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Ji-Huan He (2004)
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Fractional differential equations
In this article, we apply a relatively modified analytic iterative method for solving a time-fractional Fokker–Planck equa- tion subject to given constraints. The utilized method is a numerical technique based on the generalization of residual error function and then applying the generalized Taylor series formula. This method can be used as an alternative to obtain analytic solutions of different types of fractional partial differential equations such as Fokker–Planck equation applied in mathematics, physics, and engineering. The solutions of our equation are calculated in the form of a rapidly convergent series with easily computable components. The validity, potentiality, and practical usefulness of the proposed method have been demonstrated by applying it to several numerical examples. The results reveal that the proposed methodology is very useful and simple in determination of solution of the Fokker–Planck equation of fractional order. Keywords Fokker–Planck equation, Mittag-Leffler function, residual power series, fractional power series Date received: 6 August 2015; accepted: 8 October 2015 Academic Editor: Xiao-Jun Yang FPE was introduced by Fokker and Planck, which Introduction is a mathematical model that arises in a wide variety of In this article, a sincere attempt has been taken to solve natural science, including solid-state physics, quantum the nonlinear Riesz time-fractional Fokker–Planck optics, chemical physics, theoretical biology, and circuit equation (FPE) of the form theory. It has been commonly used to describe the Brownian motion of particles as well as the change of < a 2 D uxðÞ , t = D AxðÞ , t, u + D BxðÞ , t, u uxðÞ , t ð1Þ 0 t x probability of a random function in space and time. Applications of FPE are widely in different branches of with the initial condition given by ∂ u(x, t )= sciences and technology such as plasma physics, surface j < a ∂t = f (x), x 2 R, j = 0, 1, 2, ... , m 1, where D is j 0 t physics, population dynamics, biophysics, neuroscience, the Riesz time-fractional derivative of order a, f (x)is given analytic function on R. In the whole article, N is the set of natural numbers, R is the set of real numbers, School of Finance & Economics, Jiangsu University, Zhenjiang, China and G is the gamma function. A(x, t, u) and B(x, t, u) are Department of Mathematics, National Institute of Technology— drift and diffusion coefficients, respectively. Here, a is Jamshedpur, Jamshedpur, India the parameter representing the order of fractional deri- Corresponding author: vatives, which satisfies m 1\a m, and t.0. When Sunil Kumar, Department of Mathematics, National Institute of a = 1, the fractional equation reduces to the classical Technology—Jamshedpur, Jamshedpur 831014, Jharkhand, India. FPE. Email: skiitbhu28@gmail.com; skumar.math@nitjsr.ac.in Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). 2 Advances in Mechanical Engineering a t ð ð nonlinear hydrodynamics, pattern formation, psychol- a1 2 3 + uxðÞ , t = ðÞ t z uxðÞ , z dz, t.0, a 2 R ogy, and marketing. In recent past, Yildirim and GðÞ a Kumar have given the numerical and analytical t 0 approximate solutions of the FPE using homotopy per- 5–7 where R is the set of positive real numbers. turbation method and homotopy perturbation trans- 8,9 form method, respectively. Recently, the authors used some novel method to Definition 3. The Riesz fractional derivative of the order solve fractional differential equations (FDEs). For m 1\a m of a function u(x, t) 2 C , m 1,is 14–16 other application of FDEs in the field of engineering, defined as especially mechanics, see Yang et al. In this present analysis, we employ a relatively modified approach < a a a D uxðÞ , t = c D + D uxðÞ , t a 0 t 0 t t t residual power series method (RPSM) to find an analytical and approximate solution of FPE. The a a where c =(1=2)cos(pa=2), a 6¼ 1, and D and D a 0 t 12,13 t t RPSM is effective and easy for constructing power are the left- and right-hand side Reimann–Liouville series expansion solutions for nonlinear equations of fractional operators, respectively. different types and orders without linearization, pertur- bation, or discretization. The main advantage of this Definition 4. For 0 m 1\a m, a power series of the method is the obtained solutions, and all their frac- form tional derivatives are applicable for each arbitrary point and multidimensional variables in the given ‘ m1 X X domain by choosing an appropriate initial guess ia + j f ðÞ xðÞ t t , t t ij 0 0 approximation. Regarding the other aspect, the RPSM i = 0 j = 0 does not require any conversion while switching from the low order to the higher order and from simple line- is called a multiple fractional power series (FPS) about arity to complex nonlinearity. t = t , where t is a variable and f (x) are functions of x 0 ij The rest of this article is organized as follows: In the called the coefficients of the series. next section, we utilize some basic definitions of frac- tional calculus and theorem of power series expansions. Theorem 1. Suppose that f has a FPS representation at In section ‘‘Construction of RPSM,’’ basic idea of the t = t of the form RPSM and its convergence analysis is presented. To determine the approximate solution for the Riesz time- ‘ m1 X X ia + j fractional FPE, RPSM has been applied in section c ðÞ xðÞ t t , 0 m 1\a m, t t\t + R ij 0 0 0 ‘‘Applications and numerical discussions.’’ Finally, sec- i = 0 j = 0 tion ‘‘Conclusion’’ concludes the whole article. ia + j Furthermore, if D f (t) are continuous on (t , t + R), n = 0, 1, 2, .. . , then the coefficient c is 0 0 ij Mathematical preliminaries of fractional given by the following formula calculus and fractional power series ia + j D ftðÞ This section describes operational properties for eluci- c = , i = 0, 1, 2, .. . ij GðÞ ia + j + 1 dating sufficient fractional calculus theory, to enable us to follow the solutions of Riesz time-fractional FPE. In ia a a a where D = D , D , .. . , D (i times) and R is the radius recent year, the FDEs have gained much attention due of convergence. to the fact that they generate fractional Brownian motion, which is generalization of Brownian 14,15 motion. Construction of RPSM In this section, we construct and obtain solution of frac- Definition 1. A real function u(x, t) issaidtobeinthe tional FPE by substituting its FPS expansion among its space C , m 2 R if there exists a real number p (.m), truncated residual function. From the resulting equa- such that u(x, t)= x u (x, t), where u (x, t) 2 C½0,‘), 1 1 tion, a recursion formula for the computation of the and it is said to be in the space C if and only if coefficients is derived, while the coefficients in the FPS (m) u 2 C , m 2 N. expansion can be computed recursively by recurrent fractional differentiation of the truncated residual Definition 2. The Reimann–Liouville fractional integral function. operator of the order m 1\a m of a function The RPSM consists in expression of the solution of 14,15 u(x, t) 2 C , m 1, is defined as equation (1) as multiple FPS expansions about the m Yao et al. 3 initial point t = t . Let us consider the following form As described in El-Ajou et al. and Abu Arqub of the solution et al., it is clear that Res(x, t)= 0 for each t 2½t , t + R), x 2 R, where R is a nonnegative real 0 0 ‘ m1 ia + j X X number. In fact, this shows that ðÞ t t uxðÞ , t = f ðÞ x , ij (i1) j < a< GðÞ ia + j + 1 D D Res(x, t)= 0 for each i = 1, 2, 3, .. . , k ð2Þ 0 t 0 t i = 0 j = 0 and j = 0, 1, 2, 3, .. . , l, since the Riesz fractional m 1\a m, x 2 R, 0 t\R derivative of a constant function is zero. Again, (i1) j < a< the fractional derivatives D D for each In fact, u(x, t) satisfies the initial conditions of equa- t t 0 0 i = 1, 2, 3, .. . , k and j = 0, 1, 2, 3, ... , l of Res(x, t) and tion (1), so from equation (2), we obtain j j Res (x, t) are matching at (x, t)=(x, t ). Therefore, ∂ u(x,t )=∂t =j!f (x)=f (x), j=0,1,2, .. .,m 1. Then, (i, j) 0 0 0j j f (x)=f (x)=j!, j=0,1,2, .. .,m 1, and the initial we have the following equation 0j j guess approximation of u(x,t) can be written as (i1) j (i1) j < a< < a< D D ResðÞ x, t = D D Res ðÞ x, t = 0, t t t t (i, j) 0 0 0 0 m1 fðÞ x j j x 2 R, i = 1, 2, 3, .. . , k, j = 0, 1, 2, ... , l ð8Þ u ðÞ x, t = ðÞ t t , (0, m1) 0 j! ð3Þ j = 0 To obtain the coefficients f in equation (7), we can vw x 2 R, t t\t + R 0 0 apply the following subroutine: substitute (v, w)-trun- cated series of u(x, t) into equation (7), find the frac- With the help of equation (3), we can reformulate (v1)a tional derivative formula D D of Res (x, t)at (v, w) t t the expansion form of equation (2) as follows t = t , and then finally solve the obtained algebraic m1 ‘ m1 ia+j equation. X XX fðÞ x ðÞ t t j 0 uxðÞ ,t = ðÞ t t + f ðÞ x , To summarize the computation process of RPSM in 0 ij j! GðÞ ia+j+1 j=0 i=1 j=0 numerical values, we apply the following steps: at first, fix i = 1 and run the counter j = 0, 1, 2, .. . , l to find x 2 R, t t\t +R ð4Þ 0 0 (1, j)-truncated series expansion of suggested solution, The RPSM provides an analytical approximate solu- and next, fix i = 2 and run the counter j = 0, 1, 2, ... , l tion in terms of an infinite multiple FPS. However, to to find (2, j)-truncated series and so on. Here, to find obtain numerical values from this series, the consequent (1, 0)-truncated series expansion of equation (1), we use series truncation and the practical procedure are con- equation (5) and write ducted to accomplish this task. In the following step, we will let u (x, t) to denote the (k, l)-truncated series (k, l) u ðÞ x, t = f ðÞ x + f ðÞ xðÞ t t + + f ðÞ x (1, 0) 0 0 1 m1 of u(x, t). That is (m1) a ðÞ ðÞ t t t t 0 0 + f ðÞ x ð9Þ ia+j ðÞ m 1 ! GðÞ a + 1 m1 k l X XX fðÞ x ðÞ tt j j 0 u ðÞ x,t = ðÞ tt + f ðÞ x , (k,l) 0 ij j! GðÞ ia+j+1 For finding the first unknown coefficient f (x)in j=0 i=1 j=0 equation (9), we should substitute equation (9) into x2R, tt ð5Þ both sides of the (1, 0) residual function that was where the indices counters k = 1, 2, 3, ... and obtained from equation (7), to obtain the following l = 0, 1, 2, 3, .. . , m 1. result According to RPSM, for finding the coefficients f (x) in the series expansion of equation (5), we must Res ðÞ x, t = f ðÞ x + D A D B 1, 0 10 x ij x define the residual function concept for equation (1) as (m1) ðÞ t t f ðÞ x + f ðÞ xðÞ t t + + f ðÞ x 0 1 m1 ðÞ m 1 ! < a ResðÞ x, t = D uxðÞ , t 0 t ðÞ t t + D AxðÞ , t, u D BxðÞ , t, u uxðÞ , t , x 2 R, t t x 0 + f ðÞ x ð10Þ GðÞ a + 1 ð6Þ Depending on the result of equation (8) for and the following (k, l)-truncated residual function (i, j)=(1, 0), equation (10) gives f (x)= < a ½D B D Af (x). Hence, using the (1, 0)-truncated Res ðÞ x, t = D u ðÞ x, t (k, l) (k, l) 0 t series expansion of equation (9), the (1, 0) residual + D AxðÞ , t, u D BxðÞ , t, u u ðÞ x, t , x 2 R, t t x (k, l) 0 power series (RPS) approximate solution for equation ð7Þ (1) can be expressed as 4 Advances in Mechanical Engineering u ðÞ x, t = f ðÞ x + f ðÞ xðÞ t t + + f ðÞ x This procedure can be repeated till the arbitrary (1, 0) 0 0 1 m1 (m1) a order coefficients of the multiple FPS solution of equa- ðÞ t t ðÞ t t 0 0 + D B D A f ðÞ x ð11Þ tion (1) are obtained. But if there is a pattern in the x 0 ðÞ m 1 ! GðÞ a + 1 series coefficients, then calculating few of the terms in series is sufficient to reach the solution. Similarly, to find out the (2, 0)-truncated series expan- In this part, we study the convergence of RPS sion for equation (1), we use equation (5) and write through the given theorem. (m1) ðÞ t t u ðÞ x, t = f ðÞ x + f ðÞ xðÞ t t + + f ðÞ x (2, 0) 0 0 1 m1 ðÞ m 1 ! Theorem 2. Suppose that u(x, t) has a FPS representation a a + 1 P P ðÞ t t ðÞ t t ‘ m1 0 0 ia + j of the form u(x, t)= f (x)t , + D B D A f ðÞ x + f ðÞ x ij x 20 x 0 i = 0 j = 0 GðÞ a + 1 GðÞ a + 2 1=(ia + j) 0 m 1\a m, and limsup f = m. Then, we ij ð12Þ have Again, to find out the form of the second unknown coefficient f (x) in equation (12), we must find and for- 1. If m = 0, the series is everywhere convergent; mulate (2, 0) residual function based on equation (7) 2. If 0\m\‘, the series is absolutely convergent and then substitute the form of u (x, t) of equation (2, 0) for all t satisfying jj t \1=m and is divergent for (12) to find new discretized form of this residual func- all t satisfying jj t .1=m; tion as follows 3. If m =‘, the series is nowhere convergent. Res ðÞ x, t = f ðÞ x + f ðÞ xðÞ t t + D A D B (2, 0) 10 20 0 x (m1) Proof 1. ðÞ t t f ðÞ x + f ðÞ xðÞ t t + + f ðÞ x 0 1 0 m1 1. Let k 6¼ 0 and e = 1=2jj k . Since 0 0 ðÞ m 1 ! 1=(ia + j) limsup f = 0, there exist a natural num- ij a a + 1 ðÞ t t ðÞ t t 0 0 1=(ia + j) + D B D A f ðÞ x + f ðÞ x ber k such that f \e for all i, j k or x 0 20 ij GðÞ a + 1 GðÞ a + 2 ia + j ia + j f k \1=2 for all i, j k. Since ij P P ð13Þ ‘ m1 ia + j 1=2 is a convergent series of i = 0 j = 0 By considering equation (8) for (i, j)=(2, 0) and positive terms, by comparison test, P P < a ‘ m1 ia + j applying the operator D on both sides of equation f k is also convergent series. 0 t ij i = 0 j = 0 0 P P (13), we obtain ‘ m1 ia + j It follows that f k is absolutely ij i = 0 j = 0 0 < a < a D Res ðÞ x, t = f ðÞ x + D (2, 0) 20 0 t 0 t " " ## (m1) a a + 1 ðÞ t t ðÞ t t ðÞ t t 0 0 0 2 2 D A D B f ðÞ x + f ðÞ xðÞ t t + + f ðÞ x + D B D A f ðÞ x + f ðÞ x ð14Þ x 0 x 20 0 1 m1 0 x x ðÞ m 1 ! GðÞ a + 1 GðÞ a + 2 (i1) j < a< Using the fact that D D Res(x, t )= 0 for 0 convergent and hence convergent. As k is arbi- 0 t 0 t 0 P P ‘ m1 ia + j (i, j)=(2, 0) from equation (8) and solving the resultant trary, the series f t is every- ij i = 0 j = 0 algebraic equation for f (x), we obtain where convergent. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ 2 2 ia+j ia+j ia+j f ðÞ x = D B D A f ðÞ x + D B D A f ðÞ x 20 x x 2. limsup( f t )=limsup( f jj t )=jj t m. x 0 x 1 ij ij ð15Þ By Cauchy’s root test, the series P P ‘ m1 ia + j f t is convergent if jj t m\1. ij i = 0 j = 0 Hence, using the (2, 0)-truncated series expansion of Therefore if jj t \1=m, the series equation (2), the (2, 0) RPS approximate solution for P P ‘ m1 ia + j equation (1) can be expressed as f t is convergent, that is, the ij i = 0 j = 0 P P ‘ m1 ia + j series f t is absolutely conver- ij i = 0 j = 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u ðÞ x, t = f ðÞ x + f ðÞ xðÞ t t + + f ðÞ x (2, 0) 0 0 1 m1 ia + j ia + j gent. If jj t m\1, limsup( f t )= (m1) a ij ðÞ t t ðÞ t t 0 0 2 qﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ + D B D A f ðÞ x x 0 x ia + j ia + j ðÞ m 1 ! GðÞ a + 1 limsup( f jj t ).1 = limsup( f jj t ).1. ij ij hi ia + j 2 2 Let v = f t . Then, limsup v .1, and this + D B D A f ðÞ x + D B D A f ðÞ x ij ij ij x x 0 1 x x implies lim v 6¼ 0. Consequently, lim v 6¼ 0, ij ij a + 1 ðÞ t t P P ‘ m1 ð16Þ and it follows that v is divergent. ij i = 0 j = 0 GðÞ a + 2 Yao et al. 5 P P ‘ m1 ia + j To determine the form of coefficient f (x), in the 3. If possible, let the series f t be ij i = 0 j = 0 expansion of equation (19), we should substitute the (1, convergent for t = k ,(k 6¼ 0). Then, 0 0 ia + j ia + j 0)-truncated series u (x, t)= x + f (x)(t =G(1 + a)) (1, 0) 10 limf k = 0. The sequence ff k g being a ij ij 0 0 into the (1, 0)-truncated residual function Res (x, t)= (1, 0) bounded sequence, there exists a positive real < a 2 2 2 D u (x,t)+(∂(xu )=∂x) (∂ ((x u (x,t))=2)=∂x (1,0) (1,0) (1,0) ia + j 0 t number B such that f k \B for all i, j 2 N. ij to obtain 1=(ia + j) This shows that the sequence f f g is ij Res ðÞ x, t = f x ð21Þ (1, 0) (10)(x) bounded sequence, and this contradicts that 1=(ia + j) limsup f =‘. Thus, the series ij Depending on the results of equation (8) in case of i = 0 m1 ia + j (i, j)=(1, 0), one can obtain f (x)= x. Hence, the f t is not convergent for t = k .As k 10 ij 0 0 j = 0 (1, 0)-truncated series solution of equation (17) could is arbitrary nonzero real number, the series P P be expressed as ‘ m1 ia + j f t is nowhere convergent. ij i = 0 j = 0 u ðÞ x, t = x + x ð22Þ (1, 0) GðÞ 1 + a Applications and numerical discussions Again, to find out the form of the second unknown The application problems are carried out using the pro- coefficient f (x), we substitute the (2, 0)-truncated posed RPSM, which is one of the modern analytical series solution u (x, t)= x + x(t =G(1 + a)) + f (x) (2, 0) 20 techniques because of its iteratively nature. In this sec- 2a (t =G(1 + 2a)) into the (2, 0)-truncated residual tion, we consider the time-fractional FPE to show < a function Res (x, t)= D u (x, t)+(∂(xu )=∂x) (2, 0) (2, 0) (2, 0) 0 t potentiality, generality, and efficiency of our method. 2 2 2 (∂ (x u (x, t)=2)=∂x ) to obtain (2, 0) Throughout this work, all the symbolic and numerical a a computations were performed by using t t Res ðÞ x, t = f ðÞ x x ð23Þ (2, 0) 20 MATHEMATICA 7 software package. GðÞ 1 + a GðÞ 1 + a < a Example 1. Consider the following time-fractional FPE Now, applying the operator D one time on both 0 t sides of equation (23) and according to equation (8) for x u (i, j)=(2, 0), we have f (x)= x. Therefore, the (2, 0)- < a 2 20 D u = DðÞ xu + D x, t.0, 0\a 1 ð17Þ 0 t x truncated series solution of equation (17) is obtained, and one can collect the previous results to obtain the with the initial condition u(x, 0)= x. The exact solution following expansion of the problem is given by u(x, t)= xe for a = 1. a 2a According to equation (4) with f (x)= x, and consider- 0 t t 2 u ðÞ x, t = x + x + x ð24Þ (2, 0) ing A(x, t, u)= x and B(x, t, u)= x =2, the series solution GðÞ 1 + a GðÞ 1 + 2a of equation (17) can be written as As the former, by applying the same procedure ia X for (i, j), i = 2, 3, 4, ... and j = 0 will yield after easy ðÞ t uxðÞ , t = x + f ðÞ x ð18Þ i0 calculations to f (x)= x. Furthermore, if we collect all ij GðÞ ia + 1 i = 1 the final results, then the RPS solution of equation (17) can be constructed in the form of infinite series as where u (x, t)= x is the initial guess approximation 0, 0 follows which is obtained directly from equation (3). Next, according to equations (5) and (7), the (k, 0)-truncated uxðÞ , t = series of u(x, t) and the (k, 0)-truncated residual function a 2a 3a of equation (17) can be defined and thus constructed, t t t x 1 + + + + respectively, as follows GðÞ 1 + a GðÞ 1 + 2a GðÞ 1 + 3a ka ‘ ia ðÞ t = x = xE ðÞ t ð25Þ u ðÞ x, t = x + f ðÞ x , GðÞ 1 + ka (k, l) i0 k = 0 GðÞ ia + 1 ð19Þ i = 1 a ka where E (t )= (t =G(1 + ka)), a.0, is the k = 1, 2, 3, .. . , l = 0 k = 0 Mittag-Leffler function in one parameter. As a x u ðÞ x, t (k, l) special case when a = 1, the RPS solution of equation ∂ xu (k, l) < a (17) has the general pattern form which coincides Res ðÞ x, t = D u ðÞ x, t + , (k, l) (k, l) 0 t ∂x ∂x with the exact solution of the standard FPE in terms of k = 1, 2, 3, .. . , l = 0 ð20Þ infinite series 6 Advances in Mechanical Engineering Figure 1. (a) Exact solution u(x, t) for equation (17), (b) the surface graph of the RPS approximation solution u (x, t) for (4, 0) equation (17) when a = 1, (c) the surface graph of the RPS approximation solution u (x, t) for equation (17) when a = 0:75, (4, 0) (d) the surface graph of the RPS approximation solution u (x, t) for equation (17) when a = 0:50, and (e) absolute error (4, 0) E (h)= u(x, t) u (x, t) . (4, 0) (4, 0) 2 3 4 t t t three-dimensional space figures of the (4, 0)-truncated uxðÞ , t = x 1 + t + + + + ) = xe ð26Þ 2! 3! 4! series solution together with the exact solution on the domain ½0, 23 ½0, 2. It is clear from the scenario of The above result is completely in agreement with the Figure 1 that Figure 1(a)–(d) is nearly coinciding and 3 4 Yildirim and Kumar. similar in behavior. Again to show the accuracy of the The geometric behavior of the solutions of equa- RPSM, we report absolute error in Figure 1(e). Of tion (17) is studied next in Figure 1 by drawing the course, the accuracy can be improved by computing Yao et al. 7 ‘ ia ðÞ t u ðÞ x, t = x + f ðÞ x , (k, l) i0 GðÞ ia + 1 ð29Þ i = 1 k = 1, 2, 3, .. . , l = 0 < a Res ðÞ x, t = D u ðÞ x, t (k, l) (k, l) 0 t xu ðÞ x, t x u ðÞ x, t (k, l) (k, l) ∂ ∂ 6 12 ∂x ∂x ð30Þ To determine the form of coefficient f (x), in the expansion of equation (29), we should substitute the (1, 0)-truncated series u (x, t)= x + f (x) (1, 0) 10 (t =G(1 + a)) into the (1, 0)-truncated residual function Res (x, t) (1, 0) < a 2 2 = D u (x, t)+(∂(xu (x, t)=6)=∂x) (∂ (x u (x, t)=12) (1, 0) (1, 0) (1, 0) 0 t =∂x ) to obtain Figure 2. Plot of u (x, t) versus time t for different value of a at x =1. Res ðÞ x, t = f ð31Þ (1, 0) (10)(x) further terms of the approximate solutions using the present methods. Depending on the results of equation (8) in case of Figure 2 shows the evaluation results of the approxi- (i, j)=(1, 0), one can obtain f (x)= x =2. Hence, the mate analytical solution of equation (17) obtained by (1, 0)-truncated series solution of equation (27) could the RPSM for different fractional Brownian motions, be expressed as that is, a = 0:7, a = 0:8, and a = 0:9, and standard 2 a motions, that is, a = 1. Our solutions obtained by the x t u ðÞ x, t = x + ð32Þ (1, 0) proposed methods increase very rapidly with the 2 GðÞ 1 + a increase in t at x = 1. Again, to find f (x), we substitute the (2, 0)-trun- 2 2 a cated series solution u (x, t)= x +(x =2)(t = 2, 0 2a Example 2. Consider the following time-fractional FPE G(1 + a)) + f (x)(t =G(1 + 2a)) into the (2, 0)-trun- < a cated residual function Res (x, t)= D u (x, t)+ (2, 0) (2, 0) 0 t 2 2 2 xu x u < a 2 (∂(xu (x, t)=6)=∂x) (∂ (x u (x, t)=12)=∂x)to (2, 0) (2, 0) D u = D + D x, t.0, 0\a 1 ð27Þ 0 t x 6 12 obtain with the initial condition u(x, 0)= x . The exact solu- a 2 a t x t 2 t=2 Res ðÞ x, t = f ðÞ x ð33Þ tion of the problem is given by u(x, t)= x e for a = 1. (2, 0) 20 GðÞ 1 + a 4 GðÞ 1 + a According to equation (4) with f (x)= x and con- < a sidering A(x, t, u)= x=6 and B(x, t, u)= x =12, the series Now, applying the operator D one time on both 0 t solution of equation (27) can be written as sides of equation (33) and according to equation (8) for (i, j)=(2, 0), we have f (x)= x =4. Therefore, the ‘ ia ðÞ t (2, 0)-truncated series solution of equation (27) is uxðÞ , t = x + f ðÞ x ð28Þ i0 GðÞ ia + 1 obtained as follows i = 1 2 a 2 2a where u (x, t)= x is the initial guess approximation x t x t 0, 0 u ðÞ x, t = x + + ð34Þ (2, 0) which is obtained directly from equation (3). Next, the 2 GðÞ 1 + a 4 GðÞ 1 + 2a (k, 0)-truncated series of u(x, t) and the (k, 0)-truncated By applying the same procedure for (i, j), residual function that are derived from equations (5) i = 2, 3, 4, .. . and j = 0, we can easily obtain the rest and (7) can be formulated, respectively, in the form of of f . Using the above collected values of f , the RPS ij ij the following solution of equation (27) can be constructed in the form of infinite series as follows a 2a 3a ka a t t t t t 2 2 2 uxðÞ , t = x 1 + + + + = x = x E ð35Þ 2 3 k 2GðÞ 1 + a 2 GðÞ 1 + 2a 2 GðÞ 1 + 3a 2 GðÞ 1 + ka 2 k = 0 8 Advances in Mechanical Engineering Figure 3. (a) Exact solution u(x, t) for equation (27), (b) the surface graph of the RPS approximation solution u (x, t) for equation 4, 0 (27) when a = 1, (c) the surface graph of the RPS approximation solution u (x, t) for equation (27) when a = 0:75, (d) the surface 4, 0 graph of the RPS approximation solution u (x, t) for equation (27) when a = 0:50, and (e) absolute error 4, 0 E (h)= u(x, t) u (x, t) . (4, 0) (4, 0) 2 3 4 t t t t As a special case when a = 1, the RPS solution of uxðÞ , t = x 1 + + + + + 2 3 4 equation (27) has the general pattern form which coin- 2 2 2! 2 3! 2 4! cides with the exact solution of the standard FPE in = x e ð36Þ terms of infinite series Yao et al. 9 with the initial condition u(x, 0)= x . The exact solu- 2 t tion of the problem is given by u(x, t)= x e for a = 1. According to equation (4) with f (x)= x and con- sidering A(x, t, u)=(4u=3) (xu=3) and B(x, t, u)= u, the series solution of equation (37) can be written as ‘ ia ðÞ t uxðÞ , t = x + f ðÞ x ð38Þ i0 GðÞ ia + 1 i = 1 where u (x, t)= x is the initial guess approximation 0, 0 which is obtained directly from equation (3). Next, the (k, 0)-truncated series of u(x, t) and the (k, 0)-truncated residual function that are derived from equations (5) and (7) can be formulated, respectively, in the form of k ia ðÞ t u ðÞ x, t = x + f ðÞ x , k = 1, 2, 3, .. . (k, l) i0 GðÞ ia + 1 i = 1 Figure 4. Plot of u (x, t) versus time t for different value of a at x =1. ð39Þ 4u xu ∂ 2 3 3 ∂ u < a Res ðÞ x, t = D u ðÞ x, t + ð40Þ (k, l) (k, l) However, the above result is same as the Yildirim 0 t ∂x ∂x and Kumar. In order to illustrate the behaviors of the RPS According to the RPSM and without the loss of gen- approximate solution of equation (27) geometrically, erality for the remaining computations and results, the the approximate solution u (x, t) for following are the first four unknown coefficients in the (4, 0) a = 0:5, 0:75, and 1 has been depicted in three- multiple FPS expansion of equation (38): f = x , 2 2 2 dimensional space as shown in Figure 3 in addition to f = x , f = x , and f = x . Therefore, the (4, 0)- 20 30 40 2 t=2 the exact solution x e when a = 1 on the domain truncated series solution of equation (37) can be written ½0, 23 ½0, 2. as It is clear from Figure 3 that, on one hand, Figure a 2a 3(a)–d is nearly coinciding and similar in behavior, t t 2 2 2 u ðÞ x, t = x + x + x (4, 0) while, on the other hand, for the special case of GðÞ 1 + a GðÞ 1 + 2a ð41Þ a = 1Figure 3(a) and (b) is nearly identical and in 3a 4a t t 2 2 + x + x excellent agreement with each other in terms of accu- GðÞ 1 + 3a GðÞ 1 + 4a racy. As a result, one can achieve a good approxima- tion with the exact solution using few terms only, and Consequently, the components of the RPS approxi- thus, it is evident that the overall errors can be made mate solution are obtained as far as we wish. In fact, smaller by adding new terms of the RPS approxima- this series is exact to the last term, as one can verify, of tions. Furthermore, the accuracy of the proposed the multiple FPS of the exact solution that can be col- method for equation (27) is shown in Figure 3(e). lected to discover that the approximate solution of The behavior of the approximate analytical solution equation (37) has the general pattern form which coin- of equation (27) obtained by the RPSM for different cides with the exact solution in terms of infinite series fractional Brownian motions, that is, a = 0:7, a = 0:8, and a = 0:9, and standard motions, that is, a = 1,is ‘ ka 2 2 a shown in Figure 4. It is seen from Figure 4 that the uxðÞ , t = x = x E ðÞ t ð42Þ GðÞ ka + 1 k = 0 solution obtained by RPSM increases very rapidly with the increase in t at x = 1. However, the above result is in complete agreement 3 4 with Yildirim and Kumar. Example 3. Consider the following time-fractional Next, the geometrical behaviors of the RPS approxi- FPE mate solution of equation (37) are shown in Figure 5, which are nearly identical and in excellent agreement 4u xu < a 2 2 with each other. To measure the accuracy of the pro- D u = D + D u x, t.0, 0\a 1 0 t x 3 3 posed method, we report the absolute error in Figure 5(e). ð37Þ 10 Advances in Mechanical Engineering Figure 5. (a) Exact solution u(x, t) for equation (37), (b) the surface graph of the RPS approximation solution u (x, t) for equation 4, 0 (37) when a = 1, (c) the surface graph of the RPS approximation solution u (x, t) for equation (37) when a = 0:75, (d) the surface 4, 0 graph of the RPS approximation solution u (x, t) for equation (37) when a = 0:50, and (e) absolute error 4, 0 E (h)= u(x, t) u (x, t) . (4, 0) (4, 0) The behavior of the approximate analytical Conclusion solution of equation (37) obtained by the RPSM for In this article, the time-fractional FPE have been solved different fractional Brownian motions, that is, using RPSM. In order to show the effectiveness and a = 0:7, a = 0:8, and a = 0:9, and standard motions, leverage of the featured method, convergence analysis that is, a = 1, is shown in Figure 6. It is seen from is presented. The numerical results obtained by RPSM Figure 6 that the solution obtained by RPSM increases for a = 1, a = 0:75, and a = 0:5 highly agree with the very rapidly with the increase in t at x = 1. Yao et al. 11 2. Frank TD. Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equa- tions. Physica A 2004; 331: 391–408. 3. Yildirim A. Analytical approach to Fokker–Planck equa- tion with space- and time-fractional derivatives by means of the homotopy perturbation method. JKingSaud Univ: Sci 2010; 22: 257–264. 4. Kumar S. A numerical study for the solution of time fractional nonlinear shallow water equation in oceans. Z Naturforsch A 2013; 68: 547–553. 5. He JH. Homotopy perturbation technique. Comput Method Appl M 1999; 178: 257–262. 6. He JH. The homotopy perturbation method for non- linear oscillators with discontinuities. Appl Math Comput 2004; 151: 287–292. 7. Momani S and Odibat Z. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys Lett A 2007; 365: 345–350. 8. Singh J, Kumar D and Kumar S. New treatment of frac- Figure 6. Plot of u (x, t) versus time t for different value of a tional Fornberg–Whitham equation via Laplace trans- at x =1. form. Ain Shams Eng J 2013; 4: 557–562. 9. Kumar S, Kumar A, Kumar D, et al. Analytical solution of Abel integral equation arising in astrophysics via exact solution. Consequently, the present scheme is Laplace transform. J Egypt Math Soc 2015; 23: 102–107. very simple, attractive, and appropriate for obtaining 10. Yang XY and Baleanu D. Fractal heat conduction prob- numerical solutions of time-fractional FPE. Therefore, lem solved by local fractional variation iteration method. without loss of generality, it can be conclude that the Therm Sci 2013; 17: 625–628. 11. Yang XJ, Baleanu D, Khan Y, et al. Local fractional var- basic idea of the proposed method can be used to solve iational iteration method for diffusion and wave equa- other nonlinear FDEs of different types and orders, as tions on cantor sets. Rom J Phys 2014; 59: 1–2. well as other scientific applications. 12. El-Ajou A, Abu Arqub O, Momani S, et al. A novel expansion iterative method for solving linear partial dif- Declaration of conflicting interests ferential equations of fractional order. Appl Math Com- put, http://dx.doi.org/10.1016/j.amc.2014.12.121 The author(s) declared no potential conflicts of interest with 13. Abu Arqub O, El-Ajou A and Momani S. Constructing respect to the research, authorship, and/or publication of this and predicting solitary pattern solutions for nonlinear article. time-fractional dispersive partial differential equations. J Comput Phys 2015; 293: 385–399. Funding 14. Podlubny I. Fractional differential equations. New York: The author(s) disclosed receipt of the following financial sup- Academic Press, 1999. port for the research, authorship, and/or publication of this 15. Miller KS and Ross B. An introduction to the fractional article: A.K. was financially supported by CSIR, New Delhi, calculus and fractional differential equations. New York: India. Wiley, 1993. 16. Shen S, Liu F, Anh V, et al. The fundamental solution and numerical solution of the Riesz fractional advection- References dispersion equation. IMA J Appl Math 2008; 73: 1. Risken H.The Fokker–Planck equation: method of solu- 850–872. tion and applications. Berlin: Springer, 1989.
Advances in Mechanical Engineering – SAGE
Published: Dec 3, 2015
Keywords: Fokker–Planck equation; Mittag-Leffler function; residual power series; fractional power series
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