# Homotopy Series Solutions to Time-Space Fractional Coupled Systems.

1. Introduction

Fractional calculus, compared to integer calculus, was mentioned in a letter from L'Hospital to Leibniz in 1695. In the letter, L'Hospital raised a question, "what is the result of [d.sup.n]y/[dx.sup.n] if n= 1/2?" The answer of Leibniz was "[d.sup.1/2]x will be equal to x[square root of (dx:x)]. This is an apparent paradox, from which, one day useful consequences will be drawn" . Furthermore, the generalization of this framework indicates that it is more appropriate to talk about integration and differentiation of arbitrary order, such as fractional order, real number order, and even complex number order just as the development of number system. Thus, there is a basic question: "what are the definitions of fractional integral and derivative?" Or "how to define the fractional integral and derivative?" More and more mathematicians focused on this problem, like Lagrange, Laplace, Fourier, and so on. Some different fractional integrals and derivatives have been given according to different needs, like Riemann-Liouville fractional integral, Caputo fractional derivative, Weyl fractional derivative, and so on . But there are no uniform definitions of fractional integral and derivative, and the frequently used definitions are Riemann-Liouville integral and Caputo derivative.

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so on, which involve derivatives of fractional order. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of fractional differential equations is gaining much more attention. For example, in electromagnetism, Sebaa et al.  studied ultrasonic wave propagation in human cancellous bone by using fractional calculus to describe the viscous interactions between fluid and solid structure. In signal processing, Assaleh and Ahmad  proposed a new approach for speech signal modeling through using fractional calculus. Magin and Ovadia  molded the cardiac tissue electrode interface using fractional calculus. In control theory, Suarez et al.  applied fractional controllers to the path-tracking problem in an autonomous electric vehicle. In fluid mechanics, Kulish and Lage  applied fractional calculus to the solution of time-dependent, viscous-diffusion fluid mechanics problems.

In this paper, we intend to construct the approximate solutions to the nonlinear time-space fractional coupled systems. There are many effective methods to solve this problem, like Adomian decomposition method [8-10], variation iteration method , differential transform method , residual power series method [13, 14], iteration method , homotopy perturbation method , homotopy analysis method , and so on. Furthermore, for the nonlinear problem, the multiple exp-function method [18, 19], the transformed rational function method [20-22], and invariant subspace method [23, 24] are three systematical approaches to handle the nonlinear terms. The first one is to propose the exact solution of nonlinear partial differential equations by using rational function transformations. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. The second one is to consider the form of solution as rational exponential functions with unknown coefficients whose advantage is direct applicability to underlying equation. The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. Motivated by these fruitful results, Singh et al.  proposed the homotopy perturbation Sumudu transform method based on the homotopy perturbation method and Sumudu transform method and applied it to nonlinear partial differential equations. The HPSTM was extended to the time-fractional PDEs in [26, 27]. It is worth mentioning that the HPSTM is applied without any using of Adomian polynomials, over restrictive assumption or linearization, and is capable of reducing the volume of computational work as compared to the classical numerical methods while still maintaining the high accuracy of the result. Meanwhile, it is appropriate not only for strongly nonlinear system but also for weakly nonlinear system.

The rest of the paper is organized as follows. In Section 2, we introduce some concepts on fractional calculus and the Sumudu transform. In Section 3, we illustrate the basic idea of HPSTM which is applied to the time-space fractional coupled systems. In Section 4, we apply HPSTM to obtain fractional power series solutions of nonlinear time-space fractional coupled systems with initial values, and some numerical results are presented as well.

2. Preliminaries

Definition 1 (see ). A real function f(x), x > 0 is said to be in the space [C.sub.[mu]], [mu] [member of] R, if there exists a real number [rho] > [mu] such that f(x) = [x.sup.p][f.sub.1])x), where [f.sub.i] (x) [member of] C[0, [infinity]) and it is said that f(x) [member of] [C.sup.n.sub.[mu]] if [f.sup.(n)] (x) [member of] [C.sub.[mu]], n [member of] N.

Definition 2 (see ). The fractional integral of f(t) in the Riemann-Liouville (left-sided) sense is defined as

[mathematical expression not reproducible] (1)

where [alpha] [greater than or equal to] 0, f [member of] [C.sub.[mu]], [mu] [greater than or equal to] -1, and [GAMMA] is the Gamma function.

Definition 3 (see ). The fractional integral of f(x) in the Riemann-Liouville sense is defined as

[mathematical expression not reproducible] (2)

where [alpha] [greater than or equal to] 0, f [member of] [C.sub.[mu]], [mu] [greater than or equal to] -1, and [GAMMA] is the Gamma function.

Definition 4 (see ). The Caputo (left-sided) fractional derivative operator of order [alpha] [greater than or equal to] 0, of a function [mathematical expression not reproducible], is defined as

[mathematical expression not reproducible] (3)

Definition 5 (see ). The Caputo fractional derivative operator of order [alpha] [greater than or equal to] 0, of a function f [member of] [C.sup.n.sub.[mu]] ([mu] [greater than or equal to] -1, n [member of] N), is defined as

[mathematical expression not reproducible] (4)

Lemma 6 (see ). If m -1 < [alpha] [less than or equal to] m, m [member of] N, and f [member of] [C.sup.m.sub.[mu]], [mu] [greater than or equal to] - 1, one has

[mathematical expression not reproducible] (5)

In 1998, a new integral transform, named Sumudu transform, was introduced by Watugala  to study solutions of ordinary differential equations in control engineering problems. The Sumudu transform is defined over the set of functions [mathematical expression not reproducible] by the following formula:

[mathematical expression not reproducible]. (6)

Property 7 (see ). (i) The Sumudu transform satisfies linear property; that is,

S [af/(t) + bh (t)] = aS [f (t)] + bS [h (t)), (7)

where a, b are constants.

(ii) S[[t.sup.n]] = [u.sup.n] [GAMMA] (n+1), n [member of] N. (8)

Lemma 8 (see ). The Sumudu transform of the Caputo fractional derivative is

[mathematical expression not reproducible] (9)

3. Homotopy Perturbation Sumudu Transform Method

In this section, to illustrate the basic idea of this method, we consider a general nonhomogeneous fractional partial differential coupled system

[mathematical expression not reproducible] (10)

with the initial conditions

[mathematical expression not reproducible] (11)

where [alpha],[beta], [gamma] [member of] (0,1], [R.sup.i], [N.sup.i], i = 1,2,3, denote linear differential operators and nonlinear differential operators, respectively, and [g.sup.i] (x, t) are the source terms. Applying the Sumudu transform on both sides of (10) yields

[mathematical expression not reproducible] (12)

It follows from the property of the Sumudu transform in (9) that

[mathematical expression not reproducible] (13)

Furthermore, applying the inverse Sumudu transform [S.sup.-1] on both sides of (13) yields

[mathematical expression not reproducible] (14)

where [M.sup.i](x, f), i = 1,2, 3, represent the terms arising from the source terms and prescribed initial conditions; that is,

[mathematical expression not reproducible] (15)

Let us construct the homotopy perturbation equations

[mathematical expression not reproducible] (16)

where homotopy parameter p [member of] [0,1]. Suppose that [U.sup.i](x, t) and the nonlinear terms [N.sup.j][U.sup.i](x, t) can be written as

[mathematical expression not reproducible] (17)

where the coefficient polynomials [U.sup.i.sub.n] can be determined below and [H.sup.j.sub.n] ([U.sup.1], [U.sup.2], [U.sup.3]) are given by the following formulae:

[mathematical expression not reproducible] (18)

Substituting (17) into (16) gives

[mathematical expression not reproducible] (19)

Comparing the coefficients of p, we obtain the following recurrence equations:

[mathematical expression not reproducible] (20)

where n= 1,2,...

According to the homotopy perturbation method, we assume that the solution of (10)-(11) can be written as

[mathematical expression not reproducible] (21)

Setting p [right arrow] 1, the approximate solution to (10)-(11) is

[mathematical expression not reproducible] (22)

The convergence of series (21) depends on the nonlinear differential operator n. Generally, the derivative with respect to U of the nonlinear part in the splitting must be sufficiently small, since the parameter p may be relatively large; in fact we take p [right arrow] 1. The series is convergent for most cases .

Remark 9. HPSTM is applied to construct homotophy series solutions for fractional coupled systems, which has not too many overstrict assumptions compared to some classical methods.

4. Application of HPSTM to Time-Space Fractional Coupled Systems

In this section, we apply HPSTM to nonlinear time-space fractional coupled systems with initial conditions.

4.1. The Time-Space Fractional Coupled Burgers System. The Burgers equation is one of the most important partial differential equations from fluid mechanics, which not only describes many phenomena, for example, modeling the motion of turbulence , but also has many applications in science and engineering . Here we apply HPSTM to solve the following nonlinear time-space fractional coupled Burgers system:

[mathematical expression not reproducible] (23)

with the initial conditions

U (x, 0) = sin x, V(x, 0) = sin x, (24)

where 0 < [alpha], [beta], [delta], [gamma] [less than or equal to] 1, (x, i) [member of] R x [0, [infinity]).

Applying the Sumudu transform on both sides of (23) with the initial conditions, we can obtain

[mathematical expression not reproducible] (25)

The inverse Sumudu transform of (25) implies that

[mathematical expression not reproducible] (26)

Now applying the homotopy perturbation method gives

[mathematical expression not reproducible] (27)

where [H.sup.U.sub.n] (x, t) and [H.sup.V.sub.n] (x, t) are polynomials which denote the homotopy coefficients of the nonlinear term and are given by

[mathematical expression not reproducible] (28)

Set

[mathematical expression not reproducible] (29)

and then

[mathematical expression not reproducible] (30)

Comparing the coefficients of p, this gives

[mathematical expression not reproducible] (31)

Generally, we have

[mathematical expression not reproducible] (32)

where

[mathematical expression not reproducible] (33)

Hence, the series solution of (23) is

[mathematical expression not reproducible] (34)

Particularly, when [alpha] = [gamma] = [beta] = [delta] = 1, the exact solution of (23) is

U (x, i) = [e.sup.-t] sin x, V (x, i) = [e.sup.-t] sin x. (35)

Using HPSTM, when [alpha] = [gamma] = [beta] = [delta] = 1, the third approximate solution of (23) is

[mathematical expression not reproducible] (36)

In general, the limit of the approximate solution is

[mathematical expression not reproducible] (37)

which is as same as the exact solution. However, if the initial values are too complex to find the limit of the approximated solution, then we replace the exact solution by the approximated solution within a certain scale, which is useful in the application of engineering.

Thus we plot the images of the approximate solution (see Figure 1(a)), the exact solution (see Figure 1(b)), and the error function (see Figures 1(c) and 1(d)). It is clear that the error function [absolute value of ([U.sub.app] - [U.sub.ex])] depends on time t. When time t is small (e.g., t = 0.01), the error function is in the scale of [10.sup.-3] (see Figure 1(c)), which indicates that this is a good approximation in the neighbour of time 0 for system (23) with some explicit parameters. However, when time becomes large (e.g., i = 0.1), the error function tends to be large as well (see Figure 1(d)); that is to say, this method is only suitable for constructing the approximated solution around the initial data.

4.2. The Time-Space Fractional Coupled KdV System of Generalized Hirota-Satsuma Type. In this subsection, consider the time-space fractional generalization of the Hirota-Satsuma coupled KdV system

[mathematical expression not reproducible] (38)

with respect to the initial conditions

[mathematical expression not reproducible] (39)

where [mathematical expression not reproducible]. The HirotaSatsuma coupled KdV equation describes the unidirectional propagation of shallow water waves, which was initiated by Wu et al. . Further (38) becomes a generalized fractional KdV equation for U = 0 and a fractional MKdV equation for V = 0.

Applying the Sumudu transform on both sides of (38) with the initial conditions, we obtain

[mathematical expression not reproducible] (40)

The inverse Sumudu transform of (40) implies that

[mathematical expression not reproducible] (41)

Via the homotopy perturbation method, it gives

[mathematical expression not reproducible] (42)

where [H.sup.U.sub.n] (x, t), [H.sup.V.sub.n] (x, t), and [H.sup.W.sub.n](x, t) are polynomials which denote the nonlinear term, and they are given by

[mathematical expression not reproducible] (43)

Set

[mathematical expression not reproducible] (44)

and then

[mathematical expression not reproducible] (45)

Comparing the coefficients of p shows

[mathematical expression not reproducible] (46)

Generally, one has

[mathematical expression not reproducible] (47)

where

[mathematical expression not reproducible] (48)

Therefore, the approximate series solution of (38) is

[mathematical expression not reproducible] (49)

Particularly, when [alpha] = [gamma] = [delta] = [lambda] = 1, [tau] = [beta] = [sigma] = [theta] = 2/3, and the special initial value of (38) is

[mathematical expression not reproducible] (50)

then the exact solutions are

[mathematical expression not reproducible], (51)

Under these special conditions, via HPSTM, the first approximate solution of (38) is [mathematical expression not reproducible] (52)

Similarly, we obtain the following numerical results: see Figures 2(a), 2(b), 2(c), 2(d), 3(a), 3(b), 3(c), 3(d), 4(a), 4(b), 4(c), and 4(d).

4.3. The Time-Space Fractional Coupled Shallow Water System. Shallow water systems are widely used in many areas of fluid dynamics, such as multiphase flows , turbulence , and viscoelasticity . It is well known that the shallow water systems can accurately predict both the hydraulic parameters under conditions of slow erosion and low sediment concentration. Let us consider the time-space fractional coupled shallow water system

[mathematical expression not reproducible] (53)

with initial values

U (x, 0) = [a.sub.0] (x), V (x, 0) = [b.sub.0] (*), (54)

where [mathematical expression not reproducible].

Applying the Sumudu transform on both sides of (53) with the initial conditions, we obtain

[mathematical expression not reproducible] (55)

The inverse Sumudu transform of (55) implies that

[mathematical expression not reproducible] (56)

According to homotopy perturbation method, we have

[mathematical expression not reproducible] (57)

where [H.sup.U.sub.n] (x, t) and [H.sup.V.sub.n] (x, t) are polynomials of the nonlinear term and are given by

[mathematical expression not reproducible] (58)

Setting

[mathematical expression not reproducible] (59)

then we arrive at

[mathematical expression not reproducible] (60)

Comparing the coefficients of p yields

[mathematical expression not reproducible] (61)

Generally, we have

[mathematical expression not reproducible] (62)

where

[mathematical expression not reproducible] (63)

Hence, the series solution is

[mathematical expression not reproducible] (64)

4.4. The Time-Space Fractional Coupled KdV System. KdV equation plays an important role in nonlinear equations for wide applications in physics and engineering. Hirota and Satsuma  firstly found coupled KdV system to describe the iterations of water waves; meanwhile, they claimed that the system exists with a soliton solution. In , Fan and Zhang settled several kinds of solutions by an improved homogeneous method. The time-space fractional coupled KdV equation is a generalization of the classical coupled KdV equation. In this subsection, we consider the following timespace fractional coupled KdV system:

[mathematical expression not reproducible] (65)

with respect to initial values

U (x, 0) = [a.sub.0] (x), V (x,0) = [b.sub.0] (x), 66)

where [mathematical expression not reproducible], and the coefficients a, b are constants.

Applying the Sumudu transform on both sides of (65) with the initial conditions, we obtain

[mathematical expression not reproducible] (67)

The inverse Sumudu transform of (67) implies that

[mathematical expression not reproducible] (68)

Analogously, using homotopy perturbation method gives

[mathematical expression not reproducible] (69)

where [H.sup.U.sub.n] (x, t) and [H.sup.V.sub.n] (x, t) are homotopy polynomials coefficients of the nonlinear term, which are given by

[mathematical expression not reproducible] (70)

Setting

[mathematical expression not reproducible] (71)

then

[mathematical expression not reproducible] (72)

Comparing the coefficients of p,

[mathematical expression not reproducible] (73)

Generally, we get

[mathematical expression not reproducible] (74)

where

[mathematical expression not reproducible] (75)

Thus, the series solution of (65) is

[mathematical expression not reproducible] (76)

4.5. The Time-Space Fractional Coupled Whitham-BroerKaup (WBK) System. Under the Boussinesq approximation, Whitham , Broer , and Kaup  obtained the following nonlinear WBK system. In this subsection, we construct the approximate solution by the HPST method to the time-space fractional coupled WBK system.

Consider the time-space fractional coupled WBK system

[mathematical expression not reproducible] (77)

with respect to the initial conditions

U (x, 0) = [a.sub.0] (x), V(x,0) = [b.sub.0] (x), (78)

where 9 < [alpha], [beta], [gamma. [delta] [less than or equal to] 1, 1/2 < [delta], [tau] [less than or equal to] 1, 2/3 < [theta] [less than or equal to] 1, (x,t) [member of] R x [0,[infinity]), a, b, [member of] R denote different dispersive power, U = U(x, t) is the field of horizontal velocity, and V = V(x, t) is the height deviating equilibrium position of liquid.

Applying the Sumudu transform on both sides of (77) with the initial conditions, we obtain

[mathematical expression not reproducible] (79)

The inverse Sumudu transform of (79) implies that

[mathematical expression not reproducible] (80)

Using homotopy perturbation method, it leads to

[mathematical expression not reproducible] (81)

where [H.sup.U.sub.n] (x, t) and [H.sup.V.sub.n] (x, t) are polynomials of the nonlinear term and are given by

[mathematical expression not reproducible] (82)

Setting

[mathematical expression not reproducible] (83)

then

[mathematical expression not reproducible] (84)

Comparing the coefficients of p, this gives

[mathematical expression not reproducible] (85)

Generally, we have

[mathematical expression not reproducible] (86)

where

[mathematical expression not reproducible] (87)

Hence, the series solution of (77) is

[mathematical expression not reproducible] (88)

Remark 10. When alpha = [tau] = [delta] = [lambda] = [theta] = 1, [beta] [not equal to] 0, and y = 0, (77) reduces to the classical long-wave system that describes the shallow water wave with diffusion.

Remark 11. When [alpha] = [tau] = [delta] = [lambda] =[theta] = 1, [beta] = 0, and [gamma =1, (77) reduces to the variant Boussinesq system.

5. Concluding Remarks

In this paper, we apply the HPSTM to the nonlinear timespace fractional coupled equations. Applying the HPSTM, we can obtain analytic and approximate solutions to different coupled systems, for example, the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The advantage of the HPSTM is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear system. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation, or restrictive assumptions. The numerical results indicate that this method is effective and simple in constructing analytic or approximate solutions to fractional coupled systems.

https://doi.org/10.1155/2017/3540364

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors thank Dr. Yixian Gao for many useful suggestions and help. This research was supported in part by NSFC Grants 201114110, 11571065, 11671071, and 11401089, NSFJL Grants 20160520094JH and 20170101044JC, EDJLP Grant TTKH20170904KJ, and the Fundamental Research Funds for the Central Universities, 2412017FZ005.

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Jin Zhang, (1) Ming Cai, (2) Bochao Chen, (2) and Hui Wei (2)

(1) School of Mathematics, Jilin University, Changchun 130012, China

(2) School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, China

Correspondence should be addressed to Jin Zhang; jinzhang@jlu.edu.cn

Received 31 August 2017; Accepted 23 October 2017; Published 25 December 2017

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