Abstract
Fuzzy Inf. Eng. (2012) 3: 293-312 DOI 10.1007/s12543-012-0117-x ORIGINAL ARTICLE A New Approach to Fuzzy Initial Value Problem by Improved Euler Method Smita Tapaswini· S. Chakraverty Received: 25 November 2011 /Revised: 25 April 2012/ Accepted: 12 July 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract This paper targets to investigate the solution of linear and nonlinear or- dinary diﬀerential equations with fuzzy initial condition. Here, two improved Euler type methods have been proposed in order to obtain numerical solution of the prob- lem. Along with this, an exact methodology is also discussed. The obtained results are depicted in term of plots to show the eﬃciency of the proposed methods. The solutions are compared with the known results and are found that those obtained by the proposed methods are tighter than the results from the existing method. Keywords Fuzzy diﬀerential equation · Fuzzy number · Improved Euler method · Fuzzy initial value problem 1. Introduction Fuzzy diﬀerential equations (FDEs) are rapidly used for modelling problems in sci- ence and engineering. Most of the science and engineering problems require the solution to FDEs. Therefore a fuzzy initial value problem (FIVP) arises. In real life application, it is too complicated to obtain the exact solution to FDEs. Chang and Zadeh [1] ﬁrst introduced the concept of a fuzzy derivative, followed by Dubois and Prade [2], who deﬁned and used the extension principle in their approach. The fuzzy diﬀerential equation and FIVP were studied by Kaleva [3, 4], Seikkala [5] and others. The numerical methods to fuzzy diﬀerential equations were introduced in [6- 11]. Allahviranloo et al [12, 13] introduced the predictor corrector method to FDEs. Javad Shokri [18] applied modiﬁed Euler’s method for ﬁrst order fuzzy diﬀerential equations and it’s an iterative solution. Smita and Chakraverty [19] introduced a new Euler type method to solve fuzzy initial value problem. S. Chakraverty ()· Smita Tapaswini Department of Mathematics, National Institute of Technology, Rourkela, Odisha-769 008, India email: sne chak@yahoo.com 294 Smita Tapaswini· S. Chakraverty (2012) This paper proposes two new methods based on improved Euler type for ﬁnding the approximate solution to the FIVP. Here we have taken all the possible combi- nations of lower and upper bounds of the variable and then solved by the proposed methods. The initial condition has been considered as triangular fuzzy and trape- zoidal fuzzy number. In the following sections, preliminaries is ﬁrst discussed followed by the exact solution method to linear fuzzy initial value problem. Then proposed methods are illustrated by solving several linear and nonlinear examples with initial condition as a triangular and trapezoidal fuzzy number and lastly the conclusion is drawn. 2. Preliminaries We begin this section with deﬁning some deﬁnitions and theorems which are used throughout this paper. 2.1. Triangular Fuzzy Number (TFN) A triangular fuzzy number U is a convex normalized fuzzy set U of the real line R such that i. There exists exactly one x ∈ R withμ (x ) = 1(x is called the mean value of 0 U 0 0 U), where μ is called the membership function of the fuzzy set. ii. μ (x) is piecewise continuous. Any arbitrary triangular fuzzy number can be represented with an ordered pair of functions through r-cut approach, i.e., [u(r) , u(r)], ¯ where r ∈ [0, 1]. This satisﬁes the following requirements: i. u(r) is a bounded left continuous non-decreasing function over [0, 1]. ii. u(r) ¯ is a bounded right continuous non-increasing function over [0, 1]. iii. u(r) ≤ u ¯(r), 0 ≤ r ≤ 1. We can deﬁne triangular fuzzy number A as A = (a , a , a ). The membership 1 2 3 function of this fuzzy number will be interpreted as follows: 0, x ≤ a , ⎪ 1 x− a , a ≤ x ≤ a , ⎪ 1 2 a − a 2 1 μ = U ⎪ ⎪ a − x ⎪ , a ≤ x ≤ a , 2 3 a − a 3 2 0, x ≥ a . 2.2. Trapezoidal Fuzzy Number (TrFN) Fuzzy Inf. Eng. (2012) 3: 293-312 295 We can deﬁne trapezoidal fuzzy number A as A = (a , a , a , a ). The membership 1 2 3 4 function of this fuzzy number will be interpreted as follows: 0, x ≤ a , x− a , a ≤ x ≤ a , ⎪ 1 2 a − a ⎪ 2 1 μ = 1, a ≤ x ≤ a , U ⎪ 2 3 ⎪ a − x ⎪ , a ≤ x ≤ a , 3 4 ⎪ a − a 4 3 0, x ≥ a . Though an r-cut approach trapezoidal fuzzy number can be represented as an or- dered pair, i.e., [(a − a )r+ a ,−(a − a )r+ a ], where r ∈ [0, 1]. When a = a , the 2 1 1 4 3 4 2 3 trapezoidal fuzzy number coincides with TFN. So, the fuzzy arithematic for trape- zoidal fuzzy number is the same as that of triangular fuzzy numbers. 2.3. Fuzzy Arithmetic Following [14] for any arbitrary fuzzy number x = [¯ u(r)], y = [¯ u(r)] and scalar k,we have the fuzzy arithmetic as i. x = y if and only if x(r) = y(r) and x(r) = y(r). ii. x+ y = (x(r)+ y(r), x(r)+ y(r)). ⎢ ⎥ ⎢ min x(r)× y(r), x(r)× y(r), x(r)× y(r), x(r)× y(r) ,⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ iii. x× y = . ⎢ ⎥ ⎣ ⎦ max x(r)× y(r), x(r)× y(r), x(r)× y(r), x(r)× y(r) kx, kx , k < 0, iv. kx = kx, kx , k ≥ 0. Lemma 1 [15] IF u(t) = (x(t), y(t), z(t)) is a triangular number of valued function and if u is Hukuhara diﬀerentiable, then u = (x , y , z ), by using this property if we intend to solve the FIVP x = f (t, x), (1) x(t ) = x , 0 0 1 1 1 with x = (x , x , x¯ ) ∈ R, x(t) = (u, u , u ¯) ∈ R and f :[t , t +a]×R −→ R, f (t, (u, u , u ¯)) 0 0 0 0 0 0 ¯ ¯ 1 1 1 1 = (f(t, (u, u , u ¯))), f (t, (u, u , u ¯)), f (t, (u, u , u ¯)). We can translate it into the system of ¯ ¯ ¯ ordinary diﬀerential equations ⎪ u = f (t, u, u , u), 1 1 u = f (t, u, u , u), (2) ⎪ u = f (t, u, u , u), 1 1 u(0) = x , u (0) = x , u(0) = x . 0 0 Theorem 1 [15] Let us consider the FIVP (2) with x = (x , x , x¯ ) and f :[t , t + 0 0 0 0 0 0 a] × R −→ R, f (t, (u, u , u ¯)) = (f(t, (u, u , u ¯))), f (t, (u, u , u ¯)), f (t, (u, u , u ¯)) such that ¯ ¯ ¯ ¯ ¯ 296 Smita Tapaswini· S. Chakraverty (2012) f, f , f are Lipschitz continuous (real-valued) functions. Then the solution to (1) is triangular-valued function x(t) = (u(t), u (t), u ¯(t)):[t , t + a] × R and the Problem 0 0 (1) is equivalent to the Problem (2). 3. A Fuzzy Cauchy Problem Consider the ﬁrst-order fuzzy diﬀerential equation x = f (t, x), where x be a fuzzy function of t, f (t, x) denotes a fuzzy function of crisp variable t and fuzzy variable x, and x be a Hukahara or Seikkala fuzzy derivative of x. If an initial value x(t ) = x 0 0 is given, then a fuzzy cauchy problem of ﬁrst-order will be obtained as follows: x (t) = f (t, x(t)), t ≤ t ≤ T, x(t ) = x . (3) 0 0 0 Suﬃcient conditions for the existence of a unique solution to Eq. (3) are that f is continuous and that a Lipschitz condition f (t, x)− f (t, y) ≤ Lx− y, L > 0 (4) is fulﬁlled. By Theorem 5.2 in [3], we may replace Eq. (3) by the equivalent system x (t) = f (t, x) = F(t, x, x), x(t ) = x , (5) x (t) = f (t, x) = G(t, x, x), x(t ) = x , 0 0 which possesses a unique solution (x, x) a fuzzy function, i.e., for each t, the pair [x(t; r), x[t; r]] is a fuzzy number. The parametric form of Eq. (5) may be written as x (t; r) = F[t, x(t; r), x(t; r))], x(t ; r) = x (r), (6) x (t; r) = G[t, x(t; r), x(t; r)], x(t ; r) = x (r) 0 0 for r ∈ [0, 1]. A solution to Eq. (6) must solve Eq. (5) as well since by using the sup norm, an equality between two fuzzy numbers yields a pointwise equality. In some cases, the above system (Eq. (6)) can be solved analytically. 4. Improved Euler Method To integrate the system cited in Eq. (6) from t a preﬁxed T > t , we replace the 0 0 interval [t , T ] by a set of discrete equally spaced grid points t < t < t <··· < t = T, (7) 0 1 2 N at which the exact solution (X(t; r), X(t; r)) is approximated by some (x(t; r), x(t; r)). (Note that through out each integration, r is unchanged). The exact and approximate solutions at t , 0 ≤ n ≤ N are denoted by X (r) = (X(t; r), X(t; r)) and x (r) = n n n (x(t; r), x(t; r)), respectively. The grid points at which the solution is calculated are t = t + nh, h = (T − t )/N;1 ≤ n ≤ N. (8) n 0 0 By using the improved Euler method, we obtain: x − x = { f [t , x(t ; r)]+ f [t , x(t ; r)]}, n n n+1 n+1 n+1 n 2 (9) x − x = { f [t , x(t ; r)]+ f [t , x(t ; r)]}. n+1 n n n n+1 n+1 2 Fuzzy Inf. Eng. (2012) 3: 293-312 297 Deﬁne F[x(t ; r)] = { f [t , x(t ; r)]+ f [t , x(t ; r)]}, n n n n+1 n+1 (10) G[x(t ; r)] = { f [t , x(t ; r)]+ f [t , x(t ; r)]}. n n n n+1 n+1 The exact and approximated solution to t , 0 ≤ n ≤ N, are denoted by [X(t ; r)] = n n [X(t ; r), X(t ; r)] and [x(t ; r)] = [x(t ; r), x(t ; r)], respectively. The solution is cal- n n n n n culated by grid points at t . From Eq. (9) and Eq. (10), we have x(t ; r) = x(t ; r)+ F[x(t ; r)], n+1 n n (11) x(t ; r) = x(t ; r)+ G[x(t ; r)]. n+1 n n Also X(t ; r) ≈ X(t ; r)+ F[X(t ; r)], n+1 n n (12) X(t ; r) ≈ X(t ; r)+ G[X(t ; r)]. n+1 n n The following Lemma [11] will be applied to show the convergence of these approx- imates, lim x(t; r) = X(t; r), lim x(t; r) = X(t; r). (13) h→0 h→0 Lemma 2 Let the sequence of numbers{W } satisfy n=0 |W | ≤ A|W |+ B, 0 ≤ n ≤ N − 1 (14) n+1 n for some given positive constants A and B. Then A − 1 |W | ≤ A |W |+ B , 0 ≤ n ≤ N − 1. (15) n 0 A− 1 N N Lemma 3 Let the sequence of numbers{W } ,{V } satisfy n n n=0 n=0 |W | ≤ |W |+ A max{|W |,|V |}+ B, n+1 n n n (16) |V | ≤ |V |+ A max{|V |,|W |}+ B n+1 n n n for some given positive constants A and B, and denote U = |W |+|V |, 0 ≤ n ≤ N. n n n Then n A − 1 U ≤ A U + B , 1 ≤ n ≤ N, (17) n 0 A− 1 where A = 1+ 2A and B = 2B. Proof See[11]. Next result determined the point wise convergence of the improved Euler approxi- mations to the exact solution. Let F[t, u, v] and G[t, u, v] be the functions F and G of Eq. (5), where u and v are constants and u ≤ v. In other words, F(t, u, v) and G(t, u, v) are obtained by substituting x = (u, v) in Eq. (5). F and G are deﬁned in the domain K = (t, u, v)|t ≤ t ≤ T,−∞ < v < ∞, −∞ < u ≤ v. (18) 0 298 Smita Tapaswini· S. Chakraverty (2012) Theorem 2 [18] Let F(u, v) and G(u, v) belong to C (R ) and let the partial deriva- tives of F and G be bounded over R . Then, for arbitrary ﬁxed r, 0 ≤ r ≤ 1, the ap- proximate solutions (11) converge to the exact solutions X(t; r) and X(t; r) uniformly in t. Proof It is suﬃcient to show lim x(t ; r) = X(t ; r), lim x(t ; r) = X(t ; r), N N N N h→0 h→0 where t = T . Let W = X(t ; r) − y(t ; r), V = X(t ; r) − x(t ; r). By using the N n n n n n n Equations (11) and (12), we get W ≤ W +Lh max W , V +Lh[2Lh max W , V +max W , V ]+h M , | | | | {| | | |} {| | | |} {| | | |} n+1 n n n n n n n 1 |V | ≤ |V |+ Lh max{|W |,|V |}+ Lh[2Lh max{|W |,|V |}+ max{|W |,|V |}]+ h M , n+1 n n n n n n n 2 where, M , M are upper bound for A (r), A (r), respectively. Hence, 1 2 1 2 |W | ≤ |W |+ Lh{1+ (1+ 2Lh)} max{|W |,|V |}+ h M, n+1 n n n |V | ≤ |V |+ Lh{1+ (1+ 2Lh)} max{|W |,|V |}+ h M, n+1 n n n where M = max{M , M }, and L > 0 is a bound for the partial derivatives of F and 1 2 G. Therefore from Lemma 3, we obtain 2n (1+ 2Lh) − 1 2n 3 |W | ≤ (1+ 2Lh) |U |+ 2h M , n 0 (1+ 2Lh) − 1 2n (1+ 2Lh) − 1 2n 3 |V | ≤ (1+ 2Lh) |U |+ 2h M , n 0 (1+ 2Lh) − 1 where|U | = |W |+|V |. In particular, 0 0 0 2(T−t ) (1+ 2Lh) − 1 2N 3 |W | ≤ (1+ 2Lh) |U |+ 2h M , N 0 (1+ 2Lh) − 1 2(T−t ) (1+ 2Lh) − 1 2N 3 |V | ≤ (1+ 2Lh) |U |+ 2h M , N 0 (1+ 2Lh) − 1 since W = V = 0, we have 0 0 4L(T−t ) 4L(T−t ) 0 0 e − 1 e − 1 2 2 W ≤ M h , V ≤ M h . | | | | N N 2L(1+ hL) 2L(1+ hL) Thus, if h → 0, we conclude W → 0 and V → 0, which completes the proof. N N 5. Exact Solution Method to FIVP Let us consider the FIVP x (t) = f (t, x), (19) x(t ) = (x , x , x ). 0 0 0 0 Here an initial condition is in term of triangular fuzzy numbers. By using Theorem 1, we get the system of ordinary diﬀerential equations as Eq. (2). Then the above system may be obtained as an Eigen value problem. Finally, using the initial conditions, we get the solution to FIVP. Similarly, the initial condition may also be taken as a Fuzzy Inf. Eng. (2012) 3: 293-312 299 trapezoidal fuzzy number and the corresponding FIVP is written as four equivalent systems of ordinary diﬀerential equation. Those again may be solved by the usual procedures. It is worth mentioning that the sign before the variables are to be used carefully to have the correct set of equivalent systems for both the TFN and TrFN cases. 6. Proposed Methods to FIVP In the following paragraph, two methods are proposed to FIVP. 6.1. Method 1: Max-Min Improved Euler Method Here a fuzzy initial value problem [11] has been considered as x (t) = x(t), x(t ) = (x , x , x ). (20) 0 0 Though an r-cut approach, the triangular fuzzy initial condition can be represented as [(x − x )r+ x , x + (x − x )r], 0 ≤ r ≤ 1. (21) 0 0 0 0 In Section 4, improved Euler method has been given by taking the combination of left and right. In such the present improved Euler type of method has been developed for the present FIVP. Here, we are considering all the possible combination of lower and upper bounds of the variable and by using the improved Euler method, we obtain (1) x (t ; r) = x + { f [t , x(t ; r)]+ f [t , x(t ; r)]}, n+1 n n n+1 n+1 n+1 n 2 (1) x (t ; r) = x + { f [t , x(t ; r)]+ f [t , x(t ; r)]}, n+1 n n n n+1 n+1 n+1 2 (22) (2) x (t ; r) = x + { f [t , x(t ; r)]+ f [t , x(t ; r)]}, n+1 n n n+1 n+1 n+1 n 2 (2) x (t ; r) = x + { f [t , x(t ; r)]+ f [t , x(t ; r)]}. n+1 n n n n+1 n+1 n+1 2 Deﬁne F[x(t ; r)] = { f [t , x(t ; r)]+ f [t , x(t ; r)]}, n n n n+1 n+1 (23) G[x(t ; r)] = { f [t , x(t ; r)]+ f [t , x(t ; r)]}. n n n n+1 n+1 The exact and approximated solution to t ;0 ≤ n ≤ N are denoted by [X(t ; r)] = n n [X(t ; r), X(t ; r)] and [x(t ; r), x(t ; r)], respectively. The solution is calculated by n n n n grid points at t . From Equations (21) and (22), we have (1) x (t ; r) = x(t ; r)+ F[x(t ; r)], n+1 n n (1) x (t ; r) = x(t ; r)+ G[x(t ; r)], n+1 n n (24) (2) x (t ; r) = x(t ; r)+ G[x(t ; r)], n+1 n n (2) x (t ; r) = x(t ; r)+ F[x(t ; r)]. n+1 n n Then for lower and upper value of the independent variable x, we take minimum and maximum from (24). x = min{x(t ; r)+ F[x(t ; r)], x(t ; r)+ G[x(t ; r)]}, n n n n n+1 (25) x = max{x(t ; r)+ G[x(t ; r)], x(t ; r)+ F[x(t ; r)]}, n+1 n n n n 300 Smita Tapaswini· S. Chakraverty (2012) we can write Eq. (25) as (1) (2) x = min{x (t ; r), x (t ; r)}, n+1 n+1 n+1 (26) (1) (2) x = max{x (t ; r), x (t ; r)}. n+1 n+1 n+1 Which are the better approximations to the exact solutions. The eﬃciency and pow- erfulness of the methodology are demonstrated by variety of examples. 6.2. Method 2: Average Improved Euler Method In the second method, similarly average of lower and upper bounds are computed respectively for a fuzzy initial value problem. Then the Eq. (24) reduces to x = {2x(t ; r)+ F[x(t ; r)]+ G[x(t ; r)}, n n n n+1 2 (27) x = {2x(t ; r)+ G[x(t ; r)+ F[x(t ; r)]}. n+1 n n n Also (1) (2) x = [x (t ; r)+ x (t ; r)], n+1 n+1 n+1 2 (28) 1 (1) (2) x = [x (t ; r)+ x (t ; r)]. n+1 n+1 n+1 7. Numerical Examples In this section, we discuss exactly a numerical solution to examples including linear and nonlinear FIVP with triangular and trapezoidal initial conditions by using the proposed method. Example 1 Let us consider a linear triangular FIVP given in [11] as x (t) = x(t), x(0) = (0.75, 1, 1.125), 0 ≤ r ≤ 1. Then using the r-cut approach, the triangular fuzzy initial condition can be repre- sented as x(0) = [0.75+ 0.25r, 1.125− 0.125r], 0 ≤ r ≤ 1. First the exact solution is obtained by using the method discussed in Section 3 for t =1. Next the same is solved by applying the approach of Duraisamy and Usha [16] and the proposed Methods 1 and 2 discussed in Section 4. The results obtained by the above methods are tabulated in Tables 1 to 3 for diﬀerent values of h=0.1, 0.01 and 0.001, respectively. Also, the error obtained by diﬀerent methods are given in Tables 4 and 5. By looking into the results, one may conclude that the solution obtained by Method 1 is the same as that of the exact solution. Example 2 Let us consider now an initial value problem with initial trapezoidal fuzzy number as below x (t) = x(t), x(0) = (0.75, 0.85, 1.1, 1.125), 0 ≤ r ≤ 1. Fuzzy Inf. Eng. (2012) 3: 293-312 301 Table 1: Solutions obtained by exact method, Duraisamy and Usha [16], Methods 1 and 2 for h = 0.1at t = 1. r Exact value Duraisamy and Usha [16] Method 1 Method 2 0 [2.0387, 3.0581] [2.0355, 3.0532] [2.0356, 3.0533] [2.3472, 2.7420] 0.1 [2.1067, 3.0241] [2.1035, 3.0193] [2.1034, 3.0194] [2.3839, 2.7388] 0.2 [2.1746, 2.9901] [2.1712, 2.9853] [2.1713, 2.9855] [2.4206, 2.7361] 0.3 [2.2426, 2.9561] [2.2393, 2.9516] [2.2391, 2.9516] [2.4574, 2.7335] 0.4 [2.3105, 2.9222] [2.3070, 2.9178] [2.3070, 2.9176] [2.4942, 2.7304] 0.5 [2.3785, 2.8882] [2.3750, 2.8837] [2.3748, 2.8837] [2.5310, 2.7282] 0.6 [2.4465, 2.8542] [2.4427, 2.8496] [2.4427, 2.8498] [2.5674, 2.7252] 0.7 [2.5144, 2.8202] [2.5104, 2.8158] [2.5105, 2.8159] [2.6042, 2.7226] 0.8 [2.5824, 2.7862] [2.5784, 2.7818] [2.5784, 2.7819] [2.6408, 2.7196] 0.9 [2.6503, 2.7523] [2.6460, 2.7478] [2.6462, 2.7480] [2.6774, 2.7168] 1 [2.7183, 2.7183] [2.7140, 2.7140] [2.7141, 2.7141] [2.7140, 2.7140] Table 2: Solutions obtained by exact method, Duraisamy and Usha [16], Methods 1 and 2 for h = 0.01 at t = 1. r Exact value Duraisamy and Usha [16] Method 1 Method 2 0 [2.0387, 3.0581] [2.0388, 3.0578] [2.0387, 3.0580] [2.3606, 2.7376] 0.1 [2.1067, 3.0241] [2.1063, 3.0245] [2.1066, 3.0240] [2.3963, 2.7350] 0.2 [2.1746, 2.9901] [2.1748, 2.9902] [2.1746, 2.9901] [2.4315, 2.7327] 0.3 [2.2426, 2.9561] [2.2424, 2.9558] [2.2425, 2.9561] [2.4680, 2.7315] 0.4 [2.3105, 2.9222] [2.3102, 2.9220] [2.3105, 2.9221] [2.5030, 2.7302] 0.5 [2.3785, 2.8882] [2.3784, 2.8889] [2.3785, 2.8881] [2.5392, 2.7274] 0.6 [2.4465, 2.8542] [2.4461, 2.8544] [2.4464, 2.8541] [2.5760, 2.7272] 0.7 [2.5144, 2.8202] [2.5146, 2.8205] [2.5144, 2.8202] [2.6106, 2.7236] 0.8 [2.5824, 2.7862] [2.5815, 2.7857] [2.5823, 2.7862] [2.6467, 2.7221] 0.9 [2.6503, 2.7523] [2.6499, 2.7520] [2.6503, 2.7522] [2.6819, 2.7201] 1 [2.7183, 2.7183] [2.7184, 2.7184] [2.7182, 2.7182] [2.7184, 2.7184] Though r-cut, the initial trapezoidal fuzzy number may be written as x(0) = [0.75+ 0.1r, 1.125− 0.025r], 0 ≤ r ≤ 1. 302 Smita Tapaswini· S. Chakraverty (2012) Table 3: Solutions obtained by exact method, Duraisamy and Usha [16], Methods 1 and 2 for h = 0.001 at t = 1. r Exact value Duraisamy and Usha [16] Method 1 Method 2 0 [2.0387, 3.0581] [2.0369, 3.0566] [2.0387, 3.0581] [2.3590, 2.7360] 0.1 [2.1067, 3.0241] [2.1053, 3.0225] [2.1067, 3.0241] [2.3938, 2.7322] 0.2 [2.1746, 2.9901] [2.1730, 2.9883] [2.1746, 2.9901] [2.4294, 2.7294] 0.3 [2.2426, 2.9561] [2.2409, 2.9543] [2.2426, 2.9561] [2.4676, 2.7286] 0.4 [2.3105, 2.9222] [2.3091, 2.9210] [2.3105, 2.9222] [2.5030, 2.7276] 0.5 [2.3785, 2.8882] [2.3769, 2.8882] [2.3785, 2.8882] [2.5380, 2.7256] 0.6 [2.4465, 2.8542] [2.4442, 2.8553] [2.4465, 2.8542] [2.5742, 2.7218] 0.7 [2.5144, 2.8202] [2.5125, 2.8202] [2.5144, 2.8202] [2.6054, 2.7164] 0.8 [2.5824, 2.7862] [2.5805, 2.7851] [2.5824, 2.7862] [2.6408, 2.7152] 0.9 [2.6503, 2.7523] [2.6481, 2.7502] [2.6503, 2.7523] [2.6776, 2.7150] 1 [2.7183, 2.7183] [2.7163, 2.7163] [2.7183, 2.7183] [2.7163, 2.7163] Table 4: Error for diﬀerent values of r and h for Example 1 of Method 1. h = 0.1 h = 0.01 h = 0.001 x xx xx x 0 0.00315074 0.00472611 0.00003372 0.00005059 0.00000034 0.0000005 0.2 0.00336078 0.00462108 0.00003597 0.00004946 0.00000036 0.0000005 0.4 0.00357083 0.00451606 0.00003822 0.00004834 0.00000038 0.00000049 0.6 0.00378089 0.00441103 0.00004047 0.00004721 0.00000041 0.00000048 0.8 0.00399094 0.00430600 0.05841204 0.00004609 0.00000043 0.00000046 1 0.00420098 0.00420098 0.00415603 0.00004497 0.00000045 0.00000045 In this example, the FIVP with trapezoidal fuzzy initial condition is solved by the proposed Methods 1 and 2 discussed in Section 4. Corresponding results are given in Tables 6 to 8 for diﬀerent values of h=0.1, 0.01 and 0.001, respectively. Errors obtained by utilizing the proposed Methods 1 and 2 and that of Duraisamy and Usha [16] are presented in Tables 9 and 10. It may very well be seen that the solution obtained by Method 1 approximately matches with exact solution. The results obtained by diﬀerent methods for both the Examples 1 and 2 are de- picted in Fig.1. Fuzzy Inf. Eng. (2012) 3: 293-312 303 Table 5: Error for diﬀerent values of r and h for Example 1 of Method 2. h= 0.1 h = 0.01 h = 0.001 x xx xx x 0 0.30865067 0.31652748 0.2119579 0.32128021 0.32208375 0.32208454 0.2 0.24608032 0.25406221 0.33849611 0.25703315 0.25766672 0.31203355 0.4 0.18350999 0.19159687 0.35579639 0.19276278 0.19324995 0.30198262 0.6 0.12093968 0.12913158 0.37309669 0.12853905 0.12883319 0.29193137 0.8 0.05836932 0.06666629 0.39039697 0.06429198 0.06441596 0.28188044 1 0.00420097 0.00420100 0.40769722 0.00004506 0.00000059 0.27182877 Table 6: Solutions obtained by exact method, Duraisamy and Usha [16], Methods 1 and 2 for h = 0.1. at t = 1 r Exact value Duraisamy and Usha [16] Method 1 Method 2 0 [2.0390, 3.0580] [2.0355, 3.0532] [2.0356, 3.0533] [2.3472, 2.7420] 0.1 [2.0662, 3.0512] [2.0626, 3.0466] [2.0627, 3.0466] [2.3642, 2.7450] 0.2 [2.0934, 3.0444] [2.0898, 3.0399] [2.0898, 3.0398] [2.3810, 2.7488] 0.3 [2.1205, 3.0376] [2.1168, 3.0329] [2.1170, 3.0330] [2.3976, 2.7524] 0.4 [2.1477, 3.0308] [2.1443, 3.0266] [2.1441, 3.0262] [2.4145, 2.7563] 0.5 [2.1749, 3.0240] [2.1712, 3.0193] [2.1713, 3.0194] [2.4310, 2.7594] 0.6 [2.2021, 3.0172] [2.1985, 3.0129] [2.1984, 3.0126] [2.4479, 2.7632] 0.7 [2.2293, 3.0104] [2.2255, 3.0059] [2.2255, 3.0058] [2.4647, 2.7670] 0.8 [2.2564, 3.0036] [2.2528, 2.9990] [2.2527, 2.9991] [2.4811, 2.7703] 0.9 [2.2836, 2.9968] [2.2799, 2.9923] [2.2798, 2.9923] [2.4984, 2.7742] 1 [2.3108, 2.9900] [2.3070, 2.9853] [2.3070, 2.9855] [2.5147, 2.7775] Example 3 Consider the nonlinear fuzzy initial value problem as given in [11] −x (t) x (t) = e , x(0) = (0.75, 1, 1.5). Using r-cut triangular fuzzy initial condition becomes x(0) = [0.75+ 0.25r, 1.5− 0.5r], 0 ≤ r ≤ 1. The above nonlinear FIVP (with TFN) is solved by the proposed Methods 1 and 2. Since the exact solution cannot be computed analytically, we obtained improved 304 Smita Tapaswini· S. Chakraverty (2012) Table 7: Solutions obtained by exact method, Duraisamy and Usha [16], Methods 1 and 2 for h = 0.01 at t = 1. r Exact value Duraisamy and Usha [16] Method 1 Method 2 0 [2.0390, 3.0580] [2.0388, 3.0578] [2.0387, 3.0580] [2.3606, 2.7376] 0.1 [2.0662, 3.0512] [2.0655, 3.0512] [2.0659, 3.0512] [2.3760, 2.7410] 0.2 [2.0934, 3.0444] [2.0926, 3.0443] [2.0930, 3.0444] [2.3938, 2.7456] 0.3 [2.1205, 3.0376] [2.1209, 3.0380] [2.1202, 3.0376] [2.4092, 2.7473] 0.4 [2.1477, 3.0308] [2.1474, 3.0300] [2.1474, 3.0308] [2.4256, 2.7522] 0.5 [2.1749, 3.0240] [2.1748, 3.0245] [2.1746, 3.0240] [2.4438, 2.7581] 0.6 [2.2021, 3.0172] [2.2017, 3.0173] [2.2018, 3.0172] [2.4586, 2.7604] 0.7 [2.2293, 3.0104] [2.2296, 3.0107] [2.2290, 3.0104] [2.4770, 2.7662] 0.8 [2.2564, 3.0036] [2.2561, 3.0031] [2.2561, 3.0037] [2.4924, 2.7683] 0.9 [2.2836, 2.9968] [2.2829, 2.9970] [2.2853, 2.9969] [2.5082, 2.7727] 1 [2.3108, 2.9900] [2.3102, 2.9902] [2.3105, 2.9901] [2.5256, 2.7776] Table 8: Solutions obtained by exact method, Duraisamy and Usha [16], Methods 1 and 2 for h = 0.001 at t = 1. r Exact value Duraisamy and Usha [16] Method 1 Method 2 0 [2.0390, 3.0580] [2.0369, 3.0566] [2.0387, 3.0581] [2.3590, 2.7360] 0.1 [2.0662, 3.0512] [2.0643, 3.0498] [2.0659, 3.0513] [2.3752, 2.7390] 0.2 [2.0934, 3.0444] [2.0919, 3.0427] [2.0931, 3.0445] [2.3914, 2.7426] 0.3 [2.1205, 3.0376] [2.1188, 3.0361] [2.1203, 3.0377] [2.4072, 2.7452] 0.4 [2.1477, 3.0308] [2.1454, 3.0293] [2.1474, 3.0309] [2.4234, 2.7488] 0.5 [2.1749, 3.0240] [2.1730, 3.0225] [2.1746, 3.0241] [2.4392, 2.7514] 0.6 [2.2021, 3.0172] [2.2002, 3.0156] [2.2018, 3.0173] [2.4564, 2.7556] 0.7 [2.2293, 3.0104] [2.2274, 3.0086] [2.2290, 3.0105] [2.4728, 2.7594] 0.8 [2.2564, 3.0036] [2.2543, 3.0020] [2.2562, 3.0037] [2.4902, 2.7642] 0.9 [2.2836, 2.9968] [2.2819, 2.9950] [2.2834, 2.9969] [2.5074, 2.7678] 1 [2.3108, 2.9900] [2.3091, 2.9883] [2.3105, 2.9901] [2.5244, 2.7720] Euler Methods 1 and 2 approximation for t=1. Besides results are included in Tables 11, 12 and 13 for h=0.1, 0.01 and 0.001, respectively. Corresponding plots of [16], Methods 1 and 2 are given in Fig.2 (a) to (c) respectively. Fuzzy Inf. Eng. (2012) 3: 293-312 305 Table 9: Error for diﬀerent values of r and h for Example 2 of Method 1. h= 0.1 h = 0.01 h = 0.001 x xx xx x 0 0.00315074 0.00472611 0.00003372 0.00005059 0.00000034 0.00000051 0.2 0.00323476 0.0047051 0.00003463 0.00005036 0.00000035 0.00000051 0.4 0.00331877 0.0046841 0.00003552 0.00005014 0.00000035 0.00000051 0.6 0.00340279 0.00466309 0.00003642 0.00004991 0.00000037 0.0000005 0.8 0.00348682 0.00464208 0.00003732 0.00004969 0.00000038 0.0000005 1 0.00357083 0.00462108 0.00003822 0.00004946 0.00000038 0.0000005 Table 10: Error for diﬀerent values of r and h for Method 2 of Example 2. h= 0.1 h = 0.01 h = 0.001 x xx xx x 0 0.30865067 0.31652748 0.32119579 0.32128021 0.32208375 0.32208454 0.2 0.28777987 0.2957173 0.29977971 0.29986465 0.30061145 0.30061237 0.4 0.26690908 0.27491197 0.27836336 0.2784492 0.27913909 0.27913993 0.6 0.24603833 0.25410419 0.25694724 0.25703359 0.25766672 0.2566787 0.8 0.22516754 0.2332964 0.23553106 0.23561798 0.23619439 0.23619545 1 0.20429676 0.21248869 0.21411487 0.21420243 0.18182876 0.22260404 Example 4 Consider the nonlinear fuzzy initial value problem [11] −x (t) x (t) = e , x(0) = (0.75, 0.85, 1.3, 1.5). Though r-cut, the initial trapezoidal fuzzy number may be written as x(0) = [0.75+ 0.1r, 1.5− 0.2r], 0 ≤ r ≤ 1. The obtained results are shown in Tables 14, 15 and 16 with h=0.1, 0.01 and 0.001 us- ing the proposed Methods 1 and 2, respectively. Results of Duraisamy and Usha [16] are also incorporated in Tables 14 to 16. From this, one may note that the results ob- tained by Method 1 are the same with Duraisamy and Usha [16]. The corresponding plots of Duraisamy and Usha, Methods 1 and 2 are given in Fig.2 (d) to (f) respec- tively. 306 Smita Tapaswini· S. Chakraverty (2012) 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2 2.5 3 3.5 2 2.5 3 3.5 (a) For h = 0.1 of Example 1 (b) For h = 0.01 of Example 1 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2 2.5 3 3.5 2 2.5 3 3.5 (c) For h = 0.001 of Example 1 (d) For h = 0.1 of Example 2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2 2.5 3 3.5 2 2.5 3 3.5 (e) For h = 0.01 of Example 2 (f) For h = 0.001 of Example 2 Fig. 1 Comparison plots of Methods 1, 2, Duraisamy [16] and the exact solution of Examples 1 and 2, dash curve: Method 1; solid curve: Method 2; square curve: Duraisamy; circle curve: exact solution Fuzzy Inf. Eng. (2012) 3: 293-312 307 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2 2.5 3 3.5 4 4.5 2 2.5 3 3.5 4 4.5 (g) For h = 0.1 of Example 3 (h) For h = 0.01 of Example 3 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 2 2.5 3 3.5 4 4.5 2 2.5 3 3.5 4 4.5 (j) For h = 0.1 of Example 4 (i) For h = 0.001 of Example 3 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 2 2.5 3 3.5 4 4.5 2 2.5 3 3.5 4 4.5 (k) For h = 0.01 of Example 4 (l) For h = 0.001 of Example 4 Fig. 2 Comparison plots of Methods 1, 2 and the solution of Duraisamy [16] of Examples 3 and 4, square curve: Duraisamy; dash curve: Method 1; solid curve: Method 2 308 Smita Tapaswini· S. Chakraverty (2012) Table 11: Solutions obtained by Duraisamy and Usha [16], Methods 1 and 2 for h = 0.1at t = 1. r Duraisamy and Usha [16] Method 1 Method 2 0 [2.0355, 4.0710] [2.0356, 4 .0711] [2.6588, 3.4476] 0.1 [2.1035, 3.9351] [2.1034, 3.9354] [2.6646, 3.3743] 0.2 [2.1712, 3.7997] [2.1713, 3.7997] [2.6702, 3.3008] 0.3 [2.2393, 3.6640] [2.2391, 3.6640] [2.6757, 3.2278] 0.4 [2.3070, 3.5284] [2.3070, 3.5283] [2.6812, 3.1541] 0.5 [2.3750, 3.3924] [2.3748, 3.3926] [2.6866, 3.0808] 0.6 [2.4427, 3.2568] [2.4427, 3.2569] [2.6921, 3.0074] 0.7 [2.5104, 3.1211] [2.5105, 3.1212] [2.6976, 2.9340] 0.8 [2.5784, 2.9853] [2.5784, 2.9855] [2.7030, 2.8608] 0.9 [2.6460, 2.8496] [2.6462, 2.8498] [2.7087, 2.7874] 1 [2.7140, 2.7140] [2.7141, 2.7141] [2.7140, 2.7140] Table 12: Solutions obtained by Duraisamy and Usha [16], Methods 1 and 2 for h = 0.01 at t = 1. r Duraisamy and Usha [16] Method 1 Method 2 0 [2.0388, 4.0779] [2.0387, 4.0774] [2.6812, 3.4346] 0.1 [2.1063, 3.9419] [2.1066, 3.9414] [2.6858, 3.3633] 0.2 [2.1748, 3.8056] [2.1746, 3.8055] [2.6886, 3.2919] 0.3 [2.2424, 3.6692] [2.2425, 3.6696] [2.6934, 3.2208] 0.4 [2.3102, 3.5338] [2.3105, 3.5337] [2.6960, 3.1491] 0.5 [2.3784, 3.3975] [2.3785, 3.3978] [2.6999, 3.0768] 0.6 [2.4461, 3.2621] [2.4464, 3.2619] [2.7031, 3.0051] 0.7 [2.5146, 3.1257] [2.5144, 3.1260] [2.7072, 2.9334] 0.8 [2.5815, 2.9902] [2.5823, 2.9901] [2.7120, 2.8630] 0.9 [2.6499, 2.8544] [2.6503, 2.8541] [2.7151, 2.7907] 1 [2.7184, 2.7184] [2.7182, 2.7182] [2.7184, 2.7184] Fuzzy Inf. Eng. (2012) 3: 293-312 309 Table 13: Solutions obtained by Duraisamy and Usha [16], Methods 1 and 2 for h = 0.001 at t = 1. r Duraisamy and Usha [16] Method 1 Method 2 0 [2.0369, 4.0760] [2.0387, 4.0774] [2.6814, 3.4336] 0.1 [2.1053, 3.9423] [2.1067, 3.9415] [2.6870, 3.3638] 0.2 [2.1730, 3.8041] [2.1746, 3.8056] [2.6884, 3.2886] 0.3 [2.2409, 3.6706] [2.2426, 3.6697] [2.6940, 3.2204] 0.4 [2.3091, 3.5320] [2.3105, 3.5338] [2.6939, 3.1461] 0.5 [2.3769, 3.3989] [2.3785, 3.3979] [2.6987, 3.0753] 0.6 [2.4442, 3.2600] [2.4465, 3.2619] [2.7006, 3.0004] 0.7 [2.5125, 3.1273] [2.5144, 3.1260] [2.7066, 2.9312] 0.8 [2.5805, 2.9883] [2.5824, 2.9901] [2.7102, 2.8578] 0.9 [2.6481, 2.8553] [2.6503, 2.8542] [2.7090, 2.7834] 1 [2.7163, 2.7163] [2.7183, 2.7183] [2.7163, 2.7163] Table 14: Solutions obtained by Duraisamy and Usha [16], Methods 1 and 2 for h = 0.1at t = 1. r Duraisamy and Usha [16] Method 1 Method 2 0 [2.0355, 4.0710] [2.0356, 4.0711] [2.6588, 3.4476] 0.1 [2.0628, 4.0168] [2.0627, 4.0168] [2.6612, 3.4182] 0.2 [2.0898, 3.9628] [2.0898, 3.9626] [2.6634, 3.3888] 0.3 [2.1168, 3.9082] [2.1170, 3.9083] [2.6658, 3.3596] 0.4 [2.1443, 3.8540] [2.1441, 3.8540] [2.6682, 3.3302] 0.5 [2.1712, 3.7997] [2.1713, 3.7997] [2.6702, 3.3008] 0.6 [2.1985, 3.7454] [2.1984, 3.7454] [2.6723, 3.2713] 0.7 [2.2255, 3.6914] [2.2255, 3.6911] [2.6745, 3.2422] 0.8 [2.2528, 3.6368] [2.2527, 3.6369] [2.6768, 3.2130] 0.9 [2.2799, 3.5825] [2.2798, 3.5826] [2.6789, 3.1835] 1 [2.3070, 3.5284] [2.3070, 3.5283] [2.6812, 3.1541] 310 Smita Tapaswini· S. Chakraverty (2012) Table 15: Solutions obtained by Duraisamy and Usha [16], Methods 1 and 2 for h = 0.01 at t = 1. r Duraisamy and Usha [16] Method 1 Method 2 0 [2.0388, 4.0779] [2.0387, 4.0774] [2.6812, 3.4346] 0.1 [2.0655, 4.0238] [2.0659, 4.0230] [2.6828, 3.4063] 0.2 [2.0926, 3.9681] [2.0930, 3.9686] [2.6838, 3.3772] 0.3 [2.1209, 3.9141] [2.1202, 3.9143] [2.6850, 3.3496] 0.4 [2.1474, 3.8608] [2.1474, 3.8599] [2.6862, 3.3200] 0.5 [2.1748, 3.8056] [2.1746, 3.8055] [2.6886, 3.2919] 0.6 [2.2017, 3.7518] [2.2018, 3.7512] [2.6899, 3.2624] 0.7 [2.2296, 3.6972] [2.2290, 3.6968] [2.6919, 3.2344] 0.8 [2.2561, 3.6420] [2.2561, 3.6424] [2.6924, 3.2056] 0.9 [2.2829, 3.5883] [2.2833, 3.5881] [2.6944, 3.1767] 1 [2.3102, 3.5338] [2.3105, 3.5337] [2.6960, 3.1491] Table 16: Solutions obtained by Duraisamy and Usha [16], Methods 1 and 2 for h = 0.001 at t = 1. r Duraisamy and Usha [16] Method 1 Method 2 0 [2.0369, 4.0760] [2.0387, 4.0774] [2.6814, 3.4346] 0.1 [2.0643, 4.0221] [2.0659, 4.0231] [2.6824, 3.4048] 0.2 [2.0919, 3.9686] [2.0931, 3.9687] [2.6848, 3.3766] 0.3 [2.1188, 3.9145] [2.1203, 3.9143] [2.6878, 3.3488] 0.4 [2.1454, 3.8587] [2.1474, 3.8600] [2.6884, 3.3188] 0.5 [2.1730, 3.8041] [2.1746, 3.8056] [2.6884, 3.2886] 0.6 [2.2002, 3.7499] [2.2018, 3.7512] [2.6910, 3.2610] 0.7 [2.2274, 3.6974] [2.2290, 3.6969] [2.6930, 3.2336] 0.8 [2.2543, 3.6423] [2.2562, 3.6425] [2.6926, 3.2046] 0.9 [2.2819, 3.5871] [2.2834, 3.5881] [2.6926, 3.1744] 1 [2.3091, 3.5320] [2.3105, 3.5338] [2.6939, 3.1461] Fuzzy Inf. Eng. (2012) 3: 293-312 311 8. Conclusion In this paper, two numerical techniques based on improved Euler method are pro- posed to solve fuzzy initial value problems. It may be noted that the known methods considering the left and right bounds of the variables in the diﬀerential equations. Moreover some authors also reported exact solution method in simple problems. In this investigation, the authors present the method considering all the possible combi- nations of lower and upper bounds of the variable and solved by the proposed meth- ods. The simulation for both linear and nonlinear diﬀerential equations with trian- gular and trapezoidal fuzzy initial conditions are discussed. It shows that the present method gives better result. For this, the solution in special cases have been compared with the known methods and found in good agreement. Corresponding plots of the solutions are also given. As such this general procedure of handling of the diﬀeren- tial equation by the present methods show an easy and eﬃcient way of getting the solution. The reliability and powerfulness of the proposed methods also can be seen by the results of diﬀerent type of fuzzy initial value problems. Acknowledgments The authors would like to thank the referees for valuable suggestions to improve the paper. The ﬁrst author thanks to UGC, Government of India for ﬁnancial support under Rajiv Gandhi National Fellowship (RGNF). References 1. Chang S L, Zadeh L A (1972) On fuzzy mapping and control. IEEE Trans. Systems Man Cybernet 2(1): 30-34 2. Dubois D, Prade H (1982) Towards fuzzy diﬀerential calculus part 3: Diﬀerentiation. Fuzzy Sets and Systems 8(3): 225-233 3. Kaleva O (1987) Fuzzy diﬀerential equations. Fuzzy Sets and Systems 24(3): 301-317 4. Kaleva O (1990) The Cauchy problem for fuzzy diﬀerential equations. Fuzzy Sets and Systems 35(3): 389-396 5. Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets and Systems 24(3): 319-330 6. Abbasbandy S, Allahviranloo T (2002) Numerical solutions of fuzzy diﬀerential equations by Taylor method. Computational Methods in Applied Mathematics 2(2): 113-124 7. Abbasbandy S, Allahviranloo T, Lq pez-Pouso O, Nieto J J (2004) Numerical methods for fuzzy diﬀerential inclusions. Journal of Computer & Mathematics with Applications 48: 1633-1641 8. Allahviranloo T, Ahmady N, Ahmady E (2008) Erratum to “Numerical solution of fuzzy diﬀerential equation by predictor-corrector method”. Information Sciences 178(6): 1780-1782 9. Allahviranloo T, Ahmady E, Ahmady N (2008) Nth-order fuzzy linear diﬀerential equations. Infor- mation Sciences 178(5): 1309-1324 10. Friedman M, Ma M, Kandel A (1999) Numerical solutions of fuzzy diﬀerential and integral equa- tions. Fuzzy Sets and Systems 106(1): 35-48 11. Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy diﬀerential equations. Fuzzy Sets and Systems 105(1): 133-138 12. Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy diﬀerential equations by predictor-corrector method. Information Sciences 177(7): 1633-1647 13. Allahviranloo T, Abbasband S, Ahmady N, Ahmady E (2009) Improved predictor-corrector method for solving fuzzy initial value problems. Information Science 179(7): 945-955 14. Jaulin L, Kieﬀer M, Didrit O, Walter E (2001) Applied interval analysis, Springer 15. Bede B (2008) Note on “Numerical solutions of fuzzy diﬀerential equations by predictor-corrector method”. Information Sciences 178(7): 1917-1922 312 Smita Tapaswini· S. Chakraverty (2012) 16. Duraisamy C, Usha B (2010) Another approach to solution of fuzzy diﬀerential equations by Modi- ﬁed Euler’s method. Proceedings of the International Conference on Communication and Computa- tional Intelligence-kongu Engineering College, Perundurai, Erode, TN, India: 52-55 17. Bhat R B, Chakraverty S (2004) Numerical analysis in engineering. Alpha Science International Ltd. 18. Javad Shokri (2007) Numerical solution of fuzzy diﬀerential equations. Applied Mathematical Sci- ences 1: 2231-2246 19. Smita Tapaswini, Chakraverty S (2011) Numerical method for solving fuzzy initial value problems. National Meet Research Scholar in Mathematical Science, IIT Kgp, India
Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Sep 1, 2012
Keywords: Fuzzy differential equation; Fuzzy number; Improved Euler method; Fuzzy initial value problem