSs) of some F-DEs. Very recently, the R-PS technique was used for solving certain and uncertain F-FDEs [34–41]such as Tariq et al. [35] combined Laplace transform method with the R-PS method in new theory and view and used it to create PSSs for the target equations. Abu Arqub [30] used the R-PS method to obtain the PSSs of F-DEs under SGD; Alaroud et al. [36] presented a novel opti- misation technique, the RPS for handling certain classes of F-FDEs of order 1 < ≤ 2 under SGD; Alshammari et al. [29] presented the analytic NSs of uncertain Riccati DEs using R-PS method; Abu Arqub and Al-Smadi [37] proposed the so-called fuzzy conformable fractional derivative (FC-FD) and integral and then used to obtain the solutions of certain fuzzy con- formable fractional differential equations (FC-FDEs) under SGD and Alshammari et al. [38] FUZZY INFORMATION AND ENGINEERING 3 proposed an accurate numeric-analytic algorithm on the R-PS to investigate the fuzzy NSs for a nonlinear fuzzy Duffing oscillator under SGD. There are many definitions for FDs and the most two important of them are the Riemann- Liouville and the Caputo definitions which are used in many applications and real-natural phenomena. While in 2014, Khalil et al. [39] defined the conformable FD (C-FD) of an order α ∈ (0, 1] of f :[t , ∞) → R at t ≥ 0 as follows: 0 0 1−α f (t + ε(t − t ) ) − f (t) α (α) T f (t) = f (t) = lim , t > t,(1) ε→0 (α) (α) and f (t ) = lim f (t). Provided f (t) is differentiable, and the limits exists. t→t For more details about the basic rules and applications of C-FD, it can be found in the literature [39–47]. The main advantages of the C-FD can be summarised as follow: (i) It can be very easily computed compared with other previous fractional definitions. (ii) It satisfies all concepts of classical derivative while other fractional definitions fail to satisfy some of them. (iii) It can be computed for a non-differentiable function. (iv) It solves F-DEs, PF-DEs, and systems easily and efficiently. (v) It modifies some important transforms such as Laplace, Sumudu, and Nature trans- forms, and they are used as efficacious tools for solving some singular F-DEs. (vi) Several applications have been remodeled using C-FD and it can be opened the door for various new applications. (vii) Several new comparison results can be established based on C-FD. Given the above-mentioned features of the C-FD, we seek through this research to integrate this concept of the FD to the fuzzy equations for studying the extent of these features being reflected in the development of those equations and improving them in terms of realism and the form of the solution. In addition, we aim to investigate the applicability of the RPS method under the assumption of SGD to provide analytical-NSs for the FC-FDEs that of the following general form: ˆ ˆ T i(t) = f (t, i(t)), t ∈ [0, b], 0 <α ≤ 1, (2) subject to the fuzzy initial condition (F-IC): ˆ ˆ i(0) = i (3) where T is the C-FD of an order α, f is an analytic function, i :[a, b] × R → R is a F F continuous F-VF, in which R stands to the group of fuzzy numbers. There are three main approaches to solve FC-FDEs with F-ICs. The first one is that if the initial value is only a fuzzy number, the solution will be a fuzzy function, where as a result, the derivatives must be regarded as fuzzy derivatives. To achieve this problem, the SGD for F-VFs must be used. The second approach is that the FC-FDE will be converted to a crisp one as a family of differential inclusions. The main drawback concerning the use of differential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a F-VF. The last approach is related to the crisp equation and the 4 Z. AL-ZHOUR ET AL. initial fuzzy values, which are in the solution. The weak point lies in having the solution in the fuzzy setting re-written, making the techniques of solution less friendly and more con- strained with a lot of steps of computation to do. To achieve this point, we will replace the crisp equation and the initial fuzzy values in terms of real constants as well as arithmetic operations regarded as operations on fuzzy numbers in the final solution. The latest solu- tions approach, which focuses on searching the fuzzy set of real-valued functions (R-VFs), not F-VFs exemplified with these R-VFs, fulfils the said restrictions. Furthermore, most of the four main interests bring us to fuzzy applications and R-PS method are: (i) They are convenient and remarkably powerful tools in solving numerous issues arising in physics, engineering, financial and management of banking and other sciences. (ii) They are useful in uncertain or estimated reasoning or discretionary thinking, espe- cially for a framework with a scientific model that is difficult to derive. (iii) Useful in assessed qualities under deficient or questionable data, especially for control theory in optimality and security. (iv) (iv) Fuzzy controllers are widespread nonlinear controllers and each one of these exam- inations is fundamental in nature and more investigations should be possible. For the sake of idealism, the optimal fuzzy control field appears to be completely open. This paper consists of five Sections. Some important definitions and concepts for the frac- tional fuzzy theory are given concisely in Section 2. In Section 3, we employ the R-PS method to construct the analytical approximate fuzzy PSSs for the initial value problem (IVP) as in Equations (2) and (3) in an r-level depiction in the sense of strongly generalised α-fuzzy conformable derivative (SG α-FCD). Some important and attractive test examples with illus- trative graphs and tables are given in Section 4 in order to validate our proposed algorithm. Finally, the conclusion is presented in Section 5. 2. Conformable Fractional Fuzzy Theory This section reviews and studies some fuzzy (fractional) theory that will be used in our investigations and findings throughout the paper. Definition 2.1 ([18,22,30]): Let u be a fuzzy set in R, then u is called a fuzzy number if (i) uisanormal (∃t ∈ R such that u(t ) = 1). 0 0 (ii) u is convex (For all t, s ∈ R and 0 ≤ r ≤ 1, then u(rt + (1 − r) s) ≥ min{u (t), u (s)}). (iii) u is an upper semi-continuous ({t ∈ R : u(t) ≥ k} is closed for all k ∈ R) (iv) u is bounded support ([u] = supp(u) is a compact subset of R). Definition 2.2 ([18]): If u ∈ R and r ∈ [0, 1], then the r-level of u is called a crisp set and defined as: [u] ={t ∈ R : u(t) ≥ r}.(4) So,ifu ∈ R , then the r-level of u is a closed interval in R and defined by: [u] = [u (r), u (r)] = [u , u ], (5) 1 2 1r 2r FUZZY INFORMATION AND ENGINEERING 5 r r where u = u (r) = min{s : s ∈ [u] } and u = u (r) = max{s : s ∈ [u] }. 1r 1 2r 2 However, there are several common forms for the fuzzy number u, one of them is the triangu- lar form which is defined by an ordered triple u = (μ , μ , μ ) ∈ R with μ <μ <μ and 1 2 3 1 2 3 whose r-level is [22]: [u] = [μ + (μ − μ )r, μ − (μ − μ )r], r ∈ [0, 1]. (6) 1 2 1 3 3 2 Some conditions must be satisfied by two functions: u , u : [0, 1] → R,sothat [u , u ] can 1r 2r 1r 2r be parametrised by a fuzzy number u for all r ∈ [0, 1]. These conditions are presented in the next theorem. Theorem 2.1 ([22]): Suppose that u , u : [0, 1] → R satisfy the following conditions: 1r 2r (i) u is a bounded monotonic nondecreasing left continuous function on r ∈ (0, 1]. 1r (ii) u is a bounded monotonic nonincreasing left continuous function on r ∈ (0, 1]. 2r (iii) u and u are right continuous at r = 0. 1r 2r (iv) u ≤ u . 1r 2r Then u : R → [0, 1] defined by u(t) = sup{r : u ≤ t ≤ u } is a fuzzy number with r-level 1r 2r r r set: [u] = [u , u ]. Conversely, if u is a fuzzy number with [u] = [u , u ], then the functions: 1r 2r 1r 2r u , u :[0,1] → R satisfy the conditions (i)-(iv). 1r 2r Definition 2.3 ([18,22,48]): Let u = [u , u ] and v = [v , v ] ∈ R , μ ∈ R/{0} and r ∈ 1r 2r 1r 2r [0, 1]. Then we have: (i) Addition operation: r r r [u + v] = [u] + [ v] = [u + v , u + v](7) 1r 1r 2γ 2r (ii) H-difference: If there exists an element z ∈ R such that u = v + z, then z is called H- difference of u and v, denoted by u v, and defined as follows: r r u v = [u] − [ v] = [u − v , u − v ]. (8) 1r 1r 2γ 2r (iii) Scalar multiplication: r γ [μu] = μ[u] = [min{μu , μu }, max{μu , μu }](9) 1r 2r 1r 2r (iv) Multiplication: [uv] = [min{u v , u v , u v , u v }, max{u v , u v , u v , u v }] (10) 1r 1r 1r 2r 2r 1r 2r 2r 1r 1r 1r 2r 2r 1r 2r 2r r r (v) Equality: Two fuzzy numbers u and v are equal if [u] = [v] ,thatisu = v and u = v . 1r 1r 2r 2r Definition 2.4 ([18]): The complete metric structure on R is defined as d : R × R → F F F R ∪{0} such that d(u, v) = sup max{|u − v |, |u − v |}. (11) 1r 1r 2r 2r r∈[0,1] Note that the metric space d in R has many nice and interesting properties on addition operation and scalar multiplication and the reader can be found in the literature [18,36,44–50]. 6 Z. AL-ZHOUR ET AL. Definition 2.5 ([30,48]): Let [a, b] ⊆ R be a vector space and R be a set of fuzzy numbers. ˆ ˆ Then a function i: [a, b] → R is called a F-VF on [a, b]. Corresponding to such a function iand ˆ ˆ r ∈ [0, 1], we denote two R-VFs i (t) and i (t) for all t ∈ [a, b] as: 1r 2r ˆ ˆ ˆ [ i(t)] = [ i (t), i (t)], (12) 1r 2r which is called the r-level representation of a F-VF. ˆ ˆ Definition 2.6 ([18]): Let i and i :[a, b] → R be F-VFs. Then the uniform distance between 1 2 ˆ ˆ i (t) and i (t) is defined as: 1 2 ˆ ˆ ˆ ˆ d (i (t), i (t)) = sup d(i (t), i (t)). (13) 1 2 1 2 t∈[a,b] Definition 2.7 ([18,30,48]): Let i :[a, b] → R be a F-VF and t ∈ [a, b].If ∀ε> 0, ∃δ> 0 F 0 ˆ ˆ such that d (i(t), L)<∈, whenever |t − t | <δ, then we say that L ∈ R is limit of iat t ,which 0 F 0 ˆ ˆ is denoted by: lim i(t) = L. Also, the F-VF i(t) is said to be continuous at t ∈ [a, b] if lim i(t) = t→t t→t 0 0 ˆ ˆ ˆ i(t ). Indeed, i is continuous on [a, b] if it is continuous ∀t ∈ [a, b]. In addition, if i :[a, b] → R 0 F is a fuzzy continuous function, then we have: ˆ ˆ ˆ d(i(t),0) = sup max{|i (t)|, |i (t)|}, ∀t ∈ [a, b]. (14) 1r 2r r∈[0,1] Definition 2.8 ([18,37,48–49]): Let i :[a, b] → R be a F-VF. Then for a fixed point t ∈ F 0 ˆ ˆ [a, b], i is called the SGD at t if there exists an element i (t ) ∈ R such that either: 0 0 F ˆ ˆ ˆ ˆ (i) The H-differences i(t + ε) i(t ) and i(t ) i(t − ε) are exist, and 0 0 0 0 ˆ ˆ ˆ ˆ i(t + ε) i(t ) i(t )  i(t − ε) 0 0 0 0 ˆ ˆ i (t ) = T i(t ) = lim = lim , (15) 0 0 (1) + + ε ε ε→0 ε→0 for all ε> 0 sufficiently near to 0 and the limits in a metric d. ˆ ˆ ˆ ˆ (ii) The H-differences: i(t ) i(t + ε) and i(t − ε) i(t ) are exist, and 0 0 0 0 ˆ ˆ ˆ ˆ i(t ) i(t + ε) i(t − ε) i(t ) 0 0 0 0 ˆ ˆ i (t ) = T i(t ) = lim = lim , (16) 0 0 (2) + + ε→0 −ε ε→0 −ε for all ε> 0 sufficiently near to 0 and the limits in a metric d. ˆ ˆ Remark 2.1 ([30,37]): (i) If i is a differentiable function (DF) for all t ∈ (a, b), then i is an SGD on (a, b). (ii) The limits in Equations (17) and (18) are considered in metric space (R , d) and at the edge-point of (a, b), we consider one direction derivative and recalling that: ˆ ˆ ˆ [i(t)] = [i (t), i (t)]. (17) 1r 2r ˆ ˆ ˆ ˆ Theorem 2.2 ([30,37]): Let i :[a, b] → R be a F-VF and let [i(t)] = [i (t), i (t)], ∀r ∈ [0,1] F 1r 2r 1 1 ˆ ˆ and T i(t) or T i(t) exists. Then (1) (2) FUZZY INFORMATION AND ENGINEERING 7 ˆ ˆ ˆ (i) If i is (1) – DF, then i (t)and i (t)are DFs and, 1r 2r 1 r r 1 1 ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [i (t)] = [i (t), i (t)] = [T i (t), T i (t)]. (18) 1r 2r (1) 1r 2r ˆ ˆ ˆ (ii) If i is (2)-DF, theni (t)and i (t)are DFs and, 1r 2r 1 r r 1 1 ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [i (t)] = [i (t), i (t)] = [T i (t), T i (t)]. (19) 2r 1r 2r 1r (2) Now, we study the α-crisp conformable derivative (α-CCD) for a crisp function i(t) of order α ∈ (0, 1] and the SGα-FCD for F-VF i(t) of order α ∈ (0, 1] as in the following two definitions [30,37,49–52]. Definition 2.9: Let i :[a, b] → R be a crisp function and α ∈ (0, 1]. Then the α-CCD of i at t ∈ [a, b] is defined by: 1−α ˆ ˆ i(t + ε(t − t ) ) − i(t) α (α) ˆ ˆ T i(t) = i (t) = lim , t > t . (20) ε→0 ˆ ˆ Note that, if i is DF for all t ∈ (a, b), then we say iis α-CCD on (a, b). Lemma 2.1 ([40]): For α ∈ (0, 1] and t > t , α 1−α ˆ ˆ ˆ (i) if i :[a, b] → R is DF for all t ∈ (a, b), then T i(t) = (t − t ) i (t). 0 0 α γ γ −α (ii) T (t − t ) = γ(t − t ) for all t ∈ (a, b). 0 0 0 ˆ ˆ Definition 2.10: Let i :[a, b] → R be an FVF and α ∈ (0, 1]. Then the SG α-FCD of iat t ∈ F 0 [a, b] is defined by one of the following subsequent is working: 1−α 1−α ˆ ˆ ˆ ˆ i(t + ε(t − t ) ) i(t) i(t) i(t + ε(t − t ) ) 0 0 (i) T i(t) = lim = lim , t > t , (1) + + ε→0 ε ε→0 ε (21) 1−α 1−α ˆ ˆ ˆ ˆ i(t + ε(t − t ) ) i(t) i(t) i(t + ε(t − t ) ) 0 0 (ii) T i(t) = lim = lim , t > t , (2) + + ε→0 −ε ε→0 −ε (22) 1−α ˆ ˆ ˆ ˆ where ε> 0 small enough and the H-differences: i(t + ε(t − t ) ) i(x) and i(t) i(t + 1−α ε(t − t ) ) exist in both cases (i) and (ii). ˆ ˆ ˆ Remark 2.2: (i) If i is DF for all t ∈ (a, b), then i is SG α-FCD on (a, b). (ii) i is said to be SG ˆ ˆ α(1)-FCD on (a, b) if i is DF in case (i) of Definition 2.10 and i is said to be SG α(2)-FCD on (a, b) if i is DF in case (ii) of Definition 2.10. Definition 2.11 ([37]): Let i :[a, b] → R and α ∈ (0, 1]. Then the α-fuzzy conformable FF integral of i(t) with reference point t is defined by: α α−1 ˆ ˆ J i(t) = (τ − t ) i(τ )dτ , (23) provided that iis α-FCD on (a, b). 8 Z. AL-ZHOUR ET AL. Theorem 2.3 ([37]): Let i :[a, b] → R be a F-VF and α ∈ (0, 1]. Then the subsequent are working: (i) If iis α(1)-FCD on (a, b), then α r α α 1−α r 1−α ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [T i (t), T i (t)] = (t − t ) [T i(t)] = (t − t ) [i (t), i (t)]. (24) 1r 2r 0 (1) 0 1r 2r (1) (ii) If iis α(2)-FCD on (a, b), then α r α α 1−α r 1−α ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [T i (t), T i (t)](t − t ) [T i(t)] = (t − t ) [i (t), i (t)]. (25) 2r 1r 0 (2) 0 (2) 2r 1r Definition 2.12 ([31]): A series of the form: kα α 2α c (t − t ) = c + c (t − t ) + c (t − t ) + ... , (26) k 0 0 1 0 2 0 k=0 where 0 ≤ m − 1 <α ≤ mandt ≥ t is called a fractional power series (F-PS) about t , where 0 0 t is a variable and c ’s are constants called the coefficients of the series. Theorem 2.4: Suppose that i(t) has an F-PS representation at t of the form kα i(t) = c (t − t ) ,0 ≤ t ≤ R,0 <α ≤ 1, (27) k 0 k=0 where R is the radius of convergence of the F-PS. Then k!α mα (k−m)α T i(t) = c (t − t ) , (28) k 0 (k − m)! k=m mα α α α ˆ ˆ where T i(t) = T · T ... T i(t) (m-times). Proof: We can prove the theorem inductively. For m = 0, the formula is correct. With respect to m = 1 and according to Lemma 2.1 the formula is also true. Assume the formula is true for m = n; i.e k!α nα (k−n)α T i(t) = c (t − t ) . (k − n)! k=n To complete the proof, we need to show that the formula is true for m = n + 1. Thus, k!α (n+1)α α nα α (k−n)α ˆ ˆ T i(t) = T T i(t) = T c (t − t ) k 0 (k − n)! k=n n+1 k!α (k−(n+1))α = c (t − t ) . (k − (n + 1))! k=n+1 FUZZY INFORMATION AND ENGINEERING 9 Theorem 2.5 ([40]): Suppose that i has a F-PS representation at t of the form nα i(t) = c (t − t ) ,0 <α ≤ 1 (29) n 0 n=0 (nα) ˆ ˆ If i(t) ∈ C[t , t + R) and i (t) ∈ C(t , t + R) for n = 0, 1, 2, ..., then the coefficients c in 0 0 0 0 n (nα) i (t ) Equation (29) will take the form c = . n n α n! 3. Construction the Residual Power Series Solution for the FCFDEs In this section, we employ the R-PS method to construct analytical fuzzy NSs in the form of a F-PS for the IVP as in Equations (2) and (3) in an r-level depiction. To accomplish this, we switch the IVP in Equations (2) and (3) to crisp system of F-DEs. The crisp systems depend on the type of differentiability, where i(t) is either α(1)-FCD or α(2)-F-CD. Without losing the generality, we construct the R-PSS to α(1)-FCD only, and in the same approach, we can construct the R-PSS to or α(2)-FCD. Now, assume that i(t) is α(1)-FCD, then the IVP in Equations (2) and (3) can be re- formulated to the following F-DEs system: α α ˆ ˆ ˆ T i (t) = f (t , i (t), i (t)), 1r 2r 1r 2r α α ˆ ˆ ˆ T i (t) = f (t , i (t), i (t)), (30) 2r 1r 1r 2r with the F-ICs: ˆ ˆ ˆ ˆ i (0) = i , i (0) = i , (31) 1r 0,1r 2r 0,2r where α ∈ (0, 1] and t ∈ [0, b]. The R-PS method supposes that the solution of the given DE has an F-PS. So, we assume that the solution of the system (30) and (31) has the following expansions: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (32) 1r k 2r k k=0 k=0 ˆ ˆ According to the F-ICs in Equation (31), it easy to see that b = i and c = i . 0 0,1r 0 0,2r Indeed, R-PS is a technique used to determine the coefficients of the F-PSS in a differ- ent approach than the traditional method. Thus, to determine the coefficients of the series in Equation (32), we need to define the so-called residual functions of the equations in Equation (32) as follows: α α ˆ ˆ ˆ Res (t) = T i (t) − f (t , i (t), i (t)), 1r 1r 2r 1r 2r α α ˆ ˆ ˆ Res (t) = T i (t) − f (t , i (t), i (t)). (33) 2r 2r 1r 1r 2r 10 Z. AL-ZHOUR ET AL. Indeed, we have the following facts for our approach via the R-PS method: (i) Res (t) = Res (t) = 0, fort ∈ [0, b]. 1r 2r mα mα (ii) T Res (t) = T Res (t) = 0, m = 0, 1, 2, ... , fort ∈ [0, b]. (34) 1r 2r mα mα (iii) T Res (0) = T Res (0) = 0, m = 0, 1, 2, ... (35) 1r 2r Now, if we substitute the F-PS in Equation (32) into Equation (33) and use the Theorem 2.4, then we obtain the following expression of the residual functions: ∞ ∞ ∞ (k−1)α α kα kα Res (t) = kαb t − f t , b t , c t , 1r k 2r k k k=1 k=0 k=0 ∞ ∞ ∞ (k−1)α α kα kα Res (t) = kαc t − f t b t , c t . (36) 2r k 1r k k k=1 k=0 k=0 Since f and f are analytic functions of F-PS expansions, Equation (36) can be expressed 1r 2r in the following F-PS expansions: ∞ ∞ (k−1)α kα Res (t) = kαb t − g (b , b , ... , b , c , c , ... , c )t , 1r k 2rk 0 1 k 0 1 k k=1 k=0 ∞ ∞ (k−1)α kα Res (t) = kαc t − g (b , b , ... , b , c , c , ... , c )t . (37) 2r k 1rk 0 1 k 0 1 k k=1 k=0 where g and g are multivariable functions of b , b , ... , b , c , c , ... , c generates 1rk 2rk 0 1 k 0 1 k according to f and f . 1r 2r mα Likewise, if we apply the C-FD T on both sides of Equation (37), then it can be expressed as an F-PS expansion as follows: ∞ ∞ m+1 m k!α b k!α g (b , b , ... , b , c , c , ... , c ) k 2rk 0 1 k 0 1 k mα (k−m−1)α (k−m)α T Res (t) = t − t , 1r (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m k!α c k!α g (b , b , ... , b , c , c , ... , c ) k 1rk 0 1 k 0 1 k mα (k−m−1)α (k−m)α T Res (t) = t − t . 2r (k − m − 1)! (k − m)! k=m+1 k=m (38) According to Equation (38) and the fact in Equation (34), we define the so-called αmth-order DE as follows: ∞ ∞ m+1 m k!α b k!α g (b , b , ... , b , c , c , ... , c ) k 2rk 0 1 k 0 1 k (k−m−1)α (k−m)α t − t = 0, (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m k!α c k!α g (b , b , ... , b , c , c , ... , c ) k 1rk 0 1 k 0 1 k (k−m−1)α (k−m)α t − t = 0. (k − m − 1)! (k − m)! k=m+1 k=m (39) FUZZY INFORMATION AND ENGINEERING 11 Substitute t = 0 into Equation (39), we obtain the following iterative equations: m+1 m (m + 1)!α b − m!α g (b , b , ... , b , c , c , ... , c ) = 0, m+1 2rm 0 1 m 0 1 m m+1 m (m + 1)!α c − m!α g (b , b , ... , b , c , c , ... , c ) = 0. (40) m+1 1rm 0 1 m 0 1 m Solving Equation (40) for b and c gives the following recurrence relations which m+1 m+1 determine the coefficients of the F-PS in Equation (32): g (b , b , ... , b , c , c , ... , c ) 2rm 0 1 m 0 1 m b = , b = i , m = 0, 1, 2, ... , m+1 0 0,1r (m + 1)α g (b , b , ... , b , c , c , ... , c ) 1rm 0 1 m 0 1 m c = , c = i , m = 0, 1, 2, ... . (41) m+1 0 0,2r (m + 1)α Therefore, the exact solution (ES) of the system (30) and (31) in an F-PS form is given by g (b , b , ... , b , c , c , ... , c ) 2r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t , 1r 0,1r kα k=1 g (b , b , ... , b , c , c , ... , c ) 1r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t . (42) 2r 0,2r kα k=1 Furthermore, the mth-truncated of the F-PS in Eq (3.13) for an appropriate number of m gives the NS of the system (30) and (31) as follows: g (b , b , ... , b , c , c , ... , c ) 2r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t , 1r,m 0,1r kα k=1 g (b , b , ... , b , c , c , ... , c ) 1r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t . (43) 2r,m 0,2r kα k=1 With the same previous approach, we can find a PSS to the system of F-DEs that corre- sponding to α(2)-FCD case. While in the next section, we consider both cases (α(1)-FCD and α(2)-FCD) in details for dealing the three important and interesting examples. 4. Illustrative Examples of FC-FDEs In this chapter, we apply the R-PS method to three examples to illustrate the construction that was prepared in Section 3 and to confirm the effectiveness and efficiency of the pro- posed method used in constructing an F-PSS to FC-FDEs. The first example includes a linear FC-FDE, the second example deals with a nonlinear FC-FDE, where as the third example illustrates our method to construct the F-PSS to FC- integral equation. Example 4.1: Given the following FC-FDE: α α ˆ ˆ T i(t) =−i(t) + sin(t ), t ∈ [0, 1], 0 <α ≤ 1, (44) subject to the F-IC: i(0) = w, (45) 12 Z. AL-ZHOUR ET AL. where 25ρ − 24, 0.96 ≤ ρ ≤ 1 w(ρ) = 101 − 100ρ,1 ≤ ρ ≤ 1.01 . 0, otherwise 24 r If we assume r = 25ρ − 24, then ρ = + . Whereas, if we assume r = 101 − 100ρ, then 25 25 101 r r 24 r 101 r r ˆ ˆ ˆ ρ = − . Therefore, [w] = + , − and [i(t)] = [i (t), i (t)]. Hence, to 1r 2r 100 100 25 25 100 100 determine the α(1) and α(2)-fuzzy RPS solutions of the FC-FDE as in Equations (44) subject to the F-IC (45), we consider the following two cases: Case 1: The system of the ODEs corresponding to α(1)-FCD is α α ˆ ˆ T i (t) =−i (t) + sin(t ), 1r 2r α α ˆ ˆ T i (t) =−i (t) + sin(t ), (46) 2r 1r subject to the F-ICs: 24 1 101 1 ˆ ˆ i (0) = + r, i (0) = − r. (47) 1r 2r 25 25 100 100 To employ the RPS method to create the PSS to the system (46) and (47), we assume that the solution of this system can be expressed as F-PS expansion about the initial point t = 0 as follows: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (48) 1r 2r k k k=0 k=0 24 1 101 1 Using the F-ICs in Equation (47) give us b = + r and c = − r. Therefore, the 0 0 25 25 100 100 solution of system (46) and (48) can be represented as: ∞ ∞ 24 1 101 1 kα kα ˆ ˆ i (t) = + r + b t , i (t) = − r + c t . (49) 1r 2r k k 25 25 100 100 k=1 k=1 Apply the C-FD T to Equation (49), we obtain: ∞ ∞ (α) (α) α (k−1)α α (k−1)α ˆ ˆ ˆ ˆ T i (t) = i (t) = kαb t , T i (t) = i (t) = kαc t . (50) 1r k 1r k 1r 1r k=1 k=1 Define the residual functions of Equation (46) as follows: α α α α ˆ ˆ ˆ ˆ Res (t) = T i (t) + i (t) − sin(t ), Res (t) = T i (t) + i (t) − sin(t ). (51) 1r 1r 2r 2r 2r 1r So, the αmth-order DEs of Equation (46) have the following forms: mα α α mα α α ˆ ˆ ˆ ˆ T (T i (t) + i (t) − sin(t )) = 0, T (T i (t) + i (t) − sin(t )) = 0, m = 0, 1, 2, ... . 1r 2r 2r 1r (52) To determine the coefficients of the expansions, b and c for k = 1, 2, ..., substitute the k k expansion formulas in Equations (49) and (50) into Equation (52), yields ∞ ∞ 24 1 mα (k−1)α kα α T (kαb t ) + + r + (c t ) − sin(t ) = 0, m = 0, 1, 2, ... , k k 25 25 k=1 k=1 FUZZY INFORMATION AND ENGINEERING 13 ∞ ∞ 101 1 mα (k−1)α kα α T (kαc t ) + − r + (b t ) − sin(t ) = 0, m = 0, 1, 2, ... . k k 100 100 k=1 k=1 (53) Through a few simple calculations, a new discretised version of Equation (53) can be obtained and given by: ∞ ∞ m+1 m α k!b α k!c k k (k−m−1)α (k−m)α t + t = χ (t), (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m α k!c α k!b k k (k−m−1)α (k−m)α t + t = χ (t), (54) (k − m − 1)! (k − m)! k=m+1 k=m where (m−1)/2 m α (−1) α cos t if m is odd χ (t) = (55) m/2 m α (−1) α sin t if m is even It is obvious that, the αmth-derivative of the F-PS representation, Equation (54) is conver- gent at least at t = 0, for m = 0, 1, 2, .... Therefore, the substituting t = 0 into Equation (54) gives the following recurrence relation which determines the values of the coefficients b and c : 24 1 χ (0) − α m!c m m b = + r, b = , m = 0, 1, 2, ... , 0 m+1 m+1 25 25 α (m + 1)! 101 1 χ (0) − α m!b m m c = − r, c = , m = 0, 1, 2, ... . (56) 0 m+1 m+1 100 100 α (m + 1)! If we collect and substitute these values of the coefficients back into Equation (49), then the ES of the system (46) and (47) has the general form, which is coinciding with the general expansion: 3 3 c α + b α + c α − α + b α − α + c 0 0 0 0 0 α 2α 3α 4α 5α i (t) = b − t + t − t + t − t 1r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + b α − α + α + c α − α + α − α + b 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + c α − α + α − α + α + b 0 0 9α 10α − t + t 9 10 9!α 10α 3 5 7 9 α − α + α − α + α + c 11α − t + ... , 11!α 3 3 b α + c α + b α − α + c α − α + b 0 0 0 0 0 α 2α 3α 4α 5α i (t) = c − t + t − t + t − t 2r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + c α − α + α + b α − α + α − α + c 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + b α − α + α − α + α + c 0 0 9α 10α − t + t 9 10 9!α 10α 14 Z. AL-ZHOUR ET AL. 3 5 7 9 α − α + α − α + α + b 11α − t + ... . (57) 11!α If we rearrange the terms in Equation (57), then it can be written as follows: b b b b b c c c 0 0 0 0 0 0 0 0 2α 4α 6α 8α 10α α 3α 5α i (t) = b + t + t + t + t + t − t − t − t 1r 0 2 4 6 8 10 3 5 2!α 4!α 6!α 8!α 10α α 3!α 5!α c c c α α α − α 0 0 0 7α 9α 11α 2α 3α 4α − t − t − t + t − t + t 7 9 11 2 3 4 7!α 9!α 11!α 2!α 3!α 4!α 3 3 5 3 5 3 5 7 α − α α − α + α α − α + α α − α + α − α 5α 6α 7α 8α − t + t − t + t 5 6 7 8 5!α 6!α 7!α 8!α 3 5 7 3 5 7 9 3 5 7 9 α − α + α − α α − α + α − α + α α − α + α − α + α 9α 10α 11α − t + t − t + ... , 9 10 11 9!α 10α 11!α c c c c c b b b 0 0 0 0 0 0 0 0 2α 4α 6α 8α 10α α 3α 5α i (t) = c + t + t + t + t + t − t − t − t 2r 0 2 4 6 8 10 3 5 2!α 4!α 6!α 8!α 10α α 3!α 5!α b b b α α α − α 0 0 0 7α 9α 11α 2α 3α 4α − t − t − t + t − t + t 7 9 11 2 3 4 7!α 9!α 11!α 2!α 3!α 4!α 3 3 5 3 5 3 5 7 α − α α − α + α α − α + α α − α + α − α 5α 6α 7α 8α − t + t − t + t 5 6 7 8 5!α 6!α 7!α 8!α 3 5 7 3 5 7 9 3 5 7 9 α − α + α − α α − α + α − α + α α − α + α − α + α 9α 10α 11α − t + t − t + ... . 9 10 11 9!α 10α 11!α (58) We can summarise the solutions in Equation (57) as ⎛ ⎞ ∞ ∞ ∞ ∞ 2kα (2k+1)α kα t t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = b − c + (−1) α , 1r 0 0 2k k k (2k)!α (2k + 1)!α k!α k=0 k=0 j=1 k=2j ⎛ ⎞ ∞ ∞ ∞ ∞ 2kα (2k+1)α kα t t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = c − b + (−1) α , (59) 2r 0 0 2k k k (2k)!α (2k + 1)!α k!α k=0 k=0 j=1 k=2j which are equivalent to: ⎛ ⎞ ∞ ∞ α α kα 24 r t 101 r t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = + cosh − − sinh + (−1) α , 1r 25 25 α 100 100 α k!α j=1 k=2j ⎛ ⎞ ∞ ∞ α α k kα 101 r t 24 r t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = − cosh − + sinh + (−1) α . 2r 100 100 α 25 25 α k!α j=1 k=2j (60) In fact, if α = 1, then the PSS to the system (46) and (47) will be as: 24 1 101 1 1 1 −t i (t) = + r cosh(t) − − r sinh(t) + (sin(t) − cos(t)) + e , 1r 25 25 100 100 2 2 101 1 24 1 1 1 −t i (t) = − r cosh(t) − + r sinh(t) + (sin(t) − cos(t)) + e . 2r 100 100 25 25 2 2 FUZZY INFORMATION AND ENGINEERING 15 Figure 1. The surface graph of the α(1)-FCD – 10th approximate R-PSS and ES for Equations (46) and (47) at different values of α, where the lower surface represents i (t), where as the upper surface repre- 1r r r r ˆ ˆ ˆ ˆ sents i (t):(a)α(1)-FCD of [i (t)] at α = 0.6, (b) α(1)-FCD of [i (t)] at α = 0.8, (c)α(1)-FCD of [i (t)] 2r 10 10 10 at α = 1, (d) α(1)-FCD of [i(t)] (exact) at α = 1. Figure 1 shows the graphical results for the approximate R-PSSs and ES of Equations (46) and (47) at different values of α corresponding to α(1)-FCD. Case 2: The system of the ODEs corresponding to α(2)-FCD is: α α ˆ ˆ T i (t) =−i (t) + sin(t ), 1r 1r α α ˆ ˆ T i (t) =−i (t) + sin(t ), (61) 2r 2r subject to F-ICs: 24 1 101 1 ˆ ˆ i (0) = + r, i (0) = − r. (62) 1r 2r 25 25 100 100 Similarly, assume that the F-PSSs of Equations (58) and (59) has the following form: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (63) 1r 2r k k k=0 k=0 Then the αm10th-order DEs corresponding to Equation (58) is: ∞ ∞ 24 1 mα (k−1)α kα α T (kαb t ) + + r + (b t ) + sin(t ) = 0, m = 0, 1, 2, ... , k k 25 25 k=1 k=1 16 Z. AL-ZHOUR ET AL. ∞ ∞ 101 1 mα (k−1)α kα α T (kαc t ) + − r + (c t ) + sin(t ) = 0, m = 0, 1, 2, ... . k k 100 100 k=1 k=1 (64) mα Operator T in Equation (64) gives a summarised form of the equation: ∞ ∞ m+1 m α k!b α k!c k k (k−m−1)α (k−m)α t + t = χ (t), (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m α k!c α k!b k k (k−m−1)α (k−m)α t + t = χ (t), (65) (k − m − 1)! (k − m)! k=m+1 k=m where χ (t) is defined as in Equation (55). The recurrence relation which determines the values of the coefficients b and c is: k k 24 1 χ (0) − α m!b m m b = + r, b = , m = 0, 1, 2, ... , 0 m+1 m+1 25 25 α (m + 1)! 101 1 χ (0) − α m!c m m c = − r, c = , m = 0, 1, 2, ... . (66) 0 m+1 m+1 100 100 α (m + 1)! So, the ESs of Equations (61) and (62) have the general form: 3 3 b α + b α + b α − α + b α − α + b 0 0 0 0 0 α 2α 3α 4α 5α i (t) = b − t + t − t + t − t 1r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + b α − α + α + b α − α + α − α + b 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + b α − α + α − α + α + b 0 0 9α 10α − t + t 9 10 9!α 10α 3 5 7 9 α − α + α − α + α + b 11α − t + ... , 11!α 3 3 c α + c α + c α − α + c α − α + c 0 0 0 0 0 α 2α 3α 4α 5α i (t) = c − t + t − t + t − t 2r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + c α − α + α + c α − α + α − α + c 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + c α − α + α − α + α + c 0 0 9α 10α − t + t 9 10 9!α 10α 3 5 7 9 α − α + α − α + α + c 11α − t + ... (67) 11!α We can formulate the ES in Equation (67) as ⎛ ⎞ ∞ ∞ kα 24 r −t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = + e + (−1) α , 1r 25 25 k!α j=1 k=2j ⎛ ⎞ ∞ ∞ kα 101 r −t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = − e + (−1) α . (68) 2r 100 100 k!α j=1 k=2j FUZZY INFORMATION AND ENGINEERING 17 Figure 2. The surface graph of the α(2)-FCD-10th approximate R-PSS and ES for Equations (61) and (62) at different values of α, where the lower surface represents i (t), where as the upper surface represents 1r r r r ˆ ˆ ˆ ˆ i (t):(a) α(2)-FCD of [i (t)] at α = 0.6, (b) α(2)-FCD of [i (t)] at α = 0.8, (c) (2)-FCD of [i (t)] at 2r 10 10 10 α = 1, (d) (2)-FCD of [i(t)] (exact) at α = 1. Thus, the ES of the system (61) and (62) when α = 1 has the following expression: 24 1 1 1 −t −t i (t) = + r e + (sin(t) − cos(t)) + e , 1r 25 25 2 2 101 1 1 1 −t −t i (t) = − r e + (sin(t) − cos(t)) + e . (69) 2r 100 100 2 2 Figure 2 shows the graphical results for the approximate R-PSSs and ESs of Equations (61) and (62) at different values of α corresponding to α(2)-FCD. Example 4.2: Given the following FC-FDE: α α α ˆ ˆ T u, t ∈ [0, 1], 0 <α ≤ 1, (70) i(t) = 2t i(t) + t subject to the F-IC: i(0) = u (71) where u = max(0, 1 −|ρ|), ρ ∈ R. Case 1: The system of the ODEs corresponding to α(1)-FCD is: α α α ˆ ˆ T i (t) = 2t i (t) + t (r − 1), 1r 1r 18 Z. AL-ZHOUR ET AL. α α α ˆ ˆ T i (t) = 2t i (t) + t (1 − r), (72) 2r 2r subject to the F-ICs: ˆ ˆ i (0) = r − 1, i (0) = 1 − r. (73) 1r 2r Assume the F-PS representation of the solution to the Equations (72) and (73) is: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (74) 1r 2r k k k=0 k=0 Then the αmth-order DEs corresponding to Equation (72) are: ∞ ∞ m+1 m α k!b α (k + 1)!b k k (k−m−1)α (k−m+1)α t − 2 t = χ , 1m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 ∞ ∞ m+1 m α k!c α (k + 1)!c k k (k−m−1)α (k−m+1)α t − 2 t = χ (75) 2m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 α α where χ = b t , χ = c t , χ = αb , χ = αc ,and χ (t) = χ (t) = 0, m = 2, 3, .... 10 0 20 0 11 0 21 0 1m 2m Therefore, by substitute t = 0 into Equation (75), the recurrence relations which determine the values of the coefficients b and c are: k k 3b 2b 0 m−1 b = r − 1, b = 0, b = , b = , m = 2, 3, 4, ... , 0 1 2 m+1 2α α(m + 1) 3c 2c 0 m−1 c = 1 − r, c = 0, c = , c = , m = 2, 3, 4, ... (76) 0 1 2 m+1 2α α(m + 1) If we substitute these values in Equation (74), then the F-PSS of Equations (72) and (73) has the general form: 2α 4α 6α 8α 10α 3 2 t t t t t i (t) = b + + + + + + ... , 1r 0 2 3 4 5 2 3 (1!)α (2!)α (3!)α (4!)α (5!)α 2α 4α 6α 8α 10α 3 2 t t t t t i (t) = c + + + + + + ... . (77) 2r 0 2 3 4 5 2 3 (1!)α (2!)α (3!)α (4!)α (5!)α The closed form of the solution in Equation (77) is 2αk 2α 3 t 1 3 1 ˆ α i (t) = b − = (r − 1) e − , 1r 0 2 (k!)α 3 2 3 k=0 2αk 2α 3 t 1 3 t 1 ˆ α i (t) = c − = (1 − r) e − . (78) 2r 0 2 (k!)α 3 2 3 k=0 FUZZY INFORMATION AND ENGINEERING 19 Thus, the ES to the Equations (70) and (71) in the sense of α(1)-FCD, can be expressed as follows: 2α 1 t ˆ α i(t) = 3e − 1 u. (79) Indeed, if α = 1, the solution of Equations (70) and (71) in this case becomes: 1 2 i(t) = (3e − 1)u, (80) which is corresponding to the solution obtained by the homotopy analysis method [33]. Case 2: The system of the ODEs corresponding to α(2)-FCD is: α α α ˆ ˆ T i (t) = 2t i (t) + t (r − 1), 1r 2r α α α ˆ ˆ T i (t) = 2t i(t) + t (1 − r), (81) 2r subject to F-ICs: ˆ ˆ i (0) = r − 1, i (0) = 1 − r. (82) 1r 2r The F-PSS of Equations (81) and (82) has the following form: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (83) 1r k 2r k k=0 k=0 Then the αmth-order DEs corresponding to Equation (81) are: ∞ ∞ m+1 m α k!b α (k + 1)!c k k (k−m−1)α (k−m+1)α t − 2 t = χ , 1m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 ∞ ∞ m+1 m α k!c α (k + 1)!b k k (k−m−1)α (k−m+1)α t − 2 t = χ (84) 2m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 α α where χ = b t , χ = c t , χ = αb , χ = αc ,and χ (t) = χ (t) = 0, m = 2, 3, .... 10 0 20 0 11 0 21 0 1m 2m Therefore, the recurrence relations which determine the values of the coefficients b and c k k are: c 2c 0 m−1 b = r − 1, b = 0, b = , b = , m = 2, 3, 4, ... , 0 1 2 m+1 2α α(m + 1) b 2b 0 m−1 c = 1 − r, c = 0, c = , c = , m = 2, 3, 4, ... . (85) 0 1 2 m+1 2α α(m + 1) Substitute these values of the coefficients into Equation (83), then the F-PSS of Equations (81) and (82) has the general form: 2α 4α 6α 8α 10α b t t t t t i (t) = 2 − + − + − + ... , 1r 2 3 4 5 2 (1!)α (2!)α (3!)α (4!)α (5!)α 2α 4α 6α 8α 10α c t t t t t i (t) = 2 − + − + − + ... . (86) 2r 2 3 4 5 2 (1!)α (2!)α (3!)α (4!)α (5!)α 20 Z. AL-ZHOUR ET AL. The closed form of the solution in Equation (86) is: 2αk 2α b t r − 1 −t i (t) = 1 + = 1 + e , 1r 2 2 (k!)α k=0 2αk 2α −t c t 1 − r ˆ α i (t) = 1 + = 1 + e . (87) 2r 2 (k!)α 2 k=0 Thus, the ES to the Equations (70) and (71) in the sense of α(2)-FCD, can be expressed as follows: 2α 1 −t i(t) = 1 + e u. (88) ˆ ˆ Figure 3 shows the graphical results for α(1)-FCD and α(2)-FCD ES set [i (t), i (t)] and its 1r 2r ( ( α α ˆ ˆ derivative [T i t), T i t)] of Equations (70) and (71) when r = 0.5 and at different values 1r 2r of α. In the next example, we apply the R-PS method to construct the PSS to a fractional con- formable fuzzy integro-differential equation (FCFI-DE). The same procedure used in the previous examples will be used with a few minor differences. ( ( α α ˆ ˆ ˆ ˆ Figure 3. The α(1)-FCD and α(2)-FCD ES set [i (t), i (t)] and its derivative [T i t), T i t)]for 1r 2r 1r 2r Equations (70) and (71) when r = 0.5 at α = 0.6 (Solid),α = 0.8 (Dash-dotted), α = 1 (Dotted),: (a) ( ( α α ˆ ˆ ˆ ˆ ˆ ˆ α(2)-FCD of [i (t), i (t)], (b) α(2)-FCD of [i (t), i (t)], (c)α(1)-FC of [T i t), T i t)], (d)α(2)-FCD 1r 2r 1r 2r 1r 2r ( ( α α ˆ ˆ of[T i t), T i t)]. 1r 2r FUZZY INFORMATION AND ENGINEERING 21 Example 4.3: Consider the inductance–resistance–capacitance, RLC-series circuit as in diagram below: Then the FCFI-DE is given by: R 1 α α−1 ˆ ˆ ˆ T i(t) =− i(t) − τ i (τ )dτ + V(t),τ< t ∈ [0, 1], 0 <α ≤ 1, (89) 2r L LC subject to the F-IC: i(0) = v, (90) where R, L,and C are the resistance, inductance corresponding to the solenoid and capaci- tance, respectively, and V(t) = sin(t ) represents the voltage function in the electric circuit.. Finally, 25ρ − 24, 0.96 ≤ ρ ≤ 1 v(ρ) = 101 − 100ρ,1 ≤ ρ ≤ 1.01 . 0, otherwise For simplicity’s sake and for numerical analysis, we assume that R = L = C = 1. Similar to Example 4.1 above, if we assume r = 25ρ − 24, firstly, and r = 101 − 100ρ, secondly, then t t t r 24 r 101 r α−1 α−1 α−1 ˆ ˆ ˆ we have [v] = + , − and τ i(τ )dτ = τ i (τ )dτ , τ i (τ )dτ . 1r 2r 25 25 100 100 0 0 0 According to Nguyen’s theorem and Zadeh’s extension principle [3,53–55], we transfer the IVP (89) and (90) to ODEs systems that we exhibit in the following two cases: Case 1: The system of the FCFI-DEs corresponding to α(1)-FCDis: α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ) 1r 2r 2r α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ), (91) 2r 1r 1r subject to the F-ICs: 24 r 101 r ˆ ˆ i (0) = + , i (0) = − . (92) 1r 2r 25 25 100 100 TheESinthiscaseat α = 1 is: √   √ 1 5 1 5 1 3 − t + t − t 2 2 2 2 i (t) = γ (r)e + γ (r)e + e γ (r)cos t 1r 1 2 3 + γ (r)sin t + sin(t), 2 22 Z. AL-ZHOUR ET AL. √   √ 1 5 1 5 − t + t − t 2 2 2 2 ˆ 2 i (t) =−γ (r)e − γ (r)e + e γ (r)cos t 2r 1 2 3 + γ (r)sin t + sin(t), (93) where √ √ 5 − 5 5 + 5 ˆ ˆ ˆ ˆ γ (r) = (i (0) − i (0)), γ (r) = (i (0) − i (0)), 1 1r 2r 2 1r 2r 20 20 1 − 3 ˆ ˆ ˆ ˆ γ (r) = (i (0) + i (0)), γ (r) = (i (0) + i (0) + 4). 3 1r 2r 4 1r 2r 2 6 Assume the F-PS representations of the solution for the Equations (91) and (92) are: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (94) 1r k 2r k k=0 k=0 α α−1 Apply the C-FD T and τ (F )dτ, respectively, on Equation (94), we obtain: ∞ ∞ α (k−1)α α (k−1)α ˆ ˆ T i (t) = kαb t , T i (t) = kαc t , 1r 2r k k k=1 k=1 ∞ ∞ t (k+1)α t (k+1)α t t α−1 α−1 ˆ ˆ τ i (τ )dτ = b , τ i (τ )dτ = c . (95) 1r 2r k k (k + 1)α (k + 1)α 0 0 k=0 k=0 Then, the αmth-order DEs corresponding to Equation (91) is: ∞ ∞ ∞ (k+1)α 101 r t mα (k−1)α kα α T (kαb t ) + − + (c t ) + c − sin(t ) = 0, k k k 100 100 (k + 1)α k=1 k=1 k=0 ∞ ∞ ∞ (k+1)α 24 r t mα (k−1)α kα α T (kαc t ) + + + (b t ) + b − sin(t ) = 0, (96) k k k 25 25 (k + 1)α k=1 k=1 k=0 for m = 0, 1, 2, ... . mα Running out the operator T on Equation (96) gives a discretised form for the equa- tions: ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!b α k!c c α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!c α k!b b α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 (97) where χ (t) is defined as in Equation (55). The recurrence relations which determine the values of the coefficients b and c are: k k m m−1 24 r c χ (0) − α m!c − α (m − 1)!c 0 m m m−1 b = + , b =− , b = , m = 1, 2, 3, ... , 0 1 m+1 m+1 25 25 α α (m + 1)! FUZZY INFORMATION AND ENGINEERING 23 m m−1 101 r b χ (0) − α m!b − α (m − 1)!b 0 m m m−1 c = − , c =− , c = , m = 1, 2, 3, ... . 0 1 m+1 m+1 100 100 α α (m + 1)! (98) So, the ES of Equations (91) and (92) has the general form: −c α + b − c −α + 2b − c −α + 2b − 3c 0 0 0 0 0 0 0 α 2α 3α 4α i (t) = b + t + t + t + t 1r 0 2 3 4 α 2α 6α 24α 3 5 3 5 α + α + 4b − 4c −α + α + 7b − 6c −α − α + 10b − 11c 0 0 0 0 0 0 5α 6α 7α + t + t + t 5 6 7 120α 720α 5040α 3 7 5 7 α + α − α + 17b − 17c −α + α + α + 28b − 27c 0 0 0 0 8α 9α + t + t 8 9 40320α 362880α 3 5 9 3 7 9 −α − α + α + 44b − 45c α + α − α − α + 72b − 72c 0 0 0 0 10α 11α + t + t + ... , 10 11 3628800α 39916800α −b α + c − b −α + 2c − b −α + 2c − 3b 0 0 0 0 0 0 0 α 2α 3α 4α i (t) = c + t + t + t + t 2r 0 2 3 4 α 2α 6α 24α 3 5 3 5 α + α + 4c − 4b −α + α + 7c − 6b −α − α + 10c − 11b 0 0 0 0 0 0 5α 6α 7α + t + t + t 5 6 7 120α 720α 5040α 3 7 5 7 α + α − α + 17c − 17b −α + α + α + 28c − 27b 0 0 0 0 8α 9α + t + t 8 9 40320α 362880α 3 5 9 3 7 9 −α − α + α + 44c − 45b α + α − α − α + 72c − 72b 0 0 0 0 10α 11α + t + t + ... . 10 11 3628800α 39916800α (99) Case 2: The system of the FCFI-DEs corresponding to α(2)-FCD is: α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ), 1r 1r 1r α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ), (100) 2r 2r 2r subject to the F-ICs: 24 1 101 1 ˆ ˆ i (0) = + r, i (0) = − r. (101) 1r 2r 25 25 100 100 TheESinthiscaseat α = 1 is: √ √ 1 3 1 3 2 i (0) 1r − t − t ˆ ˆ 2 2 i (t) = sin(t) + i (0)e cos t + e sin t − √ − √ , 1r 1r 2 2 3 3 √ √ 1 3 1 3 2 i (0) 2r − t − t ˆ ˆ 2 2 i (t) = sin(t) + (i (0))e cos t + e sin t − √ − √ . (102) 2r 2r 2 2 3 3 24 Z. AL-ZHOUR ET AL. Assume the F-PS representation of the solutions for the Equations (99) and (100) are: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t , (103) 1r 2r k k k=0 k=0 Then the αmth-order DEs corresponding to Equation (100) is: ∞ ∞ ∞ (k+1)α 101 r t mα (k−1)α kα α T (kαb t ) + − + (b t ) + b − sin(t ) = 0, k k k 100 100 (k + 1)α k=1 k=1 k=0 m = 0, 1, 2, ... , ∞ ∞ ∞ (k+1)α 24 r t mα (k−1)α kα α T (kαc t ) + + + (c t ) + c − sin(t ) = 0, k k k 25 25 (k + 1)α k=1 k=1 k=0 m = 0, 1, 2, ... . (104) mα Running out the operator T on Equation (104) gives a discretised form for the equations: ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!b α k!b b α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!c α k!c c α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 (105) where χ (t) is defined as in Equation (55). The recurrence relations which determine the values of the coefficients b and c are: k k m m−1 24 r b χ (0) − α m!b − α (m − 1)!b 0 m m m−1 b = + , b =− , b = , m = 1, 2, 3, ... , 0 1 m+1 m+1 25 25 α α (m + 1)! m m−1 101 r c χ (0) − α m!c − α (m − 1)!c 0 m m m−1 c = − , c =− , c = , m = 1, 2, 3, ... . 0 1 m+1 m+1 100 100 α α (m + 1)! (106) So, the ES of Equation (100) and (101) has the general form: 3 3 −b 1 −α + b −α − b α + α 0 0 0 α 2α 3α 4α 5α i (t) = b + t + t + t + t + t 1r 0 3 4 5 α 2!α 3!α 4!α 5!α 5 3 5 3 7 5 7 −α + α + b −α − α − b α + α − α −α + α + α + b 0 0 0 6α 7α 8α 9α + t + t + t + t 6 7 8 9 6!α 7!α 8!α 9!α 3 5 9 3 7 9 −α − α + α − b α + α − α − α 10α 11α + t + t + ... , 10 11 10!α 11!α 3 3 5 −c 1 −α + c −α − c α + α −α + α + c 0 0 0 0 α 2α 3α 4α 5α 6α i (t) = c + t + t + t + t + t + t 2r 0 3 4 5 6 α 2!α 3!α 4!α 5!α 6!α 3 5 3 7 5 7 −α − α − c α + α − α −α + α + α + c 0 0 7α 8α 9α + t + t + t 7 8 9 7!α 8!α 9!α 3 5 9 3 7 9 −α − α + α − c α + α − α − α 10α 11α + t + t + ... . (107) 10 11 10!α 11!α FUZZY INFORMATION AND ENGINEERING 25 Our goal here is to illustrate some numerical results of the R-PSSs of Equations (89) and (90) to show the validity and efficiency of the proposed method. Anyhow, Tables 1 and 2 show the exact error of α(1)-FCD and α(2)-FCD 10th-approximate R-PSSs, respectively, of Equations (89) and (90) at α = 1and variousvaluesof r and t. While Tables 3 and 4 show the relative error of α(1)-FCD and α(2)-FCD 10th-approximate R-PSSs, respectively, of Equations (89) and (90) at different values of α, r and t. The exact and relative errors are defined, respectively, as follows: ˆ ˆ ˆ ˆ Exact error: = [Ext(t)] = [|i (t) − (i (t)) |, |i (t) − (i (t)) |], 1r 1r 10 2r 2r 10 ˆ ˆ ˆ ˆ Relative error := [Rel(t)] = [|(i (t)) − (i (t)) |, |(i (t)) − (i (t)) |]. (108) 1r 1r 2r 2r 11 10 11 10 Example 4.3 was discussed in [37] by the reproducing kernel Hilbert space (RKHS) method. Tables 5 and 6 show the exact error of α(1)-FCD and α(2)-FCD solutions of Equations (89) and (90) at α = 1and variousvaluesof r and t that were obtained by the RKHS and R-PS methods. The data shows that there is a slight improvement in error when using the R-PS method. Table 1. The exact error of the α(1)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at α = 1 and various values of r and t. r r r r r [Ext(0.25)] [Ext(0.5)] [Ext(0.75)] [Ext(1)] −16 −12 −10 −8 0 [6.66, 8.88] × 10 [2.68, 3.64] × 10 [3.62, 4.86] × 10 [1.19, 1.57] × 10 −16 −12 −10 −8 0.25 [3.33, 6.66] × 10 [1.88, 2.85] × 10 [2.56, 3.80] × 10 [0.84, 1.23] × 10 −16 −12 −10 −9 0.50 [2.22, 4.44] × 10 [1.09, 2.06] × 10 [1.50, 2.74] × 10 [5.01, 8.85] × 10 −16 −12 −10 −9 0.75 [0.00, 2.22] × 10 [0.30, 1.28] × 10 [0.43, 1.68] × 10 [1.53, 5.39] × 10 −13 −11 −9 1 [0, 0] [4.90, 4.90] × 10 [6.25, 6.25] × 10 [1.93, 1.93] × 10 Table 2. The exact error of the α(2)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at α = 1 and various values of r and t. r r r r r [Ext(0.25)] [Ext(0.5)] [Ext(0.75)] [Ext(1)] −16 −13 −11 −9 0 [3.33, 1.11] × 10 [4.71, 4.95] × 10 [6.00, 6.31] × 10 [1.86, 1.95] × 10 −16 −13 −11 −9 0.25 [1.11, 2.22] × 10 [4.76, 4.94] × 10 [6.06, 6.29] × 10 [1.88, 1.95] × 10 −16 −13 −11 −9 0.50 [1.11, 0.00] × 10 [4.83, 4.93] × 10 [6.12, 6.28] × 10 [1.89, 1.94] × 10 −16 −13 −10 −9 0.75 [3.33, 2.22] × 10 [4.85, 4.94] × 10 [6.18, 6.26] × 10 [1.91, 1.94] × 10 −16 −13 −11 −9 1 [1.11, 1.11] × 10 [4.90, 4.90] × 10 [6.25, 6.25] × 10 [1.93, 1.93] × 10 Table 3. The relative error of the α(1)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at various values of α, r and t. α = 0.6 α = 0.8 r r r r r [Rel(0.25)] [Rel(0.75)] [Rel(0.25)] [Rel(0.75)] −9 −5 −12 −7 0 [5.48, 6.38] × 10 [7.56, 1.14] × 10 [6.36, 9.74] × 10 [2.12, 2.22] × 10 −9 −5 −12 −7 0.25 [3.57, 5.22] × 10 [2.40, 1.02] × 10 [4.10, 7.91] × 10 [1.28, 1.87] × 10 −9 −6 −12 −7 0.50 [1.82, 3.98] × 10 [8.33, 8.75] × 10 [1.93, 6.04] × 10 [0.57, 1.49] × 10 −9 −6 −12 −8 0.75 [0.20, 2.67] × 10 [0.73, 6.67] × 10 [0.15, 4.12] × 10 [0.43, 10.6] × 10 −9 −6 −12 −8 1 [1.28, 1.28] × 10 [3.74, 3.74] × 10 [2.16, 2.16] × 10 [5.85, 5.85] × 10 26 Z. AL-ZHOUR ET AL. Table 4. The relative error of the α(2)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at various values of α, r and t. α = 0.6 α = 0.8 r r r r r [Rel(0.25)] [Rel(0.75)] [Rel(0.25)] [Rel(0.75)] −9 −6 −12 −8 0 [1.32, 1.27] × 10 [3.67, 3.75] × 10 [2.24, 2.14] × 10 [5.93, 5.83] × 10 −9 −6 −12 −8 0.25 [1.31, 1.27] × 10 [3.69, 3.75] × 10 [2.22, 2.15] × 10 [5.91, 5.83] × 10 −9 −6 −12 −8 0.50 [1.30, 1.27] × 10 [3.70, 3.74] × 10 [2.20, 2.15] × 10 [5.89, 5.84] × 10 −9 −6 −12 −8 0.75 [1.29, 1.27] × 10 [3.72, 3.74] × 10 [2.18, 2.16] × 10 [5.87, 5.84] × 10 −9 −6 −12 −8 1 [1.28, 1.28] × 10 [3.74, 3.74] × 10 [2.16, 2.16] × 10 [5.85, 5.85] × 10 Table 5. The absolute error of the α(1)-FCD RKHS and approximate R-PSSs of the Equations (89) and (90) at α = 1 and various values of r and t. RKHS RPS r r r r r [Ext(0.25)] [Ext(0.75)] [Ext(0.25)] [Ext(0.75)] −16 −9 −16 −10 0 [5.36, 9.81] × 10 [2.68, 3.64] × 10 [6.66, 8.88] × 10 [3.62, 4.86] × 10 −16 −9 −16 −10 0.25 [4.32, 7.21] × 10 [1.88, 2.85] × 10 [3.33, 6.66] × 10 [2.56, 3.80] × 10 −15 −9 −16 −10 0.50 [3.22, 5.15] × 10 [1.29, 2.06] × 10 [2.22, 4.44] × 10 [1.50, 2.74] × 10 −15 −9 −16 −10 0.75 [3.01, 4.28] × 10 [1.01, 1.28] × 10 [0.00, 2.22] × 10 [0.43, 1.68] × 10 −16 −10 −11 1 [1.12, 4.14] × 10 [4.90, 4.90] × 10 [0, 0] [6.25, 6.25] × 10 Table 6. The absolute error of the α(2)-FCD RKHS and approximate R-PSSs of the Equations (89) and (90) at α = 1 and various values of r and t. RKHS RPS r r r r r [Ext(0.25)] [Ext(0.75)] [Ext(0.25)] [Ext(0.75)] −16 −10 −16 −11 0 [4.52, 7.28] × 10 [4.77, 5.13] × 10 [3.33, 1.11] × 10 [6.00, 6.31] × 10 −16 −10 −16 −11 0.25 [3.23, 5.42] × 10 [4.98, 5.07] × 10 [1.11, 2.22] × 10 [6.06, 6.29] × 10 −15 −10 −16 −11 0.50 [2.42, 3.75] × 10 [5.04, 5.12] × 10 [1.11, 0.00] × 10 [6.12, 6.28] × 10 −15 −10 −16 −10 0.75 [1.21, 3.18] × 10 [8.35, 7.45] × 10 [3.33, 2.22] × 10 [6.18, 6.26] × 10 −16 −11 −16 −11 1 [1.98, 4.29] × 10 [7.90, 8.08] × 10 [3.33, 1.11] × 10 [6.00, 6.31] × 10 5. Conclusions and Discussions In recent years, FDs have been used to model some natural phenomena using fuzzy equa- tions. It has been observed that the order of the FD has a close relationship in controlling the shape of the solution so that it expresses the phenomenon more realistically. Our goal in this paper was to use C-FD in the fractional fuzzy equations instead of other previously used FDs, as the Caputo derivative. It was observed from the graphs that the solutions we obtained were smooth and simulating the ordinary derivative. Also, the mathematical calculations in finding the solutions were easier than previous equations in which other types of FDs were used. This is due to the advantages that the C-FD has over the other types. On the other hand, the RPS method was used in finding series solutions of initial problems on FC-FDEs inthesenseofSGα-FCD due to their novelty and ease. Indeed, the choice of this method was successful, as an accurate approximate solution was found in general, and sometimes the ES was obtained as we presented in the first and second examples. The results show that the power series analysis method is a powerful and easy-to-use analytic tool to solve initial problems on FC-FDEs. How to apply this new approach for solving problems related to FC-PDEs still need further research. FUZZY INFORMATION AND ENGINEERING 27 Acknowledgements The authors express their sincere thanks to the referees for careful reading of the manuscript and for valuable helpful suggestions. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on Contributors Dr. Zeyad Al-Zhour received his Ph.D. in applied mathematics from University Putra Malaysia (Malaysia) in 2007. Then he started working at Zarqa University (2007-2010) as an Assistant Pro- fessor, and then at Imam Abdulrahman Bin Faisal University (2010- Present) and was promoted to Associate Professor in 2018. Dr. Al-Zhour supervised several MSc. and Ph.D Thesis, published more than 60 articles, and achieved several projects. Dr. Al-Zhour has received many awards (e.g. Distin- guished Scientist in Applied Mathematics-VIRA -2018) and letters of appreciation during his work from many universities and companies. His research interests are in matrix operators and algebraic systems, fractional-order systems, fuzzy differential equations, numerical methods, mathematical engineering, and mathematical physics. Ahmad El-Ajou earned his Ph.D. degree in mathematics from the university of Jordan (Jordan) in 2009. He then began work at Al-Balqa Applied university in 2011 as assistant professor of applied mathe- matics, in 2015 received the rank of associate professor, and in 2020 received the rank of professor from the same university. His research interests focus on numerical analysis, fractional differential and integral equations, fuzzy differential and integral equations, and simulating and modeling. Dr. Moa’ath N. Oqielat received his Ph.D. in applied mathematics from Queensland University of Tech- nology (Australia) in 2010. He then began work at Al-Balqa Applied University as assistant professor of applied mathematics and promoted to associate professor in 2020. His research interests focus on modelling and simulating, fractional calculus, and numerical analysis. Dr. Oqielat has written over 35 research papers. Dr. Osama N. Oqily received his Ph.D. in applied mathematics from the University of Southern Queens- land (Australia) in 2014. He began work at Jerash University as assistant professor from 2014 to 2017and then he joined to work at Alahliyaa Amman University since 2017 till now. Dr. Oqily promoted to associate professor in 2020. His research interests focus on modelling and simulating, fractional calculus and fuzzy differential equations. Dr. Shadi Salem received his Ph.D. in Physics from Goethe University Frankfurt (Germany) in 2010. He began work at Al-Balqa Applied university as assistant professor in 2011 and got his promotion as associate professor in 2017. Now he is working at Imam Abdulrahman bin Faisal University. Mousa M. Imran received his PhD degree in physics in 2001 from the University of Rajasthan, in India. He visited laboratories in USA and Europe as postdoctoral researcher. Since 2001, he has been appointed in the Faculty of Science at Al Balqa Applied University, Jordan. He held several admin- istrative positions in the University and guided many MSc and Phd students. His research includes preparation and characterization of bulk, thin films and nano-metric semiconducting chalcogenide materials using different technique. ORCID Zeyad Al-Zhour http://orcid.org/0000-0002-8543-5497 28 Z. AL-ZHOUR ET AL. References [1] Zadeh LA. Fuzzy sets. Inform Control. 1965;8:338–353. [2] Chang SSL, Zadeh LA. On fuzzy mapping and control. Transactions Syst Man Cybernetics. 1972;2:30–34. [3] Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning. Inform Sci. 1975;2(8):301–357. [4] Dubois D, Prade H. Towards fuzzy differential calculus. Fuzzy Sets Syst. 1982;8:1–7. [5] Stefanini L. A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. 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Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations

Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations

Abstract

Purpose: We construct the analytical approximate resiual power fuzzy series solutions of fuzzy conformable fractional differential equations in an -level depiction in the sense of strongly generalized -fuzzy conformable derivative in which of the all initial conditions are taken to be fuzzy numbers. Methodology: The certain fuzzy conformable fractional differential equation under strongly generalized -fuzzy derivative is converted to a crisp one as a family of differential inclusions and...
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© 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China.
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1616-8666
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1616-8658
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10.1080/16168658.2022.2119041
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FUZZY INFORMATION AND ENGINEERING https://doi.org/10.1080/16168658.2022.2119041 RESEARCH ARTICLE Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations a b b c Zeyad Al-Zhour , Ahmad El-Ajou , Moa’ath N. Oqielat , Osama N. Al-Oqily ,Shadi a d Salem and Mousa Imran Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia; Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan; Department of Basic Sciences, Faculty of Arts and Sciences, Al-Ahliyyah Amman University, Amman, Jordan; Department of Physics, Faculty of Science, Al-Balqa Applied University, Salt, Jordan ABSTRACT ARTICLE HISTORY Received 10 February 2021 Purpose: We construct the analytical approximate resiual power Revised 16 May 2021 fuzzy series solutions of fuzzy conformable fractional differential Accepted 23 August 2022 equations in an r-level depiction in the sense of strongly generalized α-fuzzy conformable derivative in which of the all initial conditions KEYWORDS are taken to be fuzzy numbers. Fuzzy fractional operator; Methodology: The certain fuzzy conformable fractional differential fuzzy fractional power series; equation under strongly generalized α-fuzzy derivative is converted strongly generalised fuzzy derivative to a crisp one as a family of differential inclusions and solved via resiual power method. The main drawback concerning the use of dif- ferential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a fuzzy valued function. Findings: (i) To show the efficiency of our proposed method: Several important and attractive test examples, which included the fractional conformable fuzzy integro-differential equation are discussed and solved in detail. (ii) To show the stability of approximate solutions to specific prob- lems: some graphical results, numerical comparisons and tabulate data are created and discussed at different values of Value: Using the residual power series analysis methos is a pow- erful and easy-to-use analytic tool to solve initial problems on fuzzy conformable fractional differential equations and it suc- cessfully applied to solve real life problems such as the induc- tance–resistance–capacitance, RLC-series circuit. 1. Introduction Most real-world optimisation problems often involve uncertainty or inaccuracy in the data due to measurement errors or some unexpected things. Fuzzy logic is a type of uncertainty that is very common in solving those kinds of problems and it is also used as an important mathematical tool to express the ambiguity and inaccuracy of human thinking [1–4]. Zadeh CONTACT Zeyad Al-Zhour zeyad1968@yahoo.com; zalzhour@iau.edu.sa © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 Z. AL-ZHOUR ET AL. established the fuzzy set theory [1], fuzzy derivative [2] and also introduced the concept of the linguistic approach and its applications in artificial intelligence, decision-making pro- cess information retrieval, and economics [3]. However, Dubois and Prade [4] proposed the concepts of fuzzy differential and differential integration for fuzzy valued functions (F-VFs) and developed by many researchers such as Stefanini [5], Dubois, Prade [6–8], and Puri, and Ralescu [9–10]. Fuzzy fractional differential equations (F-FDEs) appeared as a generalisation of fuzzy differential equations (F-DEs) and they are playing a fundamental role in modelling many real-life fuzzy difficulties and various phenomena under uncertainty that arise many applications such as in quantum field theory and optics, doffing oscillator and population models, electronic, dynamical and control systems, artificial intelligence, industrial engi- neering, financial and management of banking, nature studies and technological processes [11–32]. For example, Guo et al. [11] introduced the fuzzy population models; Agarwal et al. [12] presented the concept of F-FDE; Fard and Salehi [13]; Soolaki et al. [14] discussed the fuzzy fractional variational problems using Caputo and combined Caputo differentiability; Zhang et al. [15] proved the generalised necessary and sufficient optimality conditions for fuzzy fractional problems based on Atangana-Baleanu fractional derivative (FD) and gen- eralised Hukuhara difference; Das and Roy [16] presented a new numerical method for solving F-FDEs using Adomian decomposition method in terms Riemann-Livoullie sense; Salahshour et al. [17] solved F-FDEs by fuzzy Laplace transforms; Ghaemi et al. [18] discussed the numerical solution (NS) of a fuzzy fractional kinetic equation in the term of fuzzy Caputo FD; Marc et al. [19] discussed the potential applications of fuzzy logic in financial and man- agement of banking; Venkat et al. [20] used fuzzy logic in financial markets for decision making; Bede and Gal [21] introduced the strongly-generalised differentiability (SGD) for a F-VF; Hasan et el. [22] investigated the analytical and NSs of fractional fuzzy hybrid system in Hilbert space are devoted to model control via Atangana-Baleanu Caputo FD; Ahma- dian et al. [23–25] used tau method for finding the NSs of a fuzzy fractional kinetic model, uncertain fractional viscoelastic model and linear F-FDEs, respectively; and Sh. Behzadi et al. [26] applied the Fuzzy Picard method for solving fuzzy quadratic Riccati and Fuzzy Painlev equations. Generally, it is not straightforward to obtain exact solutions (ESs) for these kinds of prob- lems because of the difficulties involved, so reliable numerical techniques are needed to deal with these types of problems. Recently, some methods were suggested for creating analytical NSs for uncertain F-FDEs in terms of SGD sense. These methods are the finite ele- ment, reproducing kernel, fuzzy Picard method, homotopy, and residual power series (R-PS) methods [21,26–33]. The R-PS method developed by some authors [30–31] which is considered as effec- tive optimisation technique to construct power series solutions (PSSs) of some F-DEs. Very recently, the R-PS technique was used for solving certain and uncertain F-FDEs [34–41]such as Tariq et al. [35] combined Laplace transform method with the R-PS method in new theory and view and used it to create PSSs for the target equations. Abu Arqub [30] used the R-PS method to obtain the PSSs of F-DEs under SGD; Alaroud et al. [36] presented a novel opti- misation technique, the RPS for handling certain classes of F-FDEs of order 1 < ≤ 2 under SGD; Alshammari et al. [29] presented the analytic NSs of uncertain Riccati DEs using R-PS method; Abu Arqub and Al-Smadi [37] proposed the so-called fuzzy conformable fractional derivative (FC-FD) and integral and then used to obtain the solutions of certain fuzzy con- formable fractional differential equations (FC-FDEs) under SGD and Alshammari et al. [38] FUZZY INFORMATION AND ENGINEERING 3 proposed an accurate numeric-analytic algorithm on the R-PS to investigate the fuzzy NSs for a nonlinear fuzzy Duffing oscillator under SGD. There are many definitions for FDs and the most two important of them are the Riemann- Liouville and the Caputo definitions which are used in many applications and real-natural phenomena. While in 2014, Khalil et al. [39] defined the conformable FD (C-FD) of an order α ∈ (0, 1] of f :[t , ∞) → R at t ≥ 0 as follows: 0 0 1−α f (t + ε(t − t ) ) − f (t) α (α) T f (t) = f (t) = lim , t > t,(1) ε→0 (α) (α) and f (t ) = lim f (t). Provided f (t) is differentiable, and the limits exists. t→t For more details about the basic rules and applications of C-FD, it can be found in the literature [39–47]. The main advantages of the C-FD can be summarised as follow: (i) It can be very easily computed compared with other previous fractional definitions. (ii) It satisfies all concepts of classical derivative while other fractional definitions fail to satisfy some of them. (iii) It can be computed for a non-differentiable function. (iv) It solves F-DEs, PF-DEs, and systems easily and efficiently. (v) It modifies some important transforms such as Laplace, Sumudu, and Nature trans- forms, and they are used as efficacious tools for solving some singular F-DEs. (vi) Several applications have been remodeled using C-FD and it can be opened the door for various new applications. (vii) Several new comparison results can be established based on C-FD. Given the above-mentioned features of the C-FD, we seek through this research to integrate this concept of the FD to the fuzzy equations for studying the extent of these features being reflected in the development of those equations and improving them in terms of realism and the form of the solution. In addition, we aim to investigate the applicability of the RPS method under the assumption of SGD to provide analytical-NSs for the FC-FDEs that of the following general form: ˆ ˆ T i(t) = f (t, i(t)), t ∈ [0, b], 0 <α ≤ 1, (2) subject to the fuzzy initial condition (F-IC): ˆ ˆ i(0) = i (3) where T is the C-FD of an order α, f is an analytic function, i :[a, b] × R → R is a F F continuous F-VF, in which R stands to the group of fuzzy numbers. There are three main approaches to solve FC-FDEs with F-ICs. The first one is that if the initial value is only a fuzzy number, the solution will be a fuzzy function, where as a result, the derivatives must be regarded as fuzzy derivatives. To achieve this problem, the SGD for F-VFs must be used. The second approach is that the FC-FDE will be converted to a crisp one as a family of differential inclusions. The main drawback concerning the use of differential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a F-VF. The last approach is related to the crisp equation and the 4 Z. AL-ZHOUR ET AL. initial fuzzy values, which are in the solution. The weak point lies in having the solution in the fuzzy setting re-written, making the techniques of solution less friendly and more con- strained with a lot of steps of computation to do. To achieve this point, we will replace the crisp equation and the initial fuzzy values in terms of real constants as well as arithmetic operations regarded as operations on fuzzy numbers in the final solution. The latest solu- tions approach, which focuses on searching the fuzzy set of real-valued functions (R-VFs), not F-VFs exemplified with these R-VFs, fulfils the said restrictions. Furthermore, most of the four main interests bring us to fuzzy applications and R-PS method are: (i) They are convenient and remarkably powerful tools in solving numerous issues arising in physics, engineering, financial and management of banking and other sciences. (ii) They are useful in uncertain or estimated reasoning or discretionary thinking, espe- cially for a framework with a scientific model that is difficult to derive. (iii) Useful in assessed qualities under deficient or questionable data, especially for control theory in optimality and security. (iv) (iv) Fuzzy controllers are widespread nonlinear controllers and each one of these exam- inations is fundamental in nature and more investigations should be possible. For the sake of idealism, the optimal fuzzy control field appears to be completely open. This paper consists of five Sections. Some important definitions and concepts for the frac- tional fuzzy theory are given concisely in Section 2. In Section 3, we employ the R-PS method to construct the analytical approximate fuzzy PSSs for the initial value problem (IVP) as in Equations (2) and (3) in an r-level depiction in the sense of strongly generalised α-fuzzy conformable derivative (SG α-FCD). Some important and attractive test examples with illus- trative graphs and tables are given in Section 4 in order to validate our proposed algorithm. Finally, the conclusion is presented in Section 5. 2. Conformable Fractional Fuzzy Theory This section reviews and studies some fuzzy (fractional) theory that will be used in our investigations and findings throughout the paper. Definition 2.1 ([18,22,30]): Let u be a fuzzy set in R, then u is called a fuzzy number if (i) uisanormal (∃t ∈ R such that u(t ) = 1). 0 0 (ii) u is convex (For all t, s ∈ R and 0 ≤ r ≤ 1, then u(rt + (1 − r) s) ≥ min{u (t), u (s)}). (iii) u is an upper semi-continuous ({t ∈ R : u(t) ≥ k} is closed for all k ∈ R) (iv) u is bounded support ([u] = supp(u) is a compact subset of R). Definition 2.2 ([18]): If u ∈ R and r ∈ [0, 1], then the r-level of u is called a crisp set and defined as: [u] ={t ∈ R : u(t) ≥ r}.(4) So,ifu ∈ R , then the r-level of u is a closed interval in R and defined by: [u] = [u (r), u (r)] = [u , u ], (5) 1 2 1r 2r FUZZY INFORMATION AND ENGINEERING 5 r r where u = u (r) = min{s : s ∈ [u] } and u = u (r) = max{s : s ∈ [u] }. 1r 1 2r 2 However, there are several common forms for the fuzzy number u, one of them is the triangu- lar form which is defined by an ordered triple u = (μ , μ , μ ) ∈ R with μ <μ <μ and 1 2 3 1 2 3 whose r-level is [22]: [u] = [μ + (μ − μ )r, μ − (μ − μ )r], r ∈ [0, 1]. (6) 1 2 1 3 3 2 Some conditions must be satisfied by two functions: u , u : [0, 1] → R,sothat [u , u ] can 1r 2r 1r 2r be parametrised by a fuzzy number u for all r ∈ [0, 1]. These conditions are presented in the next theorem. Theorem 2.1 ([22]): Suppose that u , u : [0, 1] → R satisfy the following conditions: 1r 2r (i) u is a bounded monotonic nondecreasing left continuous function on r ∈ (0, 1]. 1r (ii) u is a bounded monotonic nonincreasing left continuous function on r ∈ (0, 1]. 2r (iii) u and u are right continuous at r = 0. 1r 2r (iv) u ≤ u . 1r 2r Then u : R → [0, 1] defined by u(t) = sup{r : u ≤ t ≤ u } is a fuzzy number with r-level 1r 2r r r set: [u] = [u , u ]. Conversely, if u is a fuzzy number with [u] = [u , u ], then the functions: 1r 2r 1r 2r u , u :[0,1] → R satisfy the conditions (i)-(iv). 1r 2r Definition 2.3 ([18,22,48]): Let u = [u , u ] and v = [v , v ] ∈ R , μ ∈ R/{0} and r ∈ 1r 2r 1r 2r [0, 1]. Then we have: (i) Addition operation: r r r [u + v] = [u] + [ v] = [u + v , u + v](7) 1r 1r 2γ 2r (ii) H-difference: If there exists an element z ∈ R such that u = v + z, then z is called H- difference of u and v, denoted by u v, and defined as follows: r r u v = [u] − [ v] = [u − v , u − v ]. (8) 1r 1r 2γ 2r (iii) Scalar multiplication: r γ [μu] = μ[u] = [min{μu , μu }, max{μu , μu }](9) 1r 2r 1r 2r (iv) Multiplication: [uv] = [min{u v , u v , u v , u v }, max{u v , u v , u v , u v }] (10) 1r 1r 1r 2r 2r 1r 2r 2r 1r 1r 1r 2r 2r 1r 2r 2r r r (v) Equality: Two fuzzy numbers u and v are equal if [u] = [v] ,thatisu = v and u = v . 1r 1r 2r 2r Definition 2.4 ([18]): The complete metric structure on R is defined as d : R × R → F F F R ∪{0} such that d(u, v) = sup max{|u − v |, |u − v |}. (11) 1r 1r 2r 2r r∈[0,1] Note that the metric space d in R has many nice and interesting properties on addition operation and scalar multiplication and the reader can be found in the literature [18,36,44–50]. 6 Z. AL-ZHOUR ET AL. Definition 2.5 ([30,48]): Let [a, b] ⊆ R be a vector space and R be a set of fuzzy numbers. ˆ ˆ Then a function i: [a, b] → R is called a F-VF on [a, b]. Corresponding to such a function iand ˆ ˆ r ∈ [0, 1], we denote two R-VFs i (t) and i (t) for all t ∈ [a, b] as: 1r 2r ˆ ˆ ˆ [ i(t)] = [ i (t), i (t)], (12) 1r 2r which is called the r-level representation of a F-VF. ˆ ˆ Definition 2.6 ([18]): Let i and i :[a, b] → R be F-VFs. Then the uniform distance between 1 2 ˆ ˆ i (t) and i (t) is defined as: 1 2 ˆ ˆ ˆ ˆ d (i (t), i (t)) = sup d(i (t), i (t)). (13) 1 2 1 2 t∈[a,b] Definition 2.7 ([18,30,48]): Let i :[a, b] → R be a F-VF and t ∈ [a, b].If ∀ε> 0, ∃δ> 0 F 0 ˆ ˆ such that d (i(t), L)<∈, whenever |t − t | <δ, then we say that L ∈ R is limit of iat t ,which 0 F 0 ˆ ˆ is denoted by: lim i(t) = L. Also, the F-VF i(t) is said to be continuous at t ∈ [a, b] if lim i(t) = t→t t→t 0 0 ˆ ˆ ˆ i(t ). Indeed, i is continuous on [a, b] if it is continuous ∀t ∈ [a, b]. In addition, if i :[a, b] → R 0 F is a fuzzy continuous function, then we have: ˆ ˆ ˆ d(i(t),0) = sup max{|i (t)|, |i (t)|}, ∀t ∈ [a, b]. (14) 1r 2r r∈[0,1] Definition 2.8 ([18,37,48–49]): Let i :[a, b] → R be a F-VF. Then for a fixed point t ∈ F 0 ˆ ˆ [a, b], i is called the SGD at t if there exists an element i (t ) ∈ R such that either: 0 0 F ˆ ˆ ˆ ˆ (i) The H-differences i(t + ε) i(t ) and i(t ) i(t − ε) are exist, and 0 0 0 0 ˆ ˆ ˆ ˆ i(t + ε) i(t ) i(t )  i(t − ε) 0 0 0 0 ˆ ˆ i (t ) = T i(t ) = lim = lim , (15) 0 0 (1) + + ε ε ε→0 ε→0 for all ε> 0 sufficiently near to 0 and the limits in a metric d. ˆ ˆ ˆ ˆ (ii) The H-differences: i(t ) i(t + ε) and i(t − ε) i(t ) are exist, and 0 0 0 0 ˆ ˆ ˆ ˆ i(t ) i(t + ε) i(t − ε) i(t ) 0 0 0 0 ˆ ˆ i (t ) = T i(t ) = lim = lim , (16) 0 0 (2) + + ε→0 −ε ε→0 −ε for all ε> 0 sufficiently near to 0 and the limits in a metric d. ˆ ˆ Remark 2.1 ([30,37]): (i) If i is a differentiable function (DF) for all t ∈ (a, b), then i is an SGD on (a, b). (ii) The limits in Equations (17) and (18) are considered in metric space (R , d) and at the edge-point of (a, b), we consider one direction derivative and recalling that: ˆ ˆ ˆ [i(t)] = [i (t), i (t)]. (17) 1r 2r ˆ ˆ ˆ ˆ Theorem 2.2 ([30,37]): Let i :[a, b] → R be a F-VF and let [i(t)] = [i (t), i (t)], ∀r ∈ [0,1] F 1r 2r 1 1 ˆ ˆ and T i(t) or T i(t) exists. Then (1) (2) FUZZY INFORMATION AND ENGINEERING 7 ˆ ˆ ˆ (i) If i is (1) – DF, then i (t)and i (t)are DFs and, 1r 2r 1 r r 1 1 ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [i (t)] = [i (t), i (t)] = [T i (t), T i (t)]. (18) 1r 2r (1) 1r 2r ˆ ˆ ˆ (ii) If i is (2)-DF, theni (t)and i (t)are DFs and, 1r 2r 1 r r 1 1 ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [i (t)] = [i (t), i (t)] = [T i (t), T i (t)]. (19) 2r 1r 2r 1r (2) Now, we study the α-crisp conformable derivative (α-CCD) for a crisp function i(t) of order α ∈ (0, 1] and the SGα-FCD for F-VF i(t) of order α ∈ (0, 1] as in the following two definitions [30,37,49–52]. Definition 2.9: Let i :[a, b] → R be a crisp function and α ∈ (0, 1]. Then the α-CCD of i at t ∈ [a, b] is defined by: 1−α ˆ ˆ i(t + ε(t − t ) ) − i(t) α (α) ˆ ˆ T i(t) = i (t) = lim , t > t . (20) ε→0 ˆ ˆ Note that, if i is DF for all t ∈ (a, b), then we say iis α-CCD on (a, b). Lemma 2.1 ([40]): For α ∈ (0, 1] and t > t , α 1−α ˆ ˆ ˆ (i) if i :[a, b] → R is DF for all t ∈ (a, b), then T i(t) = (t − t ) i (t). 0 0 α γ γ −α (ii) T (t − t ) = γ(t − t ) for all t ∈ (a, b). 0 0 0 ˆ ˆ Definition 2.10: Let i :[a, b] → R be an FVF and α ∈ (0, 1]. Then the SG α-FCD of iat t ∈ F 0 [a, b] is defined by one of the following subsequent is working: 1−α 1−α ˆ ˆ ˆ ˆ i(t + ε(t − t ) ) i(t) i(t) i(t + ε(t − t ) ) 0 0 (i) T i(t) = lim = lim , t > t , (1) + + ε→0 ε ε→0 ε (21) 1−α 1−α ˆ ˆ ˆ ˆ i(t + ε(t − t ) ) i(t) i(t) i(t + ε(t − t ) ) 0 0 (ii) T i(t) = lim = lim , t > t , (2) + + ε→0 −ε ε→0 −ε (22) 1−α ˆ ˆ ˆ ˆ where ε> 0 small enough and the H-differences: i(t + ε(t − t ) ) i(x) and i(t) i(t + 1−α ε(t − t ) ) exist in both cases (i) and (ii). ˆ ˆ ˆ Remark 2.2: (i) If i is DF for all t ∈ (a, b), then i is SG α-FCD on (a, b). (ii) i is said to be SG ˆ ˆ α(1)-FCD on (a, b) if i is DF in case (i) of Definition 2.10 and i is said to be SG α(2)-FCD on (a, b) if i is DF in case (ii) of Definition 2.10. Definition 2.11 ([37]): Let i :[a, b] → R and α ∈ (0, 1]. Then the α-fuzzy conformable FF integral of i(t) with reference point t is defined by: α α−1 ˆ ˆ J i(t) = (τ − t ) i(τ )dτ , (23) provided that iis α-FCD on (a, b). 8 Z. AL-ZHOUR ET AL. Theorem 2.3 ([37]): Let i :[a, b] → R be a F-VF and α ∈ (0, 1]. Then the subsequent are working: (i) If iis α(1)-FCD on (a, b), then α r α α 1−α r 1−α ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [T i (t), T i (t)] = (t − t ) [T i(t)] = (t − t ) [i (t), i (t)]. (24) 1r 2r 0 (1) 0 1r 2r (1) (ii) If iis α(2)-FCD on (a, b), then α r α α 1−α r 1−α ˆ ˆ ˆ ˆ ˆ ˆ [T i(t)] = [T i (t), T i (t)](t − t ) [T i(t)] = (t − t ) [i (t), i (t)]. (25) 2r 1r 0 (2) 0 (2) 2r 1r Definition 2.12 ([31]): A series of the form: kα α 2α c (t − t ) = c + c (t − t ) + c (t − t ) + ... , (26) k 0 0 1 0 2 0 k=0 where 0 ≤ m − 1 <α ≤ mandt ≥ t is called a fractional power series (F-PS) about t , where 0 0 t is a variable and c ’s are constants called the coefficients of the series. Theorem 2.4: Suppose that i(t) has an F-PS representation at t of the form kα i(t) = c (t − t ) ,0 ≤ t ≤ R,0 <α ≤ 1, (27) k 0 k=0 where R is the radius of convergence of the F-PS. Then k!α mα (k−m)α T i(t) = c (t − t ) , (28) k 0 (k − m)! k=m mα α α α ˆ ˆ where T i(t) = T · T ... T i(t) (m-times). Proof: We can prove the theorem inductively. For m = 0, the formula is correct. With respect to m = 1 and according to Lemma 2.1 the formula is also true. Assume the formula is true for m = n; i.e k!α nα (k−n)α T i(t) = c (t − t ) . (k − n)! k=n To complete the proof, we need to show that the formula is true for m = n + 1. Thus, k!α (n+1)α α nα α (k−n)α ˆ ˆ T i(t) = T T i(t) = T c (t − t ) k 0 (k − n)! k=n n+1 k!α (k−(n+1))α = c (t − t ) . (k − (n + 1))! k=n+1 FUZZY INFORMATION AND ENGINEERING 9 Theorem 2.5 ([40]): Suppose that i has a F-PS representation at t of the form nα i(t) = c (t − t ) ,0 <α ≤ 1 (29) n 0 n=0 (nα) ˆ ˆ If i(t) ∈ C[t , t + R) and i (t) ∈ C(t , t + R) for n = 0, 1, 2, ..., then the coefficients c in 0 0 0 0 n (nα) i (t ) Equation (29) will take the form c = . n n α n! 3. Construction the Residual Power Series Solution for the FCFDEs In this section, we employ the R-PS method to construct analytical fuzzy NSs in the form of a F-PS for the IVP as in Equations (2) and (3) in an r-level depiction. To accomplish this, we switch the IVP in Equations (2) and (3) to crisp system of F-DEs. The crisp systems depend on the type of differentiability, where i(t) is either α(1)-FCD or α(2)-F-CD. Without losing the generality, we construct the R-PSS to α(1)-FCD only, and in the same approach, we can construct the R-PSS to or α(2)-FCD. Now, assume that i(t) is α(1)-FCD, then the IVP in Equations (2) and (3) can be re- formulated to the following F-DEs system: α α ˆ ˆ ˆ T i (t) = f (t , i (t), i (t)), 1r 2r 1r 2r α α ˆ ˆ ˆ T i (t) = f (t , i (t), i (t)), (30) 2r 1r 1r 2r with the F-ICs: ˆ ˆ ˆ ˆ i (0) = i , i (0) = i , (31) 1r 0,1r 2r 0,2r where α ∈ (0, 1] and t ∈ [0, b]. The R-PS method supposes that the solution of the given DE has an F-PS. So, we assume that the solution of the system (30) and (31) has the following expansions: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (32) 1r k 2r k k=0 k=0 ˆ ˆ According to the F-ICs in Equation (31), it easy to see that b = i and c = i . 0 0,1r 0 0,2r Indeed, R-PS is a technique used to determine the coefficients of the F-PSS in a differ- ent approach than the traditional method. Thus, to determine the coefficients of the series in Equation (32), we need to define the so-called residual functions of the equations in Equation (32) as follows: α α ˆ ˆ ˆ Res (t) = T i (t) − f (t , i (t), i (t)), 1r 1r 2r 1r 2r α α ˆ ˆ ˆ Res (t) = T i (t) − f (t , i (t), i (t)). (33) 2r 2r 1r 1r 2r 10 Z. AL-ZHOUR ET AL. Indeed, we have the following facts for our approach via the R-PS method: (i) Res (t) = Res (t) = 0, fort ∈ [0, b]. 1r 2r mα mα (ii) T Res (t) = T Res (t) = 0, m = 0, 1, 2, ... , fort ∈ [0, b]. (34) 1r 2r mα mα (iii) T Res (0) = T Res (0) = 0, m = 0, 1, 2, ... (35) 1r 2r Now, if we substitute the F-PS in Equation (32) into Equation (33) and use the Theorem 2.4, then we obtain the following expression of the residual functions: ∞ ∞ ∞ (k−1)α α kα kα Res (t) = kαb t − f t , b t , c t , 1r k 2r k k k=1 k=0 k=0 ∞ ∞ ∞ (k−1)α α kα kα Res (t) = kαc t − f t b t , c t . (36) 2r k 1r k k k=1 k=0 k=0 Since f and f are analytic functions of F-PS expansions, Equation (36) can be expressed 1r 2r in the following F-PS expansions: ∞ ∞ (k−1)α kα Res (t) = kαb t − g (b , b , ... , b , c , c , ... , c )t , 1r k 2rk 0 1 k 0 1 k k=1 k=0 ∞ ∞ (k−1)α kα Res (t) = kαc t − g (b , b , ... , b , c , c , ... , c )t . (37) 2r k 1rk 0 1 k 0 1 k k=1 k=0 where g and g are multivariable functions of b , b , ... , b , c , c , ... , c generates 1rk 2rk 0 1 k 0 1 k according to f and f . 1r 2r mα Likewise, if we apply the C-FD T on both sides of Equation (37), then it can be expressed as an F-PS expansion as follows: ∞ ∞ m+1 m k!α b k!α g (b , b , ... , b , c , c , ... , c ) k 2rk 0 1 k 0 1 k mα (k−m−1)α (k−m)α T Res (t) = t − t , 1r (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m k!α c k!α g (b , b , ... , b , c , c , ... , c ) k 1rk 0 1 k 0 1 k mα (k−m−1)α (k−m)α T Res (t) = t − t . 2r (k − m − 1)! (k − m)! k=m+1 k=m (38) According to Equation (38) and the fact in Equation (34), we define the so-called αmth-order DE as follows: ∞ ∞ m+1 m k!α b k!α g (b , b , ... , b , c , c , ... , c ) k 2rk 0 1 k 0 1 k (k−m−1)α (k−m)α t − t = 0, (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m k!α c k!α g (b , b , ... , b , c , c , ... , c ) k 1rk 0 1 k 0 1 k (k−m−1)α (k−m)α t − t = 0. (k − m − 1)! (k − m)! k=m+1 k=m (39) FUZZY INFORMATION AND ENGINEERING 11 Substitute t = 0 into Equation (39), we obtain the following iterative equations: m+1 m (m + 1)!α b − m!α g (b , b , ... , b , c , c , ... , c ) = 0, m+1 2rm 0 1 m 0 1 m m+1 m (m + 1)!α c − m!α g (b , b , ... , b , c , c , ... , c ) = 0. (40) m+1 1rm 0 1 m 0 1 m Solving Equation (40) for b and c gives the following recurrence relations which m+1 m+1 determine the coefficients of the F-PS in Equation (32): g (b , b , ... , b , c , c , ... , c ) 2rm 0 1 m 0 1 m b = , b = i , m = 0, 1, 2, ... , m+1 0 0,1r (m + 1)α g (b , b , ... , b , c , c , ... , c ) 1rm 0 1 m 0 1 m c = , c = i , m = 0, 1, 2, ... . (41) m+1 0 0,2r (m + 1)α Therefore, the exact solution (ES) of the system (30) and (31) in an F-PS form is given by g (b , b , ... , b , c , c , ... , c ) 2r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t , 1r 0,1r kα k=1 g (b , b , ... , b , c , c , ... , c ) 1r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t . (42) 2r 0,2r kα k=1 Furthermore, the mth-truncated of the F-PS in Eq (3.13) for an appropriate number of m gives the NS of the system (30) and (31) as follows: g (b , b , ... , b , c , c , ... , c ) 2r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t , 1r,m 0,1r kα k=1 g (b , b , ... , b , c , c , ... , c ) 1r(k−1) 0 1 k−1 0 1 k−1 kα ˆ ˆ i (t) = i + t . (43) 2r,m 0,2r kα k=1 With the same previous approach, we can find a PSS to the system of F-DEs that corre- sponding to α(2)-FCD case. While in the next section, we consider both cases (α(1)-FCD and α(2)-FCD) in details for dealing the three important and interesting examples. 4. Illustrative Examples of FC-FDEs In this chapter, we apply the R-PS method to three examples to illustrate the construction that was prepared in Section 3 and to confirm the effectiveness and efficiency of the pro- posed method used in constructing an F-PSS to FC-FDEs. The first example includes a linear FC-FDE, the second example deals with a nonlinear FC-FDE, where as the third example illustrates our method to construct the F-PSS to FC- integral equation. Example 4.1: Given the following FC-FDE: α α ˆ ˆ T i(t) =−i(t) + sin(t ), t ∈ [0, 1], 0 <α ≤ 1, (44) subject to the F-IC: i(0) = w, (45) 12 Z. AL-ZHOUR ET AL. where 25ρ − 24, 0.96 ≤ ρ ≤ 1 w(ρ) = 101 − 100ρ,1 ≤ ρ ≤ 1.01 . 0, otherwise 24 r If we assume r = 25ρ − 24, then ρ = + . Whereas, if we assume r = 101 − 100ρ, then 25 25 101 r r 24 r 101 r r ˆ ˆ ˆ ρ = − . Therefore, [w] = + , − and [i(t)] = [i (t), i (t)]. Hence, to 1r 2r 100 100 25 25 100 100 determine the α(1) and α(2)-fuzzy RPS solutions of the FC-FDE as in Equations (44) subject to the F-IC (45), we consider the following two cases: Case 1: The system of the ODEs corresponding to α(1)-FCD is α α ˆ ˆ T i (t) =−i (t) + sin(t ), 1r 2r α α ˆ ˆ T i (t) =−i (t) + sin(t ), (46) 2r 1r subject to the F-ICs: 24 1 101 1 ˆ ˆ i (0) = + r, i (0) = − r. (47) 1r 2r 25 25 100 100 To employ the RPS method to create the PSS to the system (46) and (47), we assume that the solution of this system can be expressed as F-PS expansion about the initial point t = 0 as follows: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (48) 1r 2r k k k=0 k=0 24 1 101 1 Using the F-ICs in Equation (47) give us b = + r and c = − r. Therefore, the 0 0 25 25 100 100 solution of system (46) and (48) can be represented as: ∞ ∞ 24 1 101 1 kα kα ˆ ˆ i (t) = + r + b t , i (t) = − r + c t . (49) 1r 2r k k 25 25 100 100 k=1 k=1 Apply the C-FD T to Equation (49), we obtain: ∞ ∞ (α) (α) α (k−1)α α (k−1)α ˆ ˆ ˆ ˆ T i (t) = i (t) = kαb t , T i (t) = i (t) = kαc t . (50) 1r k 1r k 1r 1r k=1 k=1 Define the residual functions of Equation (46) as follows: α α α α ˆ ˆ ˆ ˆ Res (t) = T i (t) + i (t) − sin(t ), Res (t) = T i (t) + i (t) − sin(t ). (51) 1r 1r 2r 2r 2r 1r So, the αmth-order DEs of Equation (46) have the following forms: mα α α mα α α ˆ ˆ ˆ ˆ T (T i (t) + i (t) − sin(t )) = 0, T (T i (t) + i (t) − sin(t )) = 0, m = 0, 1, 2, ... . 1r 2r 2r 1r (52) To determine the coefficients of the expansions, b and c for k = 1, 2, ..., substitute the k k expansion formulas in Equations (49) and (50) into Equation (52), yields ∞ ∞ 24 1 mα (k−1)α kα α T (kαb t ) + + r + (c t ) − sin(t ) = 0, m = 0, 1, 2, ... , k k 25 25 k=1 k=1 FUZZY INFORMATION AND ENGINEERING 13 ∞ ∞ 101 1 mα (k−1)α kα α T (kαc t ) + − r + (b t ) − sin(t ) = 0, m = 0, 1, 2, ... . k k 100 100 k=1 k=1 (53) Through a few simple calculations, a new discretised version of Equation (53) can be obtained and given by: ∞ ∞ m+1 m α k!b α k!c k k (k−m−1)α (k−m)α t + t = χ (t), (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m α k!c α k!b k k (k−m−1)α (k−m)α t + t = χ (t), (54) (k − m − 1)! (k − m)! k=m+1 k=m where (m−1)/2 m α (−1) α cos t if m is odd χ (t) = (55) m/2 m α (−1) α sin t if m is even It is obvious that, the αmth-derivative of the F-PS representation, Equation (54) is conver- gent at least at t = 0, for m = 0, 1, 2, .... Therefore, the substituting t = 0 into Equation (54) gives the following recurrence relation which determines the values of the coefficients b and c : 24 1 χ (0) − α m!c m m b = + r, b = , m = 0, 1, 2, ... , 0 m+1 m+1 25 25 α (m + 1)! 101 1 χ (0) − α m!b m m c = − r, c = , m = 0, 1, 2, ... . (56) 0 m+1 m+1 100 100 α (m + 1)! If we collect and substitute these values of the coefficients back into Equation (49), then the ES of the system (46) and (47) has the general form, which is coinciding with the general expansion: 3 3 c α + b α + c α − α + b α − α + c 0 0 0 0 0 α 2α 3α 4α 5α i (t) = b − t + t − t + t − t 1r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + b α − α + α + c α − α + α − α + b 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + c α − α + α − α + α + b 0 0 9α 10α − t + t 9 10 9!α 10α 3 5 7 9 α − α + α − α + α + c 11α − t + ... , 11!α 3 3 b α + c α + b α − α + c α − α + b 0 0 0 0 0 α 2α 3α 4α 5α i (t) = c − t + t − t + t − t 2r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + c α − α + α + b α − α + α − α + c 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + b α − α + α − α + α + c 0 0 9α 10α − t + t 9 10 9!α 10α 14 Z. AL-ZHOUR ET AL. 3 5 7 9 α − α + α − α + α + b 11α − t + ... . (57) 11!α If we rearrange the terms in Equation (57), then it can be written as follows: b b b b b c c c 0 0 0 0 0 0 0 0 2α 4α 6α 8α 10α α 3α 5α i (t) = b + t + t + t + t + t − t − t − t 1r 0 2 4 6 8 10 3 5 2!α 4!α 6!α 8!α 10α α 3!α 5!α c c c α α α − α 0 0 0 7α 9α 11α 2α 3α 4α − t − t − t + t − t + t 7 9 11 2 3 4 7!α 9!α 11!α 2!α 3!α 4!α 3 3 5 3 5 3 5 7 α − α α − α + α α − α + α α − α + α − α 5α 6α 7α 8α − t + t − t + t 5 6 7 8 5!α 6!α 7!α 8!α 3 5 7 3 5 7 9 3 5 7 9 α − α + α − α α − α + α − α + α α − α + α − α + α 9α 10α 11α − t + t − t + ... , 9 10 11 9!α 10α 11!α c c c c c b b b 0 0 0 0 0 0 0 0 2α 4α 6α 8α 10α α 3α 5α i (t) = c + t + t + t + t + t − t − t − t 2r 0 2 4 6 8 10 3 5 2!α 4!α 6!α 8!α 10α α 3!α 5!α b b b α α α − α 0 0 0 7α 9α 11α 2α 3α 4α − t − t − t + t − t + t 7 9 11 2 3 4 7!α 9!α 11!α 2!α 3!α 4!α 3 3 5 3 5 3 5 7 α − α α − α + α α − α + α α − α + α − α 5α 6α 7α 8α − t + t − t + t 5 6 7 8 5!α 6!α 7!α 8!α 3 5 7 3 5 7 9 3 5 7 9 α − α + α − α α − α + α − α + α α − α + α − α + α 9α 10α 11α − t + t − t + ... . 9 10 11 9!α 10α 11!α (58) We can summarise the solutions in Equation (57) as ⎛ ⎞ ∞ ∞ ∞ ∞ 2kα (2k+1)α kα t t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = b − c + (−1) α , 1r 0 0 2k k k (2k)!α (2k + 1)!α k!α k=0 k=0 j=1 k=2j ⎛ ⎞ ∞ ∞ ∞ ∞ 2kα (2k+1)α kα t t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = c − b + (−1) α , (59) 2r 0 0 2k k k (2k)!α (2k + 1)!α k!α k=0 k=0 j=1 k=2j which are equivalent to: ⎛ ⎞ ∞ ∞ α α kα 24 r t 101 r t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = + cosh − − sinh + (−1) α , 1r 25 25 α 100 100 α k!α j=1 k=2j ⎛ ⎞ ∞ ∞ α α k kα 101 r t 24 r t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = − cosh − + sinh + (−1) α . 2r 100 100 α 25 25 α k!α j=1 k=2j (60) In fact, if α = 1, then the PSS to the system (46) and (47) will be as: 24 1 101 1 1 1 −t i (t) = + r cosh(t) − − r sinh(t) + (sin(t) − cos(t)) + e , 1r 25 25 100 100 2 2 101 1 24 1 1 1 −t i (t) = − r cosh(t) − + r sinh(t) + (sin(t) − cos(t)) + e . 2r 100 100 25 25 2 2 FUZZY INFORMATION AND ENGINEERING 15 Figure 1. The surface graph of the α(1)-FCD – 10th approximate R-PSS and ES for Equations (46) and (47) at different values of α, where the lower surface represents i (t), where as the upper surface repre- 1r r r r ˆ ˆ ˆ ˆ sents i (t):(a)α(1)-FCD of [i (t)] at α = 0.6, (b) α(1)-FCD of [i (t)] at α = 0.8, (c)α(1)-FCD of [i (t)] 2r 10 10 10 at α = 1, (d) α(1)-FCD of [i(t)] (exact) at α = 1. Figure 1 shows the graphical results for the approximate R-PSSs and ES of Equations (46) and (47) at different values of α corresponding to α(1)-FCD. Case 2: The system of the ODEs corresponding to α(2)-FCD is: α α ˆ ˆ T i (t) =−i (t) + sin(t ), 1r 1r α α ˆ ˆ T i (t) =−i (t) + sin(t ), (61) 2r 2r subject to F-ICs: 24 1 101 1 ˆ ˆ i (0) = + r, i (0) = − r. (62) 1r 2r 25 25 100 100 Similarly, assume that the F-PSSs of Equations (58) and (59) has the following form: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (63) 1r 2r k k k=0 k=0 Then the αm10th-order DEs corresponding to Equation (58) is: ∞ ∞ 24 1 mα (k−1)α kα α T (kαb t ) + + r + (b t ) + sin(t ) = 0, m = 0, 1, 2, ... , k k 25 25 k=1 k=1 16 Z. AL-ZHOUR ET AL. ∞ ∞ 101 1 mα (k−1)α kα α T (kαc t ) + − r + (c t ) + sin(t ) = 0, m = 0, 1, 2, ... . k k 100 100 k=1 k=1 (64) mα Operator T in Equation (64) gives a summarised form of the equation: ∞ ∞ m+1 m α k!b α k!c k k (k−m−1)α (k−m)α t + t = χ (t), (k − m − 1)! (k − m)! k=m+1 k=m ∞ ∞ m+1 m α k!c α k!b k k (k−m−1)α (k−m)α t + t = χ (t), (65) (k − m − 1)! (k − m)! k=m+1 k=m where χ (t) is defined as in Equation (55). The recurrence relation which determines the values of the coefficients b and c is: k k 24 1 χ (0) − α m!b m m b = + r, b = , m = 0, 1, 2, ... , 0 m+1 m+1 25 25 α (m + 1)! 101 1 χ (0) − α m!c m m c = − r, c = , m = 0, 1, 2, ... . (66) 0 m+1 m+1 100 100 α (m + 1)! So, the ESs of Equations (61) and (62) have the general form: 3 3 b α + b α + b α − α + b α − α + b 0 0 0 0 0 α 2α 3α 4α 5α i (t) = b − t + t − t + t − t 1r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + b α − α + α + b α − α + α − α + b 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + b α − α + α − α + α + b 0 0 9α 10α − t + t 9 10 9!α 10α 3 5 7 9 α − α + α − α + α + b 11α − t + ... , 11!α 3 3 c α + c α + c α − α + c α − α + c 0 0 0 0 0 α 2α 3α 4α 5α i (t) = c − t + t − t + t − t 2r 0 2 3 4 5 α 2!α 3!α 4!α 5!α 3 5 3 5 3 5 7 α − α + α + c α − α + α + c α − α + α − α + c 0 0 0 6α 7α 8α + t − t + t 6 7 8 6!α 7!α 8!α 3 5 7 3 5 7 9 α − α + α − α + c α − α + α − α + α + c 0 0 9α 10α − t + t 9 10 9!α 10α 3 5 7 9 α − α + α − α + α + c 11α − t + ... (67) 11!α We can formulate the ES in Equation (67) as ⎛ ⎞ ∞ ∞ kα 24 r −t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = + e + (−1) α , 1r 25 25 k!α j=1 k=2j ⎛ ⎞ ∞ ∞ kα 101 r −t (−1) t j−1 2j−1 ˆ ⎝ ⎠ i (t) = − e + (−1) α . (68) 2r 100 100 k!α j=1 k=2j FUZZY INFORMATION AND ENGINEERING 17 Figure 2. The surface graph of the α(2)-FCD-10th approximate R-PSS and ES for Equations (61) and (62) at different values of α, where the lower surface represents i (t), where as the upper surface represents 1r r r r ˆ ˆ ˆ ˆ i (t):(a) α(2)-FCD of [i (t)] at α = 0.6, (b) α(2)-FCD of [i (t)] at α = 0.8, (c) (2)-FCD of [i (t)] at 2r 10 10 10 α = 1, (d) (2)-FCD of [i(t)] (exact) at α = 1. Thus, the ES of the system (61) and (62) when α = 1 has the following expression: 24 1 1 1 −t −t i (t) = + r e + (sin(t) − cos(t)) + e , 1r 25 25 2 2 101 1 1 1 −t −t i (t) = − r e + (sin(t) − cos(t)) + e . (69) 2r 100 100 2 2 Figure 2 shows the graphical results for the approximate R-PSSs and ESs of Equations (61) and (62) at different values of α corresponding to α(2)-FCD. Example 4.2: Given the following FC-FDE: α α α ˆ ˆ T u, t ∈ [0, 1], 0 <α ≤ 1, (70) i(t) = 2t i(t) + t subject to the F-IC: i(0) = u (71) where u = max(0, 1 −|ρ|), ρ ∈ R. Case 1: The system of the ODEs corresponding to α(1)-FCD is: α α α ˆ ˆ T i (t) = 2t i (t) + t (r − 1), 1r 1r 18 Z. AL-ZHOUR ET AL. α α α ˆ ˆ T i (t) = 2t i (t) + t (1 − r), (72) 2r 2r subject to the F-ICs: ˆ ˆ i (0) = r − 1, i (0) = 1 − r. (73) 1r 2r Assume the F-PS representation of the solution to the Equations (72) and (73) is: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (74) 1r 2r k k k=0 k=0 Then the αmth-order DEs corresponding to Equation (72) are: ∞ ∞ m+1 m α k!b α (k + 1)!b k k (k−m−1)α (k−m+1)α t − 2 t = χ , 1m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 ∞ ∞ m+1 m α k!c α (k + 1)!c k k (k−m−1)α (k−m+1)α t − 2 t = χ (75) 2m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 α α where χ = b t , χ = c t , χ = αb , χ = αc ,and χ (t) = χ (t) = 0, m = 2, 3, .... 10 0 20 0 11 0 21 0 1m 2m Therefore, by substitute t = 0 into Equation (75), the recurrence relations which determine the values of the coefficients b and c are: k k 3b 2b 0 m−1 b = r − 1, b = 0, b = , b = , m = 2, 3, 4, ... , 0 1 2 m+1 2α α(m + 1) 3c 2c 0 m−1 c = 1 − r, c = 0, c = , c = , m = 2, 3, 4, ... (76) 0 1 2 m+1 2α α(m + 1) If we substitute these values in Equation (74), then the F-PSS of Equations (72) and (73) has the general form: 2α 4α 6α 8α 10α 3 2 t t t t t i (t) = b + + + + + + ... , 1r 0 2 3 4 5 2 3 (1!)α (2!)α (3!)α (4!)α (5!)α 2α 4α 6α 8α 10α 3 2 t t t t t i (t) = c + + + + + + ... . (77) 2r 0 2 3 4 5 2 3 (1!)α (2!)α (3!)α (4!)α (5!)α The closed form of the solution in Equation (77) is 2αk 2α 3 t 1 3 1 ˆ α i (t) = b − = (r − 1) e − , 1r 0 2 (k!)α 3 2 3 k=0 2αk 2α 3 t 1 3 t 1 ˆ α i (t) = c − = (1 − r) e − . (78) 2r 0 2 (k!)α 3 2 3 k=0 FUZZY INFORMATION AND ENGINEERING 19 Thus, the ES to the Equations (70) and (71) in the sense of α(1)-FCD, can be expressed as follows: 2α 1 t ˆ α i(t) = 3e − 1 u. (79) Indeed, if α = 1, the solution of Equations (70) and (71) in this case becomes: 1 2 i(t) = (3e − 1)u, (80) which is corresponding to the solution obtained by the homotopy analysis method [33]. Case 2: The system of the ODEs corresponding to α(2)-FCD is: α α α ˆ ˆ T i (t) = 2t i (t) + t (r − 1), 1r 2r α α α ˆ ˆ T i (t) = 2t i(t) + t (1 − r), (81) 2r subject to F-ICs: ˆ ˆ i (0) = r − 1, i (0) = 1 − r. (82) 1r 2r The F-PSS of Equations (81) and (82) has the following form: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (83) 1r k 2r k k=0 k=0 Then the αmth-order DEs corresponding to Equation (81) are: ∞ ∞ m+1 m α k!b α (k + 1)!c k k (k−m−1)α (k−m+1)α t − 2 t = χ , 1m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 ∞ ∞ m+1 m α k!c α (k + 1)!b k k (k−m−1)α (k−m+1)α t − 2 t = χ (84) 2m (k − m − 1)! (k − m + 1)! k=m+1 k=m−1 α α where χ = b t , χ = c t , χ = αb , χ = αc ,and χ (t) = χ (t) = 0, m = 2, 3, .... 10 0 20 0 11 0 21 0 1m 2m Therefore, the recurrence relations which determine the values of the coefficients b and c k k are: c 2c 0 m−1 b = r − 1, b = 0, b = , b = , m = 2, 3, 4, ... , 0 1 2 m+1 2α α(m + 1) b 2b 0 m−1 c = 1 − r, c = 0, c = , c = , m = 2, 3, 4, ... . (85) 0 1 2 m+1 2α α(m + 1) Substitute these values of the coefficients into Equation (83), then the F-PSS of Equations (81) and (82) has the general form: 2α 4α 6α 8α 10α b t t t t t i (t) = 2 − + − + − + ... , 1r 2 3 4 5 2 (1!)α (2!)α (3!)α (4!)α (5!)α 2α 4α 6α 8α 10α c t t t t t i (t) = 2 − + − + − + ... . (86) 2r 2 3 4 5 2 (1!)α (2!)α (3!)α (4!)α (5!)α 20 Z. AL-ZHOUR ET AL. The closed form of the solution in Equation (86) is: 2αk 2α b t r − 1 −t i (t) = 1 + = 1 + e , 1r 2 2 (k!)α k=0 2αk 2α −t c t 1 − r ˆ α i (t) = 1 + = 1 + e . (87) 2r 2 (k!)α 2 k=0 Thus, the ES to the Equations (70) and (71) in the sense of α(2)-FCD, can be expressed as follows: 2α 1 −t i(t) = 1 + e u. (88) ˆ ˆ Figure 3 shows the graphical results for α(1)-FCD and α(2)-FCD ES set [i (t), i (t)] and its 1r 2r ( ( α α ˆ ˆ derivative [T i t), T i t)] of Equations (70) and (71) when r = 0.5 and at different values 1r 2r of α. In the next example, we apply the R-PS method to construct the PSS to a fractional con- formable fuzzy integro-differential equation (FCFI-DE). The same procedure used in the previous examples will be used with a few minor differences. ( ( α α ˆ ˆ ˆ ˆ Figure 3. The α(1)-FCD and α(2)-FCD ES set [i (t), i (t)] and its derivative [T i t), T i t)]for 1r 2r 1r 2r Equations (70) and (71) when r = 0.5 at α = 0.6 (Solid),α = 0.8 (Dash-dotted), α = 1 (Dotted),: (a) ( ( α α ˆ ˆ ˆ ˆ ˆ ˆ α(2)-FCD of [i (t), i (t)], (b) α(2)-FCD of [i (t), i (t)], (c)α(1)-FC of [T i t), T i t)], (d)α(2)-FCD 1r 2r 1r 2r 1r 2r ( ( α α ˆ ˆ of[T i t), T i t)]. 1r 2r FUZZY INFORMATION AND ENGINEERING 21 Example 4.3: Consider the inductance–resistance–capacitance, RLC-series circuit as in diagram below: Then the FCFI-DE is given by: R 1 α α−1 ˆ ˆ ˆ T i(t) =− i(t) − τ i (τ )dτ + V(t),τ< t ∈ [0, 1], 0 <α ≤ 1, (89) 2r L LC subject to the F-IC: i(0) = v, (90) where R, L,and C are the resistance, inductance corresponding to the solenoid and capaci- tance, respectively, and V(t) = sin(t ) represents the voltage function in the electric circuit.. Finally, 25ρ − 24, 0.96 ≤ ρ ≤ 1 v(ρ) = 101 − 100ρ,1 ≤ ρ ≤ 1.01 . 0, otherwise For simplicity’s sake and for numerical analysis, we assume that R = L = C = 1. Similar to Example 4.1 above, if we assume r = 25ρ − 24, firstly, and r = 101 − 100ρ, secondly, then t t t r 24 r 101 r α−1 α−1 α−1 ˆ ˆ ˆ we have [v] = + , − and τ i(τ )dτ = τ i (τ )dτ , τ i (τ )dτ . 1r 2r 25 25 100 100 0 0 0 According to Nguyen’s theorem and Zadeh’s extension principle [3,53–55], we transfer the IVP (89) and (90) to ODEs systems that we exhibit in the following two cases: Case 1: The system of the FCFI-DEs corresponding to α(1)-FCDis: α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ) 1r 2r 2r α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ), (91) 2r 1r 1r subject to the F-ICs: 24 r 101 r ˆ ˆ i (0) = + , i (0) = − . (92) 1r 2r 25 25 100 100 TheESinthiscaseat α = 1 is: √   √ 1 5 1 5 1 3 − t + t − t 2 2 2 2 i (t) = γ (r)e + γ (r)e + e γ (r)cos t 1r 1 2 3 + γ (r)sin t + sin(t), 2 22 Z. AL-ZHOUR ET AL. √   √ 1 5 1 5 − t + t − t 2 2 2 2 ˆ 2 i (t) =−γ (r)e − γ (r)e + e γ (r)cos t 2r 1 2 3 + γ (r)sin t + sin(t), (93) where √ √ 5 − 5 5 + 5 ˆ ˆ ˆ ˆ γ (r) = (i (0) − i (0)), γ (r) = (i (0) − i (0)), 1 1r 2r 2 1r 2r 20 20 1 − 3 ˆ ˆ ˆ ˆ γ (r) = (i (0) + i (0)), γ (r) = (i (0) + i (0) + 4). 3 1r 2r 4 1r 2r 2 6 Assume the F-PS representations of the solution for the Equations (91) and (92) are: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t . (94) 1r k 2r k k=0 k=0 α α−1 Apply the C-FD T and τ (F )dτ, respectively, on Equation (94), we obtain: ∞ ∞ α (k−1)α α (k−1)α ˆ ˆ T i (t) = kαb t , T i (t) = kαc t , 1r 2r k k k=1 k=1 ∞ ∞ t (k+1)α t (k+1)α t t α−1 α−1 ˆ ˆ τ i (τ )dτ = b , τ i (τ )dτ = c . (95) 1r 2r k k (k + 1)α (k + 1)α 0 0 k=0 k=0 Then, the αmth-order DEs corresponding to Equation (91) is: ∞ ∞ ∞ (k+1)α 101 r t mα (k−1)α kα α T (kαb t ) + − + (c t ) + c − sin(t ) = 0, k k k 100 100 (k + 1)α k=1 k=1 k=0 ∞ ∞ ∞ (k+1)α 24 r t mα (k−1)α kα α T (kαc t ) + + + (b t ) + b − sin(t ) = 0, (96) k k k 25 25 (k + 1)α k=1 k=1 k=0 for m = 0, 1, 2, ... . mα Running out the operator T on Equation (96) gives a discretised form for the equa- tions: ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!b α k!c c α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!c α k!b b α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 (97) where χ (t) is defined as in Equation (55). The recurrence relations which determine the values of the coefficients b and c are: k k m m−1 24 r c χ (0) − α m!c − α (m − 1)!c 0 m m m−1 b = + , b =− , b = , m = 1, 2, 3, ... , 0 1 m+1 m+1 25 25 α α (m + 1)! FUZZY INFORMATION AND ENGINEERING 23 m m−1 101 r b χ (0) − α m!b − α (m − 1)!b 0 m m m−1 c = − , c =− , c = , m = 1, 2, 3, ... . 0 1 m+1 m+1 100 100 α α (m + 1)! (98) So, the ES of Equations (91) and (92) has the general form: −c α + b − c −α + 2b − c −α + 2b − 3c 0 0 0 0 0 0 0 α 2α 3α 4α i (t) = b + t + t + t + t 1r 0 2 3 4 α 2α 6α 24α 3 5 3 5 α + α + 4b − 4c −α + α + 7b − 6c −α − α + 10b − 11c 0 0 0 0 0 0 5α 6α 7α + t + t + t 5 6 7 120α 720α 5040α 3 7 5 7 α + α − α + 17b − 17c −α + α + α + 28b − 27c 0 0 0 0 8α 9α + t + t 8 9 40320α 362880α 3 5 9 3 7 9 −α − α + α + 44b − 45c α + α − α − α + 72b − 72c 0 0 0 0 10α 11α + t + t + ... , 10 11 3628800α 39916800α −b α + c − b −α + 2c − b −α + 2c − 3b 0 0 0 0 0 0 0 α 2α 3α 4α i (t) = c + t + t + t + t 2r 0 2 3 4 α 2α 6α 24α 3 5 3 5 α + α + 4c − 4b −α + α + 7c − 6b −α − α + 10c − 11b 0 0 0 0 0 0 5α 6α 7α + t + t + t 5 6 7 120α 720α 5040α 3 7 5 7 α + α − α + 17c − 17b −α + α + α + 28c − 27b 0 0 0 0 8α 9α + t + t 8 9 40320α 362880α 3 5 9 3 7 9 −α − α + α + 44c − 45b α + α − α − α + 72c − 72b 0 0 0 0 10α 11α + t + t + ... . 10 11 3628800α 39916800α (99) Case 2: The system of the FCFI-DEs corresponding to α(2)-FCD is: α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ), 1r 1r 1r α α−1 α ˆ ˆ ˆ T i (t) =−i (t) − τ i (τ )dτ + sin(t ), (100) 2r 2r 2r subject to the F-ICs: 24 1 101 1 ˆ ˆ i (0) = + r, i (0) = − r. (101) 1r 2r 25 25 100 100 TheESinthiscaseat α = 1 is: √ √ 1 3 1 3 2 i (0) 1r − t − t ˆ ˆ 2 2 i (t) = sin(t) + i (0)e cos t + e sin t − √ − √ , 1r 1r 2 2 3 3 √ √ 1 3 1 3 2 i (0) 2r − t − t ˆ ˆ 2 2 i (t) = sin(t) + (i (0))e cos t + e sin t − √ − √ . (102) 2r 2r 2 2 3 3 24 Z. AL-ZHOUR ET AL. Assume the F-PS representation of the solutions for the Equations (99) and (100) are: ∞ ∞ kα kα ˆ ˆ i (t) = b t , i (t) = c t , (103) 1r 2r k k k=0 k=0 Then the αmth-order DEs corresponding to Equation (100) is: ∞ ∞ ∞ (k+1)α 101 r t mα (k−1)α kα α T (kαb t ) + − + (b t ) + b − sin(t ) = 0, k k k 100 100 (k + 1)α k=1 k=1 k=0 m = 0, 1, 2, ... , ∞ ∞ ∞ (k+1)α 24 r t mα (k−1)α kα α T (kαc t ) + + + (c t ) + c − sin(t ) = 0, k k k 25 25 (k + 1)α k=1 k=1 k=0 m = 0, 1, 2, ... . (104) mα Running out the operator T on Equation (104) gives a discretised form for the equations: ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!b α k!b b α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 ∞ ∞ ∞ m+1 m m−1 (k−m+1)α α k!c α k!c c α k!t k k k (k−m−1)α (k−m)α t + t + = χ (t), (k − m − 1)! (k − m)! (k − m + 1)! k=m+1 k=m k=m−1 (105) where χ (t) is defined as in Equation (55). The recurrence relations which determine the values of the coefficients b and c are: k k m m−1 24 r b χ (0) − α m!b − α (m − 1)!b 0 m m m−1 b = + , b =− , b = , m = 1, 2, 3, ... , 0 1 m+1 m+1 25 25 α α (m + 1)! m m−1 101 r c χ (0) − α m!c − α (m − 1)!c 0 m m m−1 c = − , c =− , c = , m = 1, 2, 3, ... . 0 1 m+1 m+1 100 100 α α (m + 1)! (106) So, the ES of Equation (100) and (101) has the general form: 3 3 −b 1 −α + b −α − b α + α 0 0 0 α 2α 3α 4α 5α i (t) = b + t + t + t + t + t 1r 0 3 4 5 α 2!α 3!α 4!α 5!α 5 3 5 3 7 5 7 −α + α + b −α − α − b α + α − α −α + α + α + b 0 0 0 6α 7α 8α 9α + t + t + t + t 6 7 8 9 6!α 7!α 8!α 9!α 3 5 9 3 7 9 −α − α + α − b α + α − α − α 10α 11α + t + t + ... , 10 11 10!α 11!α 3 3 5 −c 1 −α + c −α − c α + α −α + α + c 0 0 0 0 α 2α 3α 4α 5α 6α i (t) = c + t + t + t + t + t + t 2r 0 3 4 5 6 α 2!α 3!α 4!α 5!α 6!α 3 5 3 7 5 7 −α − α − c α + α − α −α + α + α + c 0 0 7α 8α 9α + t + t + t 7 8 9 7!α 8!α 9!α 3 5 9 3 7 9 −α − α + α − c α + α − α − α 10α 11α + t + t + ... . (107) 10 11 10!α 11!α FUZZY INFORMATION AND ENGINEERING 25 Our goal here is to illustrate some numerical results of the R-PSSs of Equations (89) and (90) to show the validity and efficiency of the proposed method. Anyhow, Tables 1 and 2 show the exact error of α(1)-FCD and α(2)-FCD 10th-approximate R-PSSs, respectively, of Equations (89) and (90) at α = 1and variousvaluesof r and t. While Tables 3 and 4 show the relative error of α(1)-FCD and α(2)-FCD 10th-approximate R-PSSs, respectively, of Equations (89) and (90) at different values of α, r and t. The exact and relative errors are defined, respectively, as follows: ˆ ˆ ˆ ˆ Exact error: = [Ext(t)] = [|i (t) − (i (t)) |, |i (t) − (i (t)) |], 1r 1r 10 2r 2r 10 ˆ ˆ ˆ ˆ Relative error := [Rel(t)] = [|(i (t)) − (i (t)) |, |(i (t)) − (i (t)) |]. (108) 1r 1r 2r 2r 11 10 11 10 Example 4.3 was discussed in [37] by the reproducing kernel Hilbert space (RKHS) method. Tables 5 and 6 show the exact error of α(1)-FCD and α(2)-FCD solutions of Equations (89) and (90) at α = 1and variousvaluesof r and t that were obtained by the RKHS and R-PS methods. The data shows that there is a slight improvement in error when using the R-PS method. Table 1. The exact error of the α(1)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at α = 1 and various values of r and t. r r r r r [Ext(0.25)] [Ext(0.5)] [Ext(0.75)] [Ext(1)] −16 −12 −10 −8 0 [6.66, 8.88] × 10 [2.68, 3.64] × 10 [3.62, 4.86] × 10 [1.19, 1.57] × 10 −16 −12 −10 −8 0.25 [3.33, 6.66] × 10 [1.88, 2.85] × 10 [2.56, 3.80] × 10 [0.84, 1.23] × 10 −16 −12 −10 −9 0.50 [2.22, 4.44] × 10 [1.09, 2.06] × 10 [1.50, 2.74] × 10 [5.01, 8.85] × 10 −16 −12 −10 −9 0.75 [0.00, 2.22] × 10 [0.30, 1.28] × 10 [0.43, 1.68] × 10 [1.53, 5.39] × 10 −13 −11 −9 1 [0, 0] [4.90, 4.90] × 10 [6.25, 6.25] × 10 [1.93, 1.93] × 10 Table 2. The exact error of the α(2)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at α = 1 and various values of r and t. r r r r r [Ext(0.25)] [Ext(0.5)] [Ext(0.75)] [Ext(1)] −16 −13 −11 −9 0 [3.33, 1.11] × 10 [4.71, 4.95] × 10 [6.00, 6.31] × 10 [1.86, 1.95] × 10 −16 −13 −11 −9 0.25 [1.11, 2.22] × 10 [4.76, 4.94] × 10 [6.06, 6.29] × 10 [1.88, 1.95] × 10 −16 −13 −11 −9 0.50 [1.11, 0.00] × 10 [4.83, 4.93] × 10 [6.12, 6.28] × 10 [1.89, 1.94] × 10 −16 −13 −10 −9 0.75 [3.33, 2.22] × 10 [4.85, 4.94] × 10 [6.18, 6.26] × 10 [1.91, 1.94] × 10 −16 −13 −11 −9 1 [1.11, 1.11] × 10 [4.90, 4.90] × 10 [6.25, 6.25] × 10 [1.93, 1.93] × 10 Table 3. The relative error of the α(1)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at various values of α, r and t. α = 0.6 α = 0.8 r r r r r [Rel(0.25)] [Rel(0.75)] [Rel(0.25)] [Rel(0.75)] −9 −5 −12 −7 0 [5.48, 6.38] × 10 [7.56, 1.14] × 10 [6.36, 9.74] × 10 [2.12, 2.22] × 10 −9 −5 −12 −7 0.25 [3.57, 5.22] × 10 [2.40, 1.02] × 10 [4.10, 7.91] × 10 [1.28, 1.87] × 10 −9 −6 −12 −7 0.50 [1.82, 3.98] × 10 [8.33, 8.75] × 10 [1.93, 6.04] × 10 [0.57, 1.49] × 10 −9 −6 −12 −8 0.75 [0.20, 2.67] × 10 [0.73, 6.67] × 10 [0.15, 4.12] × 10 [0.43, 10.6] × 10 −9 −6 −12 −8 1 [1.28, 1.28] × 10 [3.74, 3.74] × 10 [2.16, 2.16] × 10 [5.85, 5.85] × 10 26 Z. AL-ZHOUR ET AL. Table 4. The relative error of the α(2)-FCD 10th-approximate R-PSS of the Equations (89) and (90) at various values of α, r and t. α = 0.6 α = 0.8 r r r r r [Rel(0.25)] [Rel(0.75)] [Rel(0.25)] [Rel(0.75)] −9 −6 −12 −8 0 [1.32, 1.27] × 10 [3.67, 3.75] × 10 [2.24, 2.14] × 10 [5.93, 5.83] × 10 −9 −6 −12 −8 0.25 [1.31, 1.27] × 10 [3.69, 3.75] × 10 [2.22, 2.15] × 10 [5.91, 5.83] × 10 −9 −6 −12 −8 0.50 [1.30, 1.27] × 10 [3.70, 3.74] × 10 [2.20, 2.15] × 10 [5.89, 5.84] × 10 −9 −6 −12 −8 0.75 [1.29, 1.27] × 10 [3.72, 3.74] × 10 [2.18, 2.16] × 10 [5.87, 5.84] × 10 −9 −6 −12 −8 1 [1.28, 1.28] × 10 [3.74, 3.74] × 10 [2.16, 2.16] × 10 [5.85, 5.85] × 10 Table 5. The absolute error of the α(1)-FCD RKHS and approximate R-PSSs of the Equations (89) and (90) at α = 1 and various values of r and t. RKHS RPS r r r r r [Ext(0.25)] [Ext(0.75)] [Ext(0.25)] [Ext(0.75)] −16 −9 −16 −10 0 [5.36, 9.81] × 10 [2.68, 3.64] × 10 [6.66, 8.88] × 10 [3.62, 4.86] × 10 −16 −9 −16 −10 0.25 [4.32, 7.21] × 10 [1.88, 2.85] × 10 [3.33, 6.66] × 10 [2.56, 3.80] × 10 −15 −9 −16 −10 0.50 [3.22, 5.15] × 10 [1.29, 2.06] × 10 [2.22, 4.44] × 10 [1.50, 2.74] × 10 −15 −9 −16 −10 0.75 [3.01, 4.28] × 10 [1.01, 1.28] × 10 [0.00, 2.22] × 10 [0.43, 1.68] × 10 −16 −10 −11 1 [1.12, 4.14] × 10 [4.90, 4.90] × 10 [0, 0] [6.25, 6.25] × 10 Table 6. The absolute error of the α(2)-FCD RKHS and approximate R-PSSs of the Equations (89) and (90) at α = 1 and various values of r and t. RKHS RPS r r r r r [Ext(0.25)] [Ext(0.75)] [Ext(0.25)] [Ext(0.75)] −16 −10 −16 −11 0 [4.52, 7.28] × 10 [4.77, 5.13] × 10 [3.33, 1.11] × 10 [6.00, 6.31] × 10 −16 −10 −16 −11 0.25 [3.23, 5.42] × 10 [4.98, 5.07] × 10 [1.11, 2.22] × 10 [6.06, 6.29] × 10 −15 −10 −16 −11 0.50 [2.42, 3.75] × 10 [5.04, 5.12] × 10 [1.11, 0.00] × 10 [6.12, 6.28] × 10 −15 −10 −16 −10 0.75 [1.21, 3.18] × 10 [8.35, 7.45] × 10 [3.33, 2.22] × 10 [6.18, 6.26] × 10 −16 −11 −16 −11 1 [1.98, 4.29] × 10 [7.90, 8.08] × 10 [3.33, 1.11] × 10 [6.00, 6.31] × 10 5. Conclusions and Discussions In recent years, FDs have been used to model some natural phenomena using fuzzy equa- tions. It has been observed that the order of the FD has a close relationship in controlling the shape of the solution so that it expresses the phenomenon more realistically. Our goal in this paper was to use C-FD in the fractional fuzzy equations instead of other previously used FDs, as the Caputo derivative. It was observed from the graphs that the solutions we obtained were smooth and simulating the ordinary derivative. Also, the mathematical calculations in finding the solutions were easier than previous equations in which other types of FDs were used. This is due to the advantages that the C-FD has over the other types. On the other hand, the RPS method was used in finding series solutions of initial problems on FC-FDEs inthesenseofSGα-FCD due to their novelty and ease. Indeed, the choice of this method was successful, as an accurate approximate solution was found in general, and sometimes the ES was obtained as we presented in the first and second examples. The results show that the power series analysis method is a powerful and easy-to-use analytic tool to solve initial problems on FC-FDEs. How to apply this new approach for solving problems related to FC-PDEs still need further research. FUZZY INFORMATION AND ENGINEERING 27 Acknowledgements The authors express their sincere thanks to the referees for careful reading of the manuscript and for valuable helpful suggestions. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on Contributors Dr. Zeyad Al-Zhour received his Ph.D. in applied mathematics from University Putra Malaysia (Malaysia) in 2007. Then he started working at Zarqa University (2007-2010) as an Assistant Pro- fessor, and then at Imam Abdulrahman Bin Faisal University (2010- Present) and was promoted to Associate Professor in 2018. Dr. Al-Zhour supervised several MSc. and Ph.D Thesis, published more than 60 articles, and achieved several projects. Dr. Al-Zhour has received many awards (e.g. Distin- guished Scientist in Applied Mathematics-VIRA -2018) and letters of appreciation during his work from many universities and companies. His research interests are in matrix operators and algebraic systems, fractional-order systems, fuzzy differential equations, numerical methods, mathematical engineering, and mathematical physics. Ahmad El-Ajou earned his Ph.D. degree in mathematics from the university of Jordan (Jordan) in 2009. He then began work at Al-Balqa Applied university in 2011 as assistant professor of applied mathe- matics, in 2015 received the rank of associate professor, and in 2020 received the rank of professor from the same university. His research interests focus on numerical analysis, fractional differential and integral equations, fuzzy differential and integral equations, and simulating and modeling. Dr. Moa’ath N. Oqielat received his Ph.D. in applied mathematics from Queensland University of Tech- nology (Australia) in 2010. He then began work at Al-Balqa Applied University as assistant professor of applied mathematics and promoted to associate professor in 2020. His research interests focus on modelling and simulating, fractional calculus, and numerical analysis. Dr. Oqielat has written over 35 research papers. Dr. Osama N. Oqily received his Ph.D. in applied mathematics from the University of Southern Queens- land (Australia) in 2014. He began work at Jerash University as assistant professor from 2014 to 2017and then he joined to work at Alahliyaa Amman University since 2017 till now. Dr. Oqily promoted to associate professor in 2020. His research interests focus on modelling and simulating, fractional calculus and fuzzy differential equations. Dr. Shadi Salem received his Ph.D. in Physics from Goethe University Frankfurt (Germany) in 2010. He began work at Al-Balqa Applied university as assistant professor in 2011 and got his promotion as associate professor in 2017. Now he is working at Imam Abdulrahman bin Faisal University. Mousa M. Imran received his PhD degree in physics in 2001 from the University of Rajasthan, in India. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Sep 9, 2022

Keywords: Fuzzy fractional operator; fuzzy fractional power series; strongly generalised fuzzy derivative

References