A novel expansion iterative method for solving linear

Applied Mathematics and Computation xxx (2015) xxx–xxx

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A novel expansion iterative method for solving linear partial differential equations of fractional order Ahmad El-Ajou a, Omar Abu Arqub a, Shaher Momani b,c, Dumitru Baleanu d,e,⇑, Ahmed Alsaedi c a

Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan c Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Turkey e Institute of Space Sciences, Magurele-Bucharest, Romania b

a r t i c l e

i n f o

Keywords: Fractional partial differential equations Fractional power series Residual power series

a b s t r a c t In this manuscript, we implement a relatively new analytic iterative technique for solving time–space-fractional linear partial differential equations subject to given constraints conditions based on the generalized Taylor series formula. The solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations. This method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics, physics, and engineering. Some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality, generality, and accuracy for solving such equations with different constraints conditions. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Fractional partial differential equations (PDEs) are found to be an effective tool to describe certain physical phenomena such as, damping laws, rheology, diffusion processes, electrostatics, electrodynamics, fluid flow, elasticity, and so on [1–7]. Problems in fractional PDEs are not only important but also quite challenging which usually involves hard mathematical solution techniques. Anyhow, in most real-life applications, it is too complicated to obtain exact solutions to PDEs of fractional order in terms of composite elementary functions in a simple manner, so an efficient, reliable numerical algorithm for the solutions of such equations is required; it is little wonder that with the development of fast, efficient digital computers, the role of numerical methods in mathematics, physics, and engineering problem solving has increased dramatically in recent years. The objective of the present letter is to extend the application of the iterative residual power series (RPS) method [8–14] to provide analytic solutions for initial value problems of linear PDEs of fractional order and to make comparisons with some of the well-known analytic methods. The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order, which unifies and generalizes the notions of integer-order differentiation and n-fold integration. Theory of fractional calculus is a significantly ⇑ Corresponding author at: Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Turkey. E-mail address: [email protected] (D. Baleanu). http://dx.doi.org/10.1016/j.amc.2014.12.121 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.

Please cite this article in press as: A. El-Ajou et al., A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2014.12.121

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A. El-Ajou et al. / Applied Mathematics and Computation xxx (2015) xxx–xxx

important and useful branch of mathematics having a broad range of applications at almost any branch of science. Techniques of fractional calculus have been employed at the modeling of many different phenomena in mathematics, physics, and engineering [15–21]. The most important advantage of using fractional calculus in these and other applications is their nonlocal property. It is well known that the integer order differential operators and the integer order integral operators are local, while on the other hand, the fractional order differential operators and the fractional order integral operators are nonlocal. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. In fact, this is main reason why differential operators of fractional order provide an excellent instrument for description of memory and hereditary properties of various mathematical, physical, and engineering processes. Series expansions are very important aids in numerical calculations, especially for quick estimates made in hand calculation, for example, in evaluating functions, integrals, or derivatives. Solutions to fractional PDEs can often be expressed in terms of series expansions. Since, the advent of computers, it has, however, become more common to treat fractional PDEs directly, using different approximation method instead of series expansions. But in connection with the development of automatic methods for formula manipulation, one can anticipate renewed interest for series methods. These methods have some advantages, especially in multidimensional solutions for PDEs of fractional order. The RPS method was developed as an efficient method for determine values of coefficients of the power series solution for the first-order and the second-order fuzzy DEs [8]. It has been successfully applied in the numerical solution of the generalized Lane-Emden equation, which is a highly nonlinear singular DE [9], in the numerical solution of higher-order regular DEs [10], in the solution of composite and noncomposite fractional DEs [11], in predict and represent the multiplicity of solutions to boundary value problems of fractional order [12], in construct and predict the solitary pattern solutions for nonlinear time-fractional dispersive PDEs [13], and in approximate solution of the nonlinear fractional KdV-Burgers equation [14]. The RPS method is effective and easy to construct power series solution for strongly linear and nonlinear equations without linearization, perturbation, or discretization. Different from the classical power series method, the RPS method does not need to compare the coefficients of the corresponding terms and a recursion relation is not required. This method computes the coefficients of the power series by a chain of algebraic equations of one or more variables. In fact, the RPS method is an alternative procedure for obtaining analytic solutions for PDEs of fractional order. By using residual error concept, we get a series solutions; in practice truncated series solutions. Moreover, the obtained solutions and all their time–space-fractional derivatives are applicable for each arbitrary point and each multidimensional variable in the given domain. On the other aspect as well, the RPS method does not require any converting while switching from the low-order to the higher-order; as a result the method can be applied directly to the given problem by choosing an appropriate initial guess approximation. In the present paper, the RPS method will investigate to construct a new algorithm for finding analytical solutions to the following classes of higher-order linear fractional PDEs of the general form

Dat uðx; tÞ ¼ L½uðx; tÞ; x 2 R; t P t0 ; 0 6 m 1 < a 6 m; m 2 N;

ð1:1Þ

subject to the nonhomogeneous initial conditions

@ j uðx; t 0 Þ ¼ uj ðxÞ; x 2 R; @t j

j ¼ 0; 1; 2; . . . ; m 1;

ð1:2Þ

and subject to the constraint linear differential operator

L½uðx; tÞ ¼

m1 X j¼0

"

@ j @ kj uðx; tÞ g j ðxÞt @xkj @t j j

!# þ

1 X m1 X

hij ðxÞ

i¼0 j¼0

Cðia þ j þ 1Þ

ðt t 0 Þjþia ;

kj 2 N;

ð1:3Þ

where Dat is the Caputo fractional derivative of order a and uj ðxÞ; g j ðxÞ; hij ðxÞ are given analytic functions on R. Throughout this paper, N the set of integer numbers, R the set of real numbers, and C is the Gamma function. In most cases, the higher-order linear fractional PDEs cannot be solved analytically and their solutions cannot be expressed in closed forms, where solutions of such equations are always demand due to physical interests. Anyhow, in the literature, a number of methods have been developed for numerical or analytical solutions for fractional PDEs. The reader is asked to refer to [22–29] in order to know more details about these methods, including their kinds and history, their modifications and conditions for use, their scientific applications, their importance and characteristics, and their relationship including the difference. The outline of the letter is as follows. In the next section, we utilize some necessary definitions and results from fractional calculus theory. In Section 3, basic idea of the RPS method is represented in order to construct and predict the multiple fractional power series expansion solution. In Section 4, some physical and mathematical linear fractional PDEs of different types and orders are performed in order to illustrate capability and simplicity of the proposed method. The conclusion is given in the final part, Section 5. 2. Overview of fractional calculus theory: mathematical tools and theories For the concept of fractional derivative we will adopt the Caputo’s definition which is a modification of the Riemann– Liouville definition and has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case in most physical and engineering processes [21]. Please cite this article in press as: A. El-Ajou et al., A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2014.12.121

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Next, some necessary definitions and essentials results from fractional calculus theory are presented, where these definitions and results are collected from the mentioned references. Definition 2.1. A real function uðx; tÞ; x 2 R; t > 0 is said to be in the space C l ; l 2 R if there exists a real number p > l such that uðx; tÞ ¼ tp u1 ðx; tÞ, where u1 ðx; tÞ 2 CðR ½0; 1ÞÞ, and it is said to be in the space C nl if

@ n uðx;tÞ @t n

Definition 2.2. The Riemann–Liouville fractional integral operator of order a > 0 of uðx; tÞ 2 C l ,

( J at uðx; tÞ ¼

1

CðaÞ

Rt s

ðt sÞ

a1

2 C l ; n 2 N.

l P 1 is defined as

uðx; sÞds; a > 0; x 2 I; t > s > s P 0;

ð2:1Þ

a ¼ 0:

uðx; tÞ;

Definition 2.3. The Caputo fractional derivative of order a > 0 of uðx; tÞ 2 C n1 ; n 2 N is defined as

8 na @ n uðx;tÞ ; 0 6 n 1 < a < n; x 2 I; t > s > s P 0; @ a uðx; tÞ < J t @tn Dt uðx; tÞ ¼ ¼ a n : @ uðx;tÞ @t ; a ¼ n 2 N: @t n a

ð2:2Þ

On the one hand, for some certain properties of the operator Dat , it is obvious that when c > 1; t > s P 0, and C 2 R, we ca cþ1Þ have Dat ðt sÞc ¼ CCðcðþ1 and Dat C ¼ 0. On the other hand, properties of the operator J as can be summarized shortly in aÞ ðt sÞ

the form of the following: for uðx; tÞ 2 C l ; l P 1, a; b P 0, C 2 R, and c P 1, we have J at C ¼ CðaCþ1Þ ðt sÞa , J at J bt uðx; tÞ aþc . ¼ J at þb uðx; tÞ ¼ J bt J at uðx; tÞ, and J at ðt sÞc ¼ CCðaðþcþ1Þ cþ1Þ ðt sÞ

Theorem 2.1. If n 1 < a 6 n, uðx; tÞ 2 C nl , n 2 N, and P @ j uðx;sþ Þ ðtsÞ j n1 , where t > s P 0. j¼0 j! @t j

l P 1, then Dat Jat uðx; tÞ ¼ uðx; tÞ and Jat Dat uðx; tÞ ¼ uðx; tÞ

The reader may go through [1–7,15–21] in order to know more details about the mathematical properties of fractional derivatives and fractional integrals, including their types and history, their motivation for use, their characteristics, and their applications. Next, some results related to the fractional power series (FPS) in the sense of the Caputo’s definition for fractional derivative are collected in order to find analytic series solution for Eqs. (1.1) and (1.2). Definition 2.4. A power series expansion of the form 1 X m1 X

f nk ðxÞðt t 0 Þkþna ;

0 6 m 1 < a 6 m; t P t 0 ;

ð2:3Þ

n¼0 k¼0

is called a multiple FPS about t ¼ t0 , where t is a variable and f nk ðxÞ are functions of x called the coefficients of the series. Notice that in writing out the term corresponding to n ¼ 0 and k ¼ 0 in Eq. (2.3) we have adopted the convention that W ðt t 0 Þ0 ¼ 1 even when t ¼ t 0 . Also, when t ¼ t 0 each of terms of Eq. (2.3) are vanished for n–0 k–0 and so. On the other hand, the multiple FPS representation of Eq. (2.3) always converges when t ¼ t0 . In the next lemma by Dnt0a we mean that Dat0 Dat0 . . . Dat0 (n -times). Lemma 2.1. Suppose that uðx; tÞ 2 CðR ½t0 ; t0 þ RÞÞ and Djta uðx; tÞ 2 CðR ðt 0 ; t 0 þ RÞÞ for j ¼ 0; 1; 2; . . . ; n þ 1 where 0 6 m 1 < a 6 m. Then a ðnþ1Þa J ðnþ1Þ Dt uðx; tÞ ¼ t

a Dðnþ1Þ uðx; fÞ t ðt t0 Þðnþ1Þa ; Cððn þ 1Þa þ 1Þ

t0 6 f 6 t < t 0 þ R:

ð2:4Þ

Proof. From the definition of the operator J at and by using the second mean value theorem for fractional integrals, one can find a ðnþ1Þa J ðnþ1Þ Dt uðx; tÞ ¼ t

¼

1 Cððn þ 1ÞaÞ

Z

t

t0

a ðt sÞðnþ1Þa1 Dðnþ1Þ uðx; sÞds ¼ t

a Dðnþ1Þ uðx; fÞ t ðt t0 Þðnþ1Þa : Cððn þ 1Þa þ 1Þ

a Dðnþ1Þ uðx; fÞ t Cððn þ 1ÞaÞ

Z

t

ðt sÞðnþ1Þa1 ds

t0

ð2:5Þ

h Theorem 2.2. Suppose that uðx; tÞ 2 CðR ½t 0 ; t 0 þ RÞÞ, Djta uðx; tÞ 2 CðR ðt0 ; t0 þ RÞÞ, and Djta uðx; tÞ can be differentiated ðm 1Þ-times with respect to t on ðt 0 ; t 0 þ RÞ for j ¼ 0; 1; 2; . . . ; n þ 1, where 0 6 m 1 < a 6 m. Then Please cite this article in press as: A. El-Ajou et al., A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2014.12.121

4


uðx; tÞ ¼

a n m1 X X Dkþja uðx; t0 Þ Dðnþ1Þ uðx; fÞ t t ðt t 0 Þkþja þ ðt t 0 Þðnþ1Þa ; C ðj a þ k þ 1Þ C ððn þ 1Þ a þ 1Þ j¼0 k¼0

t 0 6 f 6 t 6 t0 þ R:

ð2:6Þ

Proof. From the certain properties of the operator J at , we have a ðnþ1Þa m na J ðnþ1Þ Dt uðx; tÞ ¼ J nt a ððJ at Dat ÞDnt a uðx; tÞÞ ¼ J nt a ððJ m t t Dt ÞDt uðx; tÞÞ ! k m 1 X @ k Dnt a uðx;tþ Þ k na na 0 @t ðt t 0 Þ ¼ J t Dt uðx; tÞ k! k¼0

¼

J nt a Dnt a uðx; tÞ

J nt a

m1 X Dk Dna uðx;t t

t

k!

! 0Þ

k

ðt t0 Þ

ð2:7Þ

k¼0

a m ðn1Þa ðJ m uðx; tÞ ¼ J ðn1Þ t t Dt ÞDt

m1 kþna X D uðx;t t

0Þ

Cðnaþkþ1Þ

ðt t 0 Þkþna

k¼0

a a ¼ J ðn1Þ Dðn1Þ uðx; tÞ t t

m1 k ðn1Þa X D D uðx;t t

k!

! 0Þ

ðt t0 Þk

k¼0

m1 kþna X D uðx;t t

0Þ

Cðnaþkþ1Þ

ðt t0 Þkþna :

k¼0

On the other direction, if we keep the repeating of this process, then after n -times of scientific computations, we will obtain a ðnþ1Þa J ðnþ1Þ Dt uðx; tÞ ¼ uðx; tÞ t

a n m 1 X X Dkþj uðx; t 0 Þ t ðt t 0 Þkþja ; C ðj a þ k þ 1Þ j¼0 k¼0

By using Lemma 2.1, the proof of the theorem will be complete.

t 0 6 t 6 t0 þ R:

ð2:8Þ

h

Remark 2.1. If one fixed m ¼ 1, then the series representation formula of Eq. (2.6) will leads to the following expansion of uðx; tÞ about t 0 :

uðx; tÞ ¼

n X Djta uðx; t0 Þ 0

j¼0

Cðja þ 1Þ

ðt t 0 Þja þ

a Dðnþ1Þ uðx; fÞ t0

Cððn þ 1Þa þ 1Þ

ðt t0 Þðnþ1Þa ;

t0 6 f 6 t < t 0 þ R;

ð2:9Þ

which is the same as of Generalized Taylor’s series formula that obtained in [30] for 0 < a 6 1. As with any convergent series, this means that uðx; tÞ is the limit of the sequence of partial sums. In the case of general kþja P P Dt uðx;t 0 Þ kþja form of generalized Taylor’s series, the partial sums are given as T n ðx; tÞ ¼ nj¼0 m1 . In general, uðx; tÞ is k¼0 Cðjaþkþ1Þ ðt t 0 Þ the sum of its general form of generalized Taylor’s series if uðx; tÞ ¼ lim T n ðx; tÞ. But on the other aspect as well, if n!1

Rn ðx; tÞ ¼ uðx; tÞ T n ðx; tÞ, then Rn ðx; tÞ is the remainder of general form of generalized Taylor’s series. That is, D

ðnþ1Þa

uðx;fÞ

ðnþ1Þa t Rn ðx; tÞ ¼ Cððnþ1Þ ; t 0 6 f 6 t < t0 þ R. aþ1Þ ðt t 0 Þ

ðnþ1Þa

Corollary 2.1. If jDt0 uðx; tÞj 6 MðxÞ on t 0 6 t 6 d, where m < a 6 m 1, then the reminder Rn ðx; tÞ of general form of generalized Taylor’s series satisfies

jRn ðx; tÞj 6

MðxÞ

Cððn þ 1Þa þ 1Þ

ðt t 0 Þðnþ1Þa ;

t 0 6 t 6 d:

ð2:10Þ

3. Construction of residual power series method In this section, we find out series solutions for linear fractional PDEs by substituting their FPS expansions among their truncated residual functions. From the resulting equations a recursion formulas for the computation of the coefficients are derived, while the coefficients in the multiple FPS expansions can be computed recursively by recurrent fractional differentiating of the truncated residual functions. The RPS method consists in expressing the solution of Eqs. (1.1) and (1.2) as a multiple FPS expansion about the initial point t ¼ t0 . To achieve our goal, suppose that the solution takes the following infinite expansion form

uðx; tÞ ¼

1 m 1 X X ðt t0 Þiaþj f ij ðxÞ ; Cðia þ j þ 1Þ i¼0 j¼0

m 1 < a 6 m; x 2 R; t 0 6 t < t0 þ R:

ð3:1Þ


5


Obviously, uðx; tÞ satisfy the initial conditions of Eq. (1.2); so from Eq. (3.1), we obtain, j ¼ 0; 1; 2; . . . ; m 1. Then f 0j ðxÞ ¼ m 1 X

uð0;m1Þ ðx; tÞ ¼

uj ðxÞ j!

j¼0

uj ðxÞ j!

@ j uðx;t 0 Þ @t j

¼ j!f 0j ðxÞ ¼ uj ðxÞ;

; j ¼ 0; 1; 2; . . . ; m 1 and the initial guess approximation of uðx; tÞ can be written as

ðt t0 Þ j ;

x 2 R; t 0 6 t < t0 þ R:

ð3:2Þ

As a result, using Eq. (3.2) we can reformulate the expansion form of Eq. (3.1) as follows:

uðx; tÞ ¼

m1 X j¼0

uj ðxÞ j!

ðt t0 Þ j þ

1 X m1 X

f ij ðxÞ

i¼1 j¼0

ðt t0 Þiaþj ; Cðia þ j þ 1Þ

x 2 R; t 0 6 t < t0 þ R:

ð3:3Þ

The RPS provides an analytical approximate solution in terms of an infinite multiple FPS. However, to obtain numerical values from this series, the consequent series truncation and the practical procedure are conducted to accomplish this task. In the following step, we will let uðk;lÞ ðx; tÞ to denote the (k, l)-truncated series of u(x, t). That is

uðk;lÞ ðx; tÞ ¼

m 1 X j¼0

uj ðxÞ j!

ðt t0 Þ j þ

k X l X ðt t0 Þiaþj f ij ðxÞ ; Cðia þ j þ 1Þ i¼1 j¼0

x 2 R; t P t 0 ;

ð3:4Þ

where the indices counters k and l whenever used mean that k ¼ 1; 2; 3; . . . and l ¼ 0; 1; 2; . . . ; m 1. Prior to applying the RPS technique for finding form of the coefficients f ij ðxÞ in the series expansion of Eq. (3.4), we must define the residual function concept for Eq. (1.1) as

Resðx; tÞ ¼ Dat uðx; tÞ L½uðx; tÞ;

x 2 R; t P t0 ;

ð3:5Þ

and the following (k, l)-truncated residual function:

Resðk;lÞ ðx; tÞ ¼ Dat uðk;lÞ ðx; tÞ L½uðk;lÞ ðx; tÞ;

x 2 R; t P t 0 :

ð3:6Þ

As in [8–14], it is clear that Resðx; tÞ ¼ 0 for each t 2 ½t0 ; t0 þ RÞ; x 2 R, where R is a nonnegative real number. In fact, this a j shows that Dði1Þ Dt Resðx; tÞ ¼ 0 for each i ¼ 1; 2; 3; . . . ; k and j ¼ 0; 1; 2; . . . ; l, since the fractional derivative of a constant t a j Dt for each i ¼ 1; 2; 3; . . . ; k and function in the Caputo sense is zero. In the mean time, the fractional derivatives Dði1Þ t j ¼ 0; 1; 2; . . . ; l of Resðx; tÞ and Resði;jÞ ðx; tÞ are matching at ðx; tÞ ¼ ðx; t0 Þ. Anyhow, one can write

a j a j Dði1Þ Dt Resðx; t 0 Þ ¼ Dði1Þ Dt Resði;jÞ ðx; t 0 Þ ¼ 0; t t

x 2 R; i ¼ 1; 2; 3; . . . ; k; j ¼ 0; 1; 2; . . . ; l:

ð3:7Þ

To obtain form of the coefficients f v w in Eq. (3.4) for v ¼ 1; 2; 3; . . . ; k and w ¼ 0; 1; 2; . . . ; l, we apply the following subrou-

tine: substitute (v, w)-truncated series of u(x, t) into Eq. (3.6), find the fractional derivative formula Dðtv 1Þa Dw t of Resðv ;wÞ ðx; tÞ at t ¼ t0 , and then finally solve the obtained algebraic equation to get the required coefficients. To summarize the computation process of RPS method in numerical values, we apply the following steps: fixed i = 1 and run the counter j ¼ 0; 1; 2; . . . ; l to find (1, j)-truncated series expansion of suggested solution, next fixed i = 2 and run the counter j ¼ 0; 1; 2; . . . ; l to obtain the (2, j)-truncated series, and so on. In fact, to obtain (1, 0)-truncated series expansion of Eqs. (1.1) and (1.2), we use Eq. (3.4) and write

uð1;0Þ ðx; tÞ ¼ u0 ðxÞ þ u1 ðxÞðt t 0 Þ þ þ um1 ðxÞ

ðt t 0 Þðm1Þ ðt t 0 Þa þ f 10 ðxÞ : ðm 1Þ! Cða þ 1Þ

ð3:8Þ

On the other aspect as well, to determine form of the first unknown coefficient f 10 ðxÞ in Eq. (3.8), we should substitute Eq. (3.8) into both sides of the (1, 0)-residual function that obtained from Eq. (3.6), to get the following result:

"

# ðt t 0 Þðm1Þ ðt t 0 Þa : Resð1;0Þ ðx; tÞ ¼ f 10 ðxÞ L u0 ðxÞ þ u1 ðxÞðt t 0 Þ þ þ um1 ðxÞ þ f 10 ðxÞ ðm 1Þ! Cða þ 1Þ

ð3:9Þ ðk Þ

Depending on the result of Eq. (3.7) for ði; jÞ ¼ ð1; 0Þ; then Eq. (3.9) gives f 10 ðxÞ ¼ L½u0 ðxÞ ¼ g 0 ðxÞu0 0 þ h00 ðxÞ, where ðk Þ

u0 0 ðxÞ denotes to the k0 th-derivative of u0 ðxÞ. Hence, using the (1, 0)-truncated series expansion of Eq. (3.8); the (1, 0)RPS approximate solution for Eqs. (1.1) and (1.2) can be expressed as uð1;0Þ ðx; tÞ ¼ u0 ðxÞ þ u1 ðxÞðt t 0 Þ þ þ um1 ðxÞum1 ðxÞ

ðt t0 Þðm1Þ ðt t0 Þa þ L½u0 ðxÞ : ðm 1Þ! Cða þ 1Þ

ð3:10Þ

Similarly, to find out the (1, 1)-truncated series expansion for Eqs. (1.1) and (1.2), we use Eq. (3.4) and write

uð1;1Þ ðx; tÞ ¼ u0 ðxÞ þ u1 ðxÞðt t 0 Þ þ þ um1 ðxÞ

ðt t 0 Þðm1Þ ðt t 0 Þa ðt t0 Þaþ1 þ L½u0 ðxÞ : þ f 11 ðxÞ ðm 1Þ! Cða þ 1Þ Cða þ 2Þ

ð3:11Þ

Again, to find out form of the second unknown coefficient f 11 ðxÞ in Eq. (3.11), we must find and formulate (1, 1)-residual function based on Eq. (3.6) and then substitute the form of uð1;1Þ ðx; tÞ of Eq. (3.11) to find new discretized form of this residual function as follows:


6


Resð1;1Þ ðx; tÞ ¼ f 10 ðxÞ þ f 11 ðxÞðt t 0 Þ "

# ðt t 0 Þðm1Þ ðt t 0 Þa ðt t0 Þaþ1 L u0 ðxÞ þ u1 ðxÞðt t 0 Þ þ þ um1 ðxÞ þ L½u0 ðxÞ þ f 11 ðxÞ : ðm 1Þ! Cða þ 1Þ Cða þ 2Þ

ð3:12Þ

By considering Eq. (3.7) for (i, j) = (1, 1) and applying the operator Dt to the both side of Eq. (3.12), we get

Dt Resð1;1Þ ðx; t 0 Þ ¼ f 11 ðxÞ "

ðt t0 Þðm1Þ ðt t0 Þa ðt t0 Þaþ1 þ L½u0 ðxÞ þ f 11 ðxÞ Dt L u0 ðxÞ þ u1 ðxÞðt t 0 Þ þ þ um1 ðxÞ ðm 1Þ! Cða þ 1Þ Cða þ 2Þ

# : t¼t0

ð3:13Þ a j Using the fact that Dði1Þ Dt Resði;jÞ ðx; t0 Þ ¼ 0 for (i, j) = (1, 1}) from Eq. (3.7) and solving the resultant algebraic equation for t f 11 ðxÞ, we can easily obtain ðk Þ

ðk Þ

f 11 ðxÞ ¼ g 1 ðxÞu0 0 ðxÞ þ g 0 ðxÞu1 1 ðxÞ þ h01 ðxÞ:

ð3:14Þ

Hence, using the (1, 1)-truncated series expansion of Eq. (3.1); the (1, 1)-RPS approximate solution for Eqs. (1.1) and (1.2) can be expressed as

u1;1 ðx; tÞ ¼

m 1 X j¼0

uj ðxÞ j!

ðt t Þa ðt t Þ1þa 0 0 ðk Þ ðk Þ ðk Þ ðt t0 Þ j þ g 0 ðxÞu0 0 þ h00 ðxÞ : þ g 1 ðxÞu0 0 ðxÞ þ g 0 ðxÞu1 1 ðxÞ þ h01 ðxÞ Cða þ 1Þ Cða þ 2Þ ð3:15Þ

This procedure can be repeated till the arbitrary order coefficients of the multiple FPS solution of Eqs. (1.1) and (1.2) are obtained. Moreover, higher accuracy can be achieved by evaluating more components of the solution. As we will see later, if there is a pattern in the series coefficients, then calculating few of terms in series is sufficient to reach the solution. 4. Applications and numerical discussions The application problems are carried out using the proposed RPS method, which is one of the modern analytical techniques because of it’s iteratively nature; it can be handle any kind of initial conditions and other constraints. In this section, we consider six applications to show potentiality, generality, and superiority of our method to solve linear fractional PDEs as compared with other well-known methods. In the first application we consider the time-fractional wave equation, in the second two applications we consider two versions of the space-fractional telegraph equations, in the fourth and five applications we consider the time-fractional Poisson and Navier–Stokes equation, respectively, while in the remaining application we construct a time-fractional equation. In the process of computation, all the symbolic and numerical computations were performed by using MATHEMATICA 7 software package. Throughout this work, we will try to give the results of the all applications; however, in some cases we will switch between the results obtained for the applications in order not to increase the length of the paper without the loss of generality for the remaining applications and results. However, by easy calculations we can collect further results and discussion for the desire application. Application 4.1. Consider the following homogeneous time-fractional wave equation:

@ a uðx; tÞ 1 2 @ 2 uðx; tÞ ¼ x ; @ta 2 @x2

x 2 R; t P 0; 1 < a 6 2;

ð4:1Þ


uðx; 0Þ ¼ x;

ut ðx; 0Þ ¼ x2 :

ð4:2Þ

According to Eq. (3.3) with u0 ðxÞ ¼ x and u1 ðxÞ ¼ x2 , and considering the linear differential operator L½uðx; tÞ ¼ 12 x2 @ we assume that the series solution of Eqs. (4.1) and (4.2) can be written as

uðx; tÞ ¼ x þ x2 t þ

1 X 1 X f ij ðxÞ i¼1 j¼0

tjþia

Cð1 þ j þ iaÞ

2

uðx;tÞ , @x2

ð4:3Þ

;

where uð0;1Þ ðx; tÞ ¼ x þ x2 t is the initial guess approximation which is obtained directly from Eq. (3.2). Next, according to Eqs. (3.4) and (3.6) the (k, l)-truncated series of u(x, t) and the (k, l)-truncated residual function of Eq. (4.1) can be defined and thus constructed, respectively, as follows:

uðk;lÞ ðx; tÞ ¼ x þ x2 t þ

k X l X f ij ðxÞ i¼1 j¼0

tjþia ; Cð1 þ j þ iaÞ

k ¼ 1; 2; 3; . . . ; l ¼ 0; 1;

ð4:4Þ


7


1 @ 2 uðk;lÞ ðx; tÞ Resðk;lÞ ðx; tÞ ¼ Dat uðk;lÞ ðx; tÞ x2 ; 2 @x2

k ¼ 1; 2; 3; . . . ; l ¼ 0; 1:

ð4:5Þ

To determine form of the coefficient f 10 ðxÞ, we find out the (1, 0)-truncated series of uðx; tÞ as 00 ta 1 2 ta uð1;0Þ ðx; tÞ ¼ x þ x2 t þ f 10 ðxÞ Cð1þ aÞ and the ð1; 0Þ -truncated residual function as Resð1;0Þ ðx; tÞ ¼ f 10 ðxÞ 2 x ð2t þ f 10 ðxÞ Cð1þaÞÞ. On the other aspect as well, by considering Eq. (3.7) for ði; jÞ ¼ ð1; 0Þ and substituting t ¼ 0, one can obtain f 10 ðxÞ ¼ 0. Similarly, to find out form of the coefficient f 11 ðxÞ, we evaluate the ð1; 1Þ -truncated series of uðx; tÞ as 1þa

1þa

00

t 1 2 t uð1;1Þ ðx; tÞ ¼ x þ x2 t þ f 11 ðxÞ Cð2þ aÞ and the ð1; 1Þ-truncated residual function as Resð1;1Þ ðx; tÞ ¼ f 11 ðxÞt 2 x ð2t þ f 11 ðxÞ Cð2þaÞÞ. 00

ta

Thus, for ði; jÞ ¼ ð1; 1Þ, we conclude that Dt Resð1;1Þ ðx; tÞ ¼ f 11 ðxÞ 12 x2 ð2 þ f 11 ðxÞ Cð1þaÞÞ. Anyhow, the substitution of t = 0 will 2

leads to f 11 ðxÞ ¼ x . 1þa

2a

t t To evaluate form of the coefficient f 20 ðxÞ, we need to write uð2;0Þ ðx; tÞ ¼ x þ x2 t þ x2 Cð2þ aÞ þ f 20 ðxÞ Cð1þ2aÞ and ta

t 1þa

00

t 2a

Resð2;0Þ ðx; yÞ ¼ f 20 ðxÞ Cð1þaÞ 12 x2 ð2 Cð2þaÞ þ f 20 ðxÞ Cð1þ2aÞÞ. Considering the fact that Dat Resð2;0Þ ðx; 0Þ ¼ 0, we can easily find f 20 ðxÞ ¼ 0. Anyhow, the continuation in the same manner will leads also to f 21 ðxÞ ¼ x2 . Moreover, for k ¼ 3; 4; 5; . . . ; we can conclude that f k0 ðxÞ ¼ 0 and f k1 ðxÞ ¼ x2 . Therefore, according to Eq. (4.3), the RPS solution of Eqs. (4.1) and (4.2) can be written in the form of infinite series as follows:

uðx; tÞ ¼ x þ x2 t þ

t1þa t 1þ2a t 1þ3a þ þ þ : Cð2 þ aÞ Cð2 þ 2aÞ Cð2 þ 3aÞ

ð4:6Þ

As a special case when a ¼ 2, the RPS solution of Eqs. (4.1) and (4.2) has the general pattern form which is coinciding with the following exact solution in terms of infinite series:

uðx; tÞ ¼ x þ x2 ðt þ

t3 t5 t7 þ þ þ Þ: 3! 5! 7!

ð4:7Þ

So, the exact solution of Eqs. (4.1) and (4.2) in a closed form of elementary function will be uðx; tÞ ¼ x þ x2 sinh t. Remark 4.1. It is to be noted that, the (3, 1)-truncated series solution of Eqs. (4.1) and (4.2) is

uð3;1Þ ðx; tÞ ¼ x þ x2 ðt þ

t1þa t 1þ2a t 1þ3a þ þ Þ; Cð2 þ aÞ Cð2 þ 2aÞ Cð2 þ 3aÞ

ð4:8Þ

which is the same as the fourth term approximate solution that obtained by the Adomian decomposition method (ADM) [23]. Whereas, the fourth term approximate solution for Eqs. (4.1) and (4.2) that obtained by the variational iteration method (VIM) [31] and the homotopy perturbation method (HPM) [32] are the same and are given as

t3 t5 t7 3t5a 2t 7a t72a : u4 ðx; tÞ ¼ x þ x2 t þ þ þ þ 2 40 7! Cð6 aÞ Cð8 aÞ Cð8 2aÞ

ð4:9Þ

In fact, the expansion form of Eq. (4.8) shows that it is the most natural solution, and the most fruitful one to Eqs. (4.1) and (4.2) in comparison with the expansion form of Eq. (4.9). Whilst the solution of Eq. (4.8) when a ¼ 2 coincides with the classical solution of Eq. (4.7) in ordinary case in contrast to the solution of Eq. (4.9). Results from numerical analysis are an approximation, in general, which can be made as accurate as desired. Because a computer has a finite word length, only a fixed number of digits are stored and used during computations. Next, the agreement between the solutions obtained using the RPS method, the ADM, the VIM, and the HPM are investigated for Application 4.1. Anyhow, Tables 1–3 show, respectively, the residual errors when a ¼ 1:5; 1:75; 2 and for various x and t. Those results are generated from the (3, 1)-truncated series approximation for the RPS method and from the fourth term approximate solution for the remaining mentioned method. It is clear from the tables that, the residual errors can be measure the extent of agreement between the (3, 1)-truncated series approximation of the RPS solutions and unknowns closed form solutions in the case of 1 < a < 2 which are inapplicable in general for such fractional equations. Anyhow, from the tables, it can be seen that the RPS technique provides us with the accurate approximate solution and explicates the rapid convergence in approximating the solution of Eqs. (4.1) and (4.2) from the other mentioned methods. Application 4.2. Consider the following homogeneous space-fractional telegraph equation:

@ a uðx; tÞ @ 2 uðx; tÞ @uðx; tÞ ¼ þ þ uðx; tÞ; @xa @t @t 2

x 2 R; t P 0; 1 < a 6 2;

ð4:10Þ


uð0; tÞ ¼ ux ð0; tÞ ¼ et :

ð4:11Þ

In this application, the initial conditions are on the independent variable x. So, we will replace x by t and t by x in Eq. (3.1). Therefore, we assume that the series solution of Eqs. (4.10) and (4.11) can be written as


8


Table 1 The comparison values of the residual errors for Application 4.1 at a ¼ 1:5 . t

x

VIM

HPM

0.2

10

5.51277

5.51277

5

4:97102 10

4:97102 105

1

5:51277 102 5.51277

4:97102 107

4:97102 107

10

5:51277 102 5.51277

5

4:97102 105

10

5.31407

5.31407

3

2:24963 103

5:31407 10 5.31407

5

2:24963 10

2:24963 105

2:24963 103

2:24963 103

0.4

1

5:31407 10 5.31407

10 0.6

0.8

2

ADM

RPS method

4:97102 10

2:24963 10 2

10

4.64626

4.64626

2:09224 102

2:09224 102

1

4:64626 102 4.64626

2:09224 104

2:09224 104

10

4:64626 102 4.64626

2

2:09224 10

2:09224 102

10 1

4.91059

4.91059

0:101806

0:101806

4:91059 102

4:91059 102

1:01806 103

1:01806 103

Table 2 The comparison values of the residual errors for Application 4.1 at a ¼ 1:75. t

x

VIM

HAM

ADM

RPS method

0.2

10

1.15719

1.15719

3:70435 106

3:70435 106

1

1:15719 102 1.15719

3:70435 108

3:70435 108

10

1:15719 102 1.15719

3:70435 106

3:70435 106

10

0.92375

0.92375

4

2:81935 10

2:81935 104

1

9:23749 103 0.92375

9:23749 103 0.92375

2:81935 106

2:81935 106

2:81935 104

2:81935 104

0.4

10 0.6

0.8

10

1.06133

1.06133

3:55402 103

3:55402 103

1

1:06133 102 1:06133

3:55402 105

3:55402 105

10

1:06133 102 1.06133

3:55402 103

3:55402 103

10

1.73713

1.73713

2

2:14579 10

2:14579 102

1

1:73713 102

1:73713 102

2:14579 104

2:14579 104

RPSmethod

Table 3 The comparison values of the residual errors for Application 4.1 at a ¼ 2. t

x

VIM

HAM

ADM

0.2

10

2:53968 107

2:53968 107

2:53968 107

2:53968 107

9

9

9

0.4

1

2:53968 10

2:53968 10

2:53968 10

2:53968 109

10

2:53968 107

2:53968 107

2:53968 107

2:53968 107

10

3:25079 105

3:25079 105

3:25079 105

3:25079 105

1

3:25079 107

3:25079 107

3:25079 107

3:25079 107

5

5

5

3:25079 105

4

5:55429 104

6

10 0.6

3:25079 10

4

uðx; tÞ ¼

3:25079 10

5:55429 10

4

5:55429 10

5:55429 10

6

5:55429 10

5:55429 106

5:55429 104

5:55429 104

5:55429 104

5:55429 104

10

4:16102 103

4:16102 103

4:16102 103

4:16102 103

1

4:16102 105

4:16102 105

4:16102 105

4:16102 105

10

0.8

3:25079 10

1 10

1 X 1 X f ij ðtÞ i¼0 j¼0

5:55429 10

6

xjþia : Cð1 þ j þ iaÞ

5:55429 10

ð4:12Þ

Using RPS method, taking f 00 ðtÞ ¼ uð0; tÞ ¼ et and f 01 ðxÞ ¼ ux ð0; tÞ ¼ et ; the initial guess approximation of uðx; tÞ is of the form

uð0;1Þ ðx; tÞ ¼ et þ xet :

ð4:13Þ


9


According to Eq. (3.3), the multiple FPS solution of Eqs. (4.10) and (4.11) will take the following infinite expansion:

uðx; tÞ ¼ et þ et x þ

1 X 1 X f ij ðtÞ i¼1 j¼0

xjþia ; Cð1 þ j þ iaÞ

ð4:14Þ

while the ðk; lÞ-truncated series of uðx; tÞ and the ðk; lÞ-truncated residual function that are derived from Eqs. (3.4) and (3.6) can be formulated, respectively, in form of the following:

uðk;lÞ ðx; tÞ ¼ et þ et x þ

k X l X f ij ðtÞ i¼1 j¼0

Resðk;lÞ ðx; tÞ ¼ Dax uðk;lÞ ðx; tÞ

xjþia ; Cð1 þ j þ iaÞ

ð4:15Þ

@ 2 uðk;lÞ ðx; tÞ @uðk;lÞ ðx; tÞ uðk;lÞ ðx; tÞ: @t @t 2

ð4:16Þ

To determine form of the first unknown coefficient f 10 ðtÞ in the expansion of Eq. (4.15) we should substitute the ð1; 0Þa xa truncated series uð1;0Þ ðx; tÞ ¼ et þ et x þ f 10 ðtÞ Cð1þ aÞ into the ð1; 0Þ-truncated residual function Resð1;0Þ ðx; tÞ ¼ Dx uð1;0Þ ðx; tÞ

@ 2 uð1;0Þ ðx;tÞ @t 2

@uð1;0Þ ðx;tÞ @t

uð1;0Þ ðx; tÞ to get 0

Resð1;0Þ ðx; tÞ ¼ et et x þ f 10 ðtÞ

00

xa ½f 10 ðtÞ þ f 10 ðtÞ þ f 10 ðtÞ : Cð1 þ aÞ

ð4:17Þ

Depending on the result of Eq. (3.7) in the case of ði; jÞ ¼ ð1; 0Þ, the substituting of x ¼ 0 back into Eq. (4.17) will yields f 10 ðtÞ ¼ et : Hence, the ð1; 0Þ -truncated series solution of Eqs. (4.10) and (4.11) could be expressed as

uð1;0Þ ðx; tÞ ¼ et þ et x þ et

xa : Cð1 þ aÞ

ð4:18Þ

Again, to find out form of the second unknown coefficient f 11 ðtÞ we substitute the (1, 1)-truncated series solution a

1þa

x x uð1;1Þ ðx; tÞ ¼ et þ et x þ et Cð1þ aÞ þ f 11 ðtÞ Cð2þaÞ

@ 2 uð1;1Þ ðx;tÞ @t 2

@uð1;1Þ ðx;tÞ @t

into

the

ð1; 1Þ-truncated

residual

function

Resð1;1Þ ðx; tÞ ¼ Dax uð1;1Þ ðx; tÞ

uð1;1Þ ðx; tÞ to obtain 0

Resð1;1Þ ðx; tÞ ¼ et x þ xf11 ðtÞ

00

et xa x1þa ½f 11 ðtÞ þ f 11 ðtÞ þ f 11 ðtÞ : Cð1 þ aÞ Cð2 þ aÞ

ð4:19Þ

Now, applying the operator Dx one time on the both sides of Eq. (4.19) will gives the following first partial derivative of Resð1;1Þ ðx; tÞ with respect to x:

Dx Resð1;1Þ ðx; tÞ ¼ et þ f 11 ðtÞ

aet xa1 ð1 þ aÞ½f 11 ðtÞ þ f 011 ðtÞ þ f 0011 ðtÞxa : Cð1 þ aÞ Cð2 þ aÞ

ð4:20Þ

According to Eq. (3.7) for ði; jÞ ¼ ð1; 1Þ, we have Dx Resð1;1Þ ð0; tÞ ¼ 0 (recall that in this application, we replace x by t and t by x). Solving the resulting equation for f 11 ðtÞ, one obtain f 11 ðtÞ ¼ et . Therefore, the ð1; 1Þ -truncated series solution of Eqs. (4.10) and (4.11) is obtained and one can collect the previous results to get the following expansion:

uð1;1Þ ðx; tÞ ¼ et þ et x þ et

xa

Cð1 þ aÞ

þ et

x1þa

Cð2 þ aÞ

:

ð4:21Þ

As the former, by applying the same procedure for ði; jÞ, j ¼ 0; 1 and i ¼ 2; 3; 4; . . . will leads after easy calculations to f ij ðtÞ ¼ et . Anyhow, if we employ the last results, then the RPS solution of Eqs. (4.10) and (4.11) can be constructed in the form of infinite series as follows:

uðx; tÞ ¼ et 1 þ x þ

xa x1þa x2a x1þ2a þ þ þ þ : Cð1 þ aÞ Cð2 þ aÞ Cð1 þ 2aÞ Cð2 þ 2aÞ

ð4:22Þ

As a special case when a ¼ 2, the RPS solution of Eqs. (4.10) and (4.11) has the general pattern form which is coinciding with the exact solution in terms of infinite series

x2 x3 x4 x5 uðx; tÞ ¼ et 1 þ x þ þ þ þ þ : 2! 3! 4! 5!

ð4:23Þ

So, the exact solution of Eqs. (4.10) and (4.11) in a closed form of elementary function will be uðx; tÞ ¼ ext . Although there are a lot of studies for the linear time-fractional PDEs and some profound results have been established, it seems the detailed results and conclusions in the RPS procedure in finding and predicting the solution of Application 4.2 that, the RPS solution is the same as the Adomian decomposition solution [22] and differential transform solution [33]. Anyhow, Table 4 shows the full agreement between those solutions and the RPS solution that obtained from the (1, 1)-truncated series


10


Table 4 The comparison values of the approximate solutions for Application 4.2. x

t

a ¼ 1:25

a ¼ 1:5

a ¼ 1:75

a¼2

0.2

0 0.25 0.5 0.75 1

1.3343810819 1.0392170315 0.8093430379 0.6303169917 0.4908913667

1.2740803840 0.9922548007 0.7727688158 0.6018329589 0.4687079796

1.2402165526 0.9658816223 0.7522293638 0.5858368176 0.4562501723

1.2214027581 0.9512294245 0.7408182206 0.5769498103 0.4493289641

0.4

0 0.25 0.5 0.75 1

1.7667126467 1.3759171927 1.0715653871 0.8345359626 0.6499372611

1.6328207772 1.2716420999 0.9903558632 0.7712899218 0.6006811950

1.5471211015 1.2048991253 0.9383763823 0.7308082613 0.5691540462

1.4918246976 1.1618342427 0.9048374180 0.7046880897 0.5488116360

0.6

0 0.25 0.5 0.75 1

2.2986560463 1.7901951289 1.3942053682 1.0858082325 0.8456283017

2.0770503324 1.6176084253 1.2597947083 0.9811291054 0.7641041155

1.9265225531 1.5003772729 1.1684949950 0.9100248171 0.7087280402

1.8221187986 1.4190675472 1.1051709170 0.8607079755 0.6703200453

0.8

0 0.25 0.5 0.75 1

2.9473348307 2.2953866741 1.7876489393 1.3922223937 1.0842638904

2.6209313794 2.0411834106 1.5896752386 1.2380403206 0.9641867712

2.3914567305 1.8624683744 1.4504918284 1.1296441718 0.8797677656

2.2255408965 1.7332529930 1.3498587882 1.0512710813 0.8187307413

1

0 0.25 0.5 0.75 1

3.7347380599 2.9086169256 2.2652331393 1.7641653427 1.3739333504

3.2840656609 2.5576329084 1.9918865118 1.5512827752 1.2081402401

2.9594178326 2.3047969255 1.7949776504 1.3979299997 1.0887089784

2.7182815255 2.1169997807 1.6487210869 1.2840252736 0.9999998885

when a ¼ 1:25; 1:5; 1:75; 2 and for various x and t on the domain ½0; 1 ½0; 1. As a result, we can easily conclude that the RPS method is an efficient tool in finding the solution. Application 4.3. Consider the following nonhomogeneous space-fractional telegraph equation:

@ a uðx; tÞ @ 2 uðx; tÞ @uðx; tÞ ¼ þ x2 t þ 1; @xa @t @t 2

x 2 R; t P 0; 1 < a 6 2;

ð4:24Þ


uð0; tÞ ¼ t; ux ð0; tÞ ¼ 0:

ð4:25Þ

As the previous application, the initial conditions are on the independent variable x. So, we will replace x by t and t by x in Eq. (3.1). Moreover, Eq. (4.24) is not exactly as the general form that suggested in Eqs. (1.1) and (1.3) because the term x2 is of a power 2 which must be of the form j þ ia; j ¼ 0; 1; :::; m 1; i ¼ 0; 1; 2; . . ., that is either of the form ia or 1 þ ia since m ¼ 2 in this application. Therefore, we assume that the solution of Eqs. (4.24) and (4.25) as in the following form:

uðx; tÞ ¼ f 00 ðtÞ þ f 01 ðtÞx þ

1 X 2 X

f ij ðtÞ

i¼1 j¼0

xjþia : Cð1 þ j þ iaÞ

ð4:26Þ

Using RPS method and according to the initial conditions of Eq. (4.25), the initial guess approximation of uðx; tÞ is uð0;1Þ ðx; tÞ ¼ t. Therefore, the RPS solution of Eqs. (4.24) and (4.25) has the following expansion form:

uðx; tÞ ¼ t þ

1 X 2 X

f ij ðtÞ

i¼1 j¼0

xjþia

Cð1 þ j þ iaÞ

ð4:27Þ

:

Whilst the ðk; lÞ -truncated series of uðx; tÞ and the ðk; lÞ-truncated residual function that are derived from Eqs. (3.4) and (3.6) can be formulated, respectively, in form of

uðk;lÞ ðx; tÞ ¼ t þ

k X l X i¼1 j¼0

f ij ðtÞ

xjþia

Cð1 þ j þ iaÞ

Resðk;lÞ ðx; tÞ ¼ Dax uðk;lÞ ðx; tÞ

;

l ¼ 0; 1; 2;

@ 2 uðk;lÞ ðx; tÞ @uðk;lÞ ðx; tÞ þ x2 þ t 1: @t @t 2

ð4:28Þ

ð4:29Þ



11

To determine form of the first unknown coefficient f 10 ðtÞ in the expansion of Eq. (4.28), we should substitute the (1, 0)truncated series of uðx; tÞ into the (1, 0)-truncated residual function to get the following result: 0

Resð1;0Þ ðx; tÞ ¼ 2 þ t þ x2 þ f 10 ðtÞ

00

xa ½f 10 ðtÞ þ f 10 ðtÞ : Cð1 þ aÞ

ð4:30Þ

Depending on the result of Eq. (3.7), we have f 10 ðtÞ ¼ 2 t. Hence, the ð1; 0Þ -truncated series solution of Eqs. (4.24) and (4.25) takes the form

uð1;0Þ ðx; tÞ ¼ t þ ð2 tÞ

xa : Cð1 þ aÞ

ð4:31Þ

Similarly, to find out form of the second unknown coefficient f 11 ðtÞ, we substitute the (1, 1)-truncated series solution of Eqs. (4.24) and (4.25) into the (1, 1)-truncated residual function to obtain 0

Resð1;1Þ ðx; tÞ ¼ x2 þ xf11 ðtÞ þ

00

xa x1þa ½f 11 ðtÞ þ f 11 ðtÞ : Cð1 þ aÞ Cð2 þ aÞ

ð4:32Þ

Hence, the application of the operator Dx on both sides of Eq. (4.32) will gives the first partial derivative of Resð1;1Þ ðx; tÞ with respect to x as

Dx Resð1;1Þ ðx; tÞ ¼ 2x þ f 11 ðtÞ þ

0 00 axa1 ð1 þ aÞ½f 11 ðtÞ þ f 11 ðtÞxa : Cð1 þ aÞ Cð2 þ aÞ

ð4:33Þ

According to Eq. (3.7) in the case of ði; jÞ ¼ ð1; 1Þ, we have Dx Resð1;1Þ ð0; tÞ ¼ 0. Solving the resulting equation for f 11 ðtÞ, one can obtain f 11 ðtÞ ¼ 0. To determine rule of the coefficient f 12 ðtÞ, we need to solve the algebraic equation a j Dði1Þ Dx Resði;jÞ ð0; tÞ ¼ 0 when ði; jÞ ¼ ð1; 2Þ. After some calculations, we get f 12 ðtÞ ¼ 2. Similarly, by applying the same prox cedure for ði; jÞ ¼ ð2; 0Þ, we have f 20 ðtÞ ¼ 1. It is to be noted that, the rest of the coefficients f ij ðtÞ ¼ 0 for i ¼ 2; 3; . . . and j ¼ 0; 1; 2. Therefore, the RPS solution of Eqs. (4.24) and (4.25) can be constructed as follows:

uðx; tÞ ¼ t þ ð2 tÞ

xa x2þa x2 a ; 2 Cð1 þ aÞ Cð3 þ aÞ Cð1 þ 2aÞ

ð4:34Þ

which is precisely the exact solution. As a special case when a ¼ 2 ., the RPS solution of Eqs. (4.24) and (4.25) has the general pattern form which is coinciding with the exact solution in terms of elementary function


x2 x4 : 2 8

ð4:35Þ

Remark 4.2. We mention here that, the author in [22] obtained the following Adomian decomposition solution of Eqs. (4.24) and (4.25):


xa xa x2þa x2a þ ð1 þ tÞ 2 t ; Cð1 þ aÞ Cð1 þ aÞ Cð3 þ aÞ Cð1 þ 2aÞ

ð4:36Þ

which is not give any visualize about the form of the exact solution. Anyhow, the expansion form of Eq. (4.34) shows that it is the most natural solution, and the most fruitful one to Eqs. (4.24) and (4.25) in comparison with the expansion form of Eq. (4.36). Whilst the solution of Eq. (4.34) when a ¼ 2 coincides with the classical solution of Eq. (4.35) in ordinary case in contrast to the solution of Eq. (4.36). The geometric behavior of the solutions of Application 4.3 are studied next in Fig. 1 by drawing the 3 -dimensional space figures of the ð2; 1)-truncated ss solution together with the exact solution expressed by Eq. (4.34). Anyhow, the scenario of the subfigures (a), (b), (c), and (d) is to plot uð2;0Þ ðx; tÞ when a ¼ 1:25, a ¼ 1:5, a ¼ 1:75, and a ¼ 2, respectively, on the domain ½1; 0 ½0; 3. It is clear from the Fig. 1 that each of the subfigures are nearly coinciding and similar in their behavior. As a result, one can achieve an excellent approximation and thus the exact solution by using a few terms only. It is to be noted from the subfigures that when the value of x. moving away to the left side of zero on the specific domain of t the representation surface graph solutions are increasing firstly and then decreases gradually. Application 4.4. Consider the following nonhomogeneous space-fractional Poisson equation:

@ a uðx; tÞ @ 2 uðx; tÞ þ ¼ xa1 et ; @xa @t 2

x; t P 0; 1 < a 6 2;

ð4:37Þ


uð0; tÞ ¼ et ; ux ð0; tÞ ¼ et :

ð4:38Þ


12


Fig. 1. The surface graph of the RPS approximate solution uð1;0Þ ðx; tÞ for Application 4.3: (a) uð2;0Þ ðx; tÞ when a ¼ 1:25, (b) uð2;0Þ ðx; tÞ when a ¼ 1:5, (c) uð2;0Þ ðx; tÞ when a ¼ 1:75, (d) uð2;0Þ ðx; tÞ when a ¼ 2.

As the previous applications, the initial conditions are on the independent variable x. So, we will replace x by t and t by x in Eq. (3.1). Moreover, Eq. (4.37) is not exactly as the general form which we suggested in Eq. (1.1) because the term xa1 is of a power a 1 which should be one of the forms f0; 1; a; a þ 1g. Therefore, we assume that the solution of Eqs. (4.37) and (4.38) as in the following expansion form:

uðx; tÞ ¼ f 00 ðtÞ þ f 01 ðtÞx þ

1 m 2 X X f ij ðtÞ i¼1 j¼0

1 X xjþia xi a f i0 ðtÞ : ¼ f 00 ðtÞ þ f 01 ðtÞx þ Cð1 þ j þ iaÞ Cð1 þ iaÞ i¼1

ð4:39Þ

Using RPS method and according to the initial conditions of Eq. (4.38), the initial guess approximation of uðx; tÞ is uð0;1Þ ðx; tÞ ¼ et þ et x. Therefore, the RPS solution of Eqs. (4.37) and (4.38) takes the form

uðx; tÞ ¼ et þ et x þ

1 X

f i0 ðtÞ

i¼1

xi a

Cð1 þ iaÞ

:

ð4:40Þ

Whilst the (k, 0)-truncated series of uðx; tÞ and the (k, 0)-truncated residual function that are derived from Eqs. (3.4) and (3.6) can be formulated, respectively, in form of

uðk;lÞ ðx; tÞ ¼ et þ et x þ

k X f i0 ðtÞ i¼1

Resðk;lÞ ðx; tÞ ¼ Dax uðk;lÞ

xi a ; Cð1 þ iaÞ

@ 2 uðk;lÞ xa1 et : @t2

ð4:41Þ

ð4:42Þ

According to the RPS procedure and without the loss of generality for the remaining computations and results, the following are the first four unknown coefficients in the multiple FPS expansion of Eq. (4.39): f 10 ðtÞ ¼ et , f 20 ðtÞ ¼ et , f 30 ðtÞ ¼ et , and f 40 ðtÞ ¼ et . Therefore, the ð4; 0Þ -truncated series solution of Eqs. (4.37) and (4.38) can be written as

uð4;0Þ ðx; tÞ ¼ et þ et x et

xa x2 a x3 a x4 a : þ et et þ et Cð1 þ aÞ Cð1 þ 2aÞ Cð1 þ 3aÞ Cð1 þ 4aÞ

ð4:43Þ


13


Consequently, the components of the RPS approximate solution are obtained as far as we like. In fact, this series is exact to the last term, as one can verify, of the multiple FPS of the exact solution can be collected to discover that the approximate solution of Eqs. (4.37) and (4.38) has the general pattern form which is coinciding with the exact solution in terms of infinite series

uðx; tÞ ¼ et x þ et

1 X ð1Þi i¼0

xia

Cð1 þ iaÞ

ð4:44Þ

:

As a special case when a ¼ 2, the RPS solution of Eqs. (4.37) and (4.38) has the general pattern form which is coinciding with the exact solution in terms of elementary function

uðx; tÞ ¼ et x þ et cos x:

ð4:45Þ

Accuracy refers to how closely a computed or measured value agrees with the true value. To show this accuracy for Application 4.4, we report the consecutive error which is defined by Conðk;lÞ ðx; tÞ ¼ juðkþ1;lÞ ðx; tÞ uðk;lÞ ðx; tÞj, where x; t P 0 and uðk;lÞ is the ðk; lÞ-truncated series of uðx; tÞ obtained from the RPS method. Anyhow, while one cannot know the error without knowing the solution, in most cases the consecutive error can be used as a reliable indicator in the iteration progresses. In Table 5, the numerical values of consecutive errors for the two consecutive approximate consecutive solutions have been calculated when a ¼ 1:25; 1:5; 1:75; 2 and for various x and t on the domain ½0; 1 ½0; 1, to measure the difference between consecutive solutions obtained from the ð8; 0Þ-RPS approximate solution. The computational results of Table 5 provide a numerical estimate for the convergence of the RPS method. Also, it is clear that the accuracy obtained using present method is in advanced by using only few terms approximations. In addition, we can conclude that higher accuracy can be achieved by evaluating more components of the solution. On the other hand, based on this heuristic, we terminate the iteration in our method. Application 4.5. Consider the following nonhomogeneous time-fractional Navier-Stokes equation:

@ a uðx; tÞ @ 2 uðx; tÞ 1 @uðx; tÞ ¼pþ þ ¼ 0; a @t @x2 x @x

p 2 R; x; t > 0; 0 < a 6 1;

ð4:46Þ

subject to the nonhomogeneous initial condition

uðx; 0Þ ¼ 1 x2 :

ð4:47Þ

Table 5 The numerical values of the consecutive errors for Application 4.4 at ðk; lÞ ¼ ð8; 0Þ.

a ¼ 1:5

a ¼ 1:75

a¼2

0

2:22045 10

16

0

0

0

0.25

2:22045 1016

0

0

0

0.5

4:44089 1016

0

0

0

0.75

4:44089 1016

0

0

0

1

4:44089 1016

0

0

0

0

4:52527 1013

1:11022 1016

0

0

0.25

5:81091 1013

2:22045 1016

0

0

0.5

7:46070 1013

2:22045 1016

0

0

0.75

9:57900 1013

2:22045 1016

0

0

1

1:23013 1012

4:44089 1016

0

0

11

14

x

t

0.2

0.4

0.6

0.8

1

a ¼ 1:25

0

4:33177 10

0

0

0.25

5:56211 1011

5:61773 1014

0

0

0.5

7:14189 1011

7:21645 1014

0

0

0.75

9:17035 1011

9:25926 1014

0

0

1

1:17750 1010

1:19016 1013

0

0

0

1:10212 109

2:12941 1012

2:88658 1015

0

0.25

1:41514 109

2:73415 1012

3:66374 1015

0

0.5

1:81708 109

3:51086 1012

4:66294 1015

0

0.75

2:33318 109

4:50795 1012

6:21725 1015

0

1

2:99586 109

5:78826 1012

7:77156 1015

0

0

11

14

1:11022 1016

13

4:37428 10

1:35664 10

8

0.25

1:74196 10

8

1:23457 10

2:22045 1016

0.5

2:23672 108

7:13969 1011

1:58540 1013

2:22045 1016

8

11

13

2:22045 1016

2:61346 1013

4:44089 1016

0.75

2:87201 10

1

3:68773 108

4:33044 10 5:56040 10

11

9:16754 10

1:17714 1010

9:61453 10

2:03504 10


14


According to Remark 2.1 and the previous discussions, we assume that the multiple FPS solution of Eqs. (4.46) and (4.47) takes the following expansion form:

uðx; tÞ ¼ 1 x2 þ

1 X f i ðxÞ i¼1

t ia : Cð1 þ iaÞ

ð4:48Þ

Depending on the RPS approach, the initial guess approximation of uðx; tÞ is of the form u0 ðx; tÞ ¼ 1 x2 , while on the other hand, the kth-truncated series of uðx; tÞ and the kth-truncated residual function are given, respectively, as follows:

uk ðx; tÞ ¼ 1 x2 þ

k X f i ðxÞ i¼1

Resk ðx; tÞ ¼ Dat uk ðx; tÞ

t ia ; Cð1 þ iaÞ

ð4:49Þ

@ 2 uðx; tÞ 1 @uðx; tÞ p: @x2 x @x

ð4:50Þ

According to the RPS procedure and without the loss of generality for the remaining computations and results, we can conclude that the unknown coefficients f k ðtÞ of Eq. (4.48) should take the following values f 1 ðxÞ ¼ ðp 4Þ and f k ðtÞ ¼ 0; k ¼ 2; 3; . . .. Consequently, the components of the RPS approximate solution are obtained as far as we like. In fact, this series is exact to the last term, as one can verify, of the multiple FPS of the exact solution can be collected to discover that the exact solution of Eqs. (4.46) and (4.47) has the general pattern form which is coinciding with the exact solution

uðx; tÞ ¼ 1 x2 þ ðp 4Þ

ta ; Cð1 þ aÞ

ð4:51Þ

which is the same solution that obtained by the ADM [26]. As a special case when a ¼ 1, the RPS solution of Eqs. (4.46) and (4.47) has the general pattern form which is coinciding with the exact solution in terms of elementary function uðx; tÞ ¼ 1 x2 þ ðp 4Þt. Application 4.6. Consider the following nonhomogeneous time-fractional equation:

@ a uðx; tÞ @ 2 uðx; tÞ @ 2 uðx; tÞ ¼ þt þ uðx; tÞ xt; a 2 @t @x @t@x

x 2 R; t P 0; 1 < a 6 2;

ð4:52Þ


uðx; 0Þ ¼ 1; ut ðx; 0Þ ¼ x:

ð4:53Þ

According to Eq. (3.3) with u0 ðxÞ ¼ 1 and u0 ðxÞ ¼ x, and considering the nonlinear differential operator L½uðx; tÞ ¼ @

2

uðx;tÞ @x2

2

uðx;tÞ þt @ @t@x þ uðx; tÞ xt, we assume that the series solution of Eqs. (4.52) and (4.53) can be written as

uðx; tÞ ¼ 1 þ xt þ

1 X 1 X f ij ðxÞ i¼1 j¼0

tjþia : Cð1 þ j þ iaÞ

ð4:54Þ

According to the RPS procedure and without the loss of generality for the remaining computations and results, we can conclude that the unknown coefficients f ij ðxÞ of Eq. (4.54) should take the following values f ij ðxÞ ¼ 1 for all i ¼ 0; 1; 2; . . . and j ¼ 0; 1. Consequently, the components of the RPS approximate solution are obtained as far as we like. In fact, this series is exact to the last term, as one can verify, of the multiple FPS of the exact solution can be collected to discover that the exact solution of Eqs. (4.52) and (4.53) has the general pattern form which is coinciding with the exact solution in terms of infinite series

uðx; tÞ ¼ 1 þ xt þ

ta t 1þa t 2a t1þ2a þ þ þ þ : Cð1 þ aÞ Cð2 þ aÞ Cð1 þ 2aÞ Cð2 þ 2aÞ

ð4:55Þ

In a particular case, as a special case when a ¼ 2, the RPS solution of Eqs. (4.52) and (4.53) has the general pattern form which is coinciding with the exact solution in terms of elementary function uðx; tÞ ¼ t þ xt þ et . 5. Conclusion In this letter, we have developed a new analytical iterative method, so-called RPS, for solving initial value problems of linear PDEs of fractional order in the Caputo sense. The present technique is performed based on mainly the generation of residual error function and then applying the generalized Taylor series formula. The RPS does not require linearization, perturbation, or discretization of the variables, it is not affected by computation round off errors, and one is not faced with necessity of large computer memory and time. The main advantage of this method is the simplicity in computing the coefficients of terms of the series solution by using the differential operators only and not as the other well-known analytic techniques such as, ADM, HAM, or VIM that need the integration operators which is difficult in the fractional case. Anyhow, using



15

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