Advances in Mathematical Physics
Volume 2017 (2017), Article ID 1535826, 14 pages
https://doi.org/10.1155/2017/1535826

Research Article

Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

Yongjin Li¹ and Kamal Shah²

¹Department of Mathematics, Sun Yat-Sen University, Guangzhou, China
²Department of Mathematics, University of Malakand, Chakdara Dir (L), Khyber Pakhtunkhwa, Pakistan

Correspondence should be addressed to Kamal Shah; kamalshah408@gmail.com

Received 26 May 2017; Revised 11 July 2017; Accepted 16 July 2017; Published 20 September 2017

Academic Editor: Rehana Naz

Copyright © 2017 Yongjin Li and Kamal Shah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

1. Introduction

Recently fractional calculus has attracted the attention of many researchers as it has many applications in various disciplines of applied sciences and engineering (we refer to [1–4] and the references therein). In fact most of the engineering and physical processes can be modeled and well explained by using a system of fractional order differential and integral equations more realistically as compared to system of conventional differential and integral equations, (see [1, 5–15] and the references therein). The area of fractional calculus devoted to the existence and uniqueness of positive solution to (FDEs) and (FPDEs) is well studied and a lot of research work is available on it (we refer to [16–20]). In the last two decades, the area dealing with numerical solutions of FDEs and FPDEs has attracted the attention of many researchers. Researchers are taking keen interests in developing numerical procedures for FPDEs. Here, it is remarkable that most of the problems of applied sciences can be modeled by using PDEs and their systems. The study of coupled systems of PDEs can widely be found in engineering and biomechanics and other disciplines. For example, in biomechanics when modeling the phenomena of electrical activity in the heart, the resulting equations are coupled systems of PDEs (see, e.g., [21–23]). Coupled systems of PDEs also occur in modeling of some chemical and material engineering processes such as system containing a continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) in series (see [24, 25]). Various applications of coupled systems can be found in solid state physics and mechanics; for example, dynamics of multideformable bodies coupled by standard light fractional order discrete continuum layers is described by coupled systems of fractional order partial differential equations [7, 26–28]. The coupled systems of PDEs also appear in modeling of some important electromagnetic and gravitational problems, (see, e.g., [29, 30]).

Most of the FPDEs do not have exact analytical solutions; therefore researchers need some appropriate numerical technique for the approximate solutions of such types (FPDEs). For the numerical solutions numerous techniques are available in the literature; for example, some of them are eigenvector expansion, Adomian decomposition method (ADM), fractional differential transform method (FDTM) [31, 32], generalized block pulse operational matrix method [33], and so on. In recent years many attentions have been devoted to develop operational matrices for fractional order differentiation and integrations. Such types of operational matrices are obtained by using Haar wavelets, Legendre wavelets, Sine Cosine, Chebyshev wavelets, and so on. Based on these operational matrices, some approximate methods for numerical solutions to FDEs have been developed (we refer to [34–36] and the references therein). In [37], the authors established a new numerical scheme based on the Haar wavelet for solving (FPDEs). More recently, the authors [38] developed some new results related to the Jacobi polynomials for operational matrices. All the operational matrices methods are used to solve FDEs and FPDEs. As much as we know they are not well studied and neither is properly applied to solve coupled systems of PFDEs. To the best of our knowledge very few articles are devoted to the numerical solutions of coupled systems of integral equations and ordinary and partial differential equations of fractional order; for details see [39–41].

This manuscript is devoted to bring out the numerical solutions to a coupled system of FPDEs on an unbounded domain. The concerned coupled system contains mixed partial order derivative with respect to , as given by

corresponding to the conditions

where represents partial derivative, , the real constants are denoted by , and

It is remarkable that the proposed system contains the coupled system investigated in [38] as a special case. If we take together with and , the proposed system becomes a coupled system of Laplace equations of fractional order by taking Further, if we involve the external (source) terms and consider , the proposed coupled system assumes the form of coupled system of Poisson’s equation of arbitrary order. Poisson’s equations have many applications in physics as well as in other various disciplines including electrostatics, mechanical engineering, and theoretical physics. For example, it has been applied to describe the potential energy field caused by a given charge or mass density distribution (see for details [42]).

With the use of shifted Legendre polynomials in two variables, some operational matrices corresponding to fractional order differentiations and integrations are developed. Thanks to these operational matrices, the coupled system under consideration is transformed to a system of Sylvester type algebraic equations. Here, we remark that no discretization like for Tau-collocations method is required. By doing so, the proposed system (1) corresponding to the initial conditions is transformed to a system of Sylvester type algebraic equations. Our proposed method is simple and there is no computational complexity in the resulting algebraic system to solve. Further, we remark that our present method is a numerical method based on shifted Jacobi polynomials while the method discussed in [43] is an iterative method based on monotone iterative technique. Further we can easily establish a simple relationship for convergence of the propped method. The aforesaid procedure has now recently being applied for some nonlinear fractional order differential equations and some fruitful results were obtained; for details see [44]. In our future work we will use operational matrices method to compute approximate solutions of nonlinear FPDEs and their system.

The manuscript is organized as follows: Section 2 is concerning some basic definitions and preliminaries results related to fractional calculus and Legendre polynomials which are needed to form the proposed method. Section 3 is concerning the operational matrices corresponding to the fractional order derivatives and integration. Further, Section 4 is related to the application of the operational matrices to covert coupled system (1) to the corresponding system of algebraic equations. Section 5 is devoted to the numerical examples, and in the last section, we have provided a brief conclusion.

2. Preliminaries

Definition 1. The Mittag-Leffler function subjected to one parameter is recalled by

The extension of (4) to two parameters is given as

Definition 2 (see [45, 46]). The fractional integral of Riemann-Liouville type of order of a function is defined as

provided that the right-hand side is pointwise defined on .

Definition 3. For a given function , the Caputo fractional derivative of order is defined as

provided that the right side is pointwise defined on , where .

Hence, it follows that

Theorem 4 (see [45]). The solution of FDE

is given by

where each is a real number for , where , such that represents integer part of .

Remark 5. We will use Caputo derivative throughout this paper, because Caputo fractional derivative has clear geometrical representations like classical derivative. Further, it turns out that the Riemann-Liouville derivatives have certain disadvantages when trying to model real-world phenomena with fractional differential equations. For further details, see [45, 47].

2.1. The Shifted Legendre Polynomials

The well-known Legendre polynomials described over are representing as

The transformation transforms the interval to and the corresponding polynomials known as shifted Legendre polynomials are given as

The orthogonality condition is

Therefore, any function can be approximated in terms of shifted Legendre polynomials as

In matrix form, we may write (14) as

where is the matrix of functions while is term column matrix. The shifted Legendre polynomials of order for two dimensions are recalled by

and the corresponding orthognathic relation is provided as

Therefore any function can be approximated in terms of as

where

In matrix notation, we may write (18) as

where the column vector of coefficients with order is represented as Also column vector of order is represented by and provided as

where .

2.2. Maximum Absolute Error ([48])

Consider a sufficiently smooth function on ; then the error in the numerical solutions is provided as

where

The above relation also holds at points for interpolating polynomial

3. Operational Matrices of Integrations and Differentiations

Extending the notion of the operational matrices of fractional order integration and differentiation from one dimension [36, 49] to two dimensions, the construction of the aforesaid operational matrices is provided as follows.

Theorem 6. The operational matrix corresponding to fractional integration of order of defined in (21) with respect to is provided as

where is the operational matrix of fractional integration given as

for ,

Proof. Consider as given in (16); then the arbitrary order integral with order of corresponding to is provided as

which gives that

Further approximating in terms of shifted Legendre polynomials, we have

where Thanks to orthogonality condition, we have

Therefore (28) yields

where

We use these notations for easiness of Thus, we received the required result

Theorem 7. The fractional order derivative of   can be written as

such that is the operational matrix corresponding to arbitrary order derivative of order with respect to provided as

where   for  ,

Proof. Take the order fractional derivative of corresponding to as

Approximate in terms of shifted Legendre polynomials as

Applying orthogonality condition, we may write

Therefore, we get

where

Denoting , such that , for easiness, we get the required.

Theorem 8. The fractional derivative with order of with respect to can be written as

such that is the operational matrix given as

  for   and

Proof. To prove this theorem it can be easily followed from the proof of Theorem 7.

Theorem 9. The fractional derivative of with respect to is provided as

where represents the operational matrix of fractional derivative given as

and   for  ,

Proof. Take given in (17) and apply fractional order derivative of corresponding to and as

We approximate in terms of shifted Legendre polynomials as

such that

Thanks to the orthogonality conditions and convolution theorem of Laplace transform, evaluating the integrals (49), we get

where

Plugging (48) in (47), we may write

Representing , and for , we obtain the required results.

4. Solutions of the Coupled Systems of Equations

Thanks to the operational matrices established in previous section, we now in position to obtain numerical solutions of the proposed coupled system (1) of FPDEs.

With the help of operational matrices, we consider the approximations as

In view of Theorem 4, we can write

Applying initial conditions, we get

where and   In simple notation, we may write

Therefore (55) yields

Thanks to (57), the other terms of system (1) can be approximated as

Putting in (1), we get

Now we write matrix form of system (59) as

which further gives

where

is a column null matrix and other null matrices are and .

Simplifying (61), we get

Therefore by using and , the aforesaid equation can be written as

where

Hence (64) is Sylvester type matrix equation, which on solving for unknown matrix and using its value in (57), we get the required numerical solution of the proposed problem.

Remark 10. Here, we remark that we have used a machine type DESKTOP-V8125H8 with processor intel(R) Core (TM) i5-3210M CPU 2.50 GHz and installed memory (RAM) is 4 GB together with 64-bit operating system for computation and numerical simulations.

5. Numerical Test Problems

This section is concerning to the numerical test problems and their visualization.

Problem 1. Let the coupled system of FPDEs given as

such that the external functions and are given as

The exact solution of the above system is

Evaluate the approximate solution of Problem 1 with our proposed method. After applying the method to the given problem, we see from Figure 1 that the considered scheme provides close agreement between numerical and exact solution. The comparison between exact and approximate solution is shown in Figure 1, while the absolute error corresponding to scale level is provided in Figure 2. Further, to check the efficiency of the method, we have also computed absolute error at various scale level and different points of the spaces as given in Table 1.
Further in Table 2, we compute the CPU time for the computational of solutions for the test Problem 1.

Table 1: Absolute error at various values of for in and of Problem 1.

Table 2: Absolute error and CPU times at various values of for in and of Problem 1.

CPU time

CPU time

Figure 1: Evaluation of numerical and exact solutions , corresponding to scale level of Problem 1.

Figure 2: Absolute error in and using of Problem 1.

Problem 2. To support the aforesaid established results, we consider the following problem:

The external source functions are provided as

where the exact solution of the Problem 2 is We approximate the given test problem by using the proposed method. In Figure 3, we have shown the comparison between exact and approximate solutions at scale level . Further, to verify the efficiency of the proposed method, also the absolute error at scale level in and has been shown in Figure 4. To demonstrate the maximum absolute error at different scale levels and various values of the space variables has been provided in Table 3. From Table 3, it is clear that the absolute error is below , which is a very small quantity. The absolute error may further be decreased by enlarging scale level. CPU time for test Problem 2 is provided in Table 4.

Table 3: Maximum absolute error at of and using various values of for Problem 2.

Table 4: Absolute error and CPU times at various values of for in and of Problem 2.

CPU time

CPU time

Figure 3: Evaluation of exact and numerical solutions , corresponding to scale level for test Problem 2.

Figure 4: The maximum absolute error in and at of Problem 2.

Problem 3. Consider another coupled system of FPDEs as

While the external source term is given by

The exact solution of Problem 3 at is

Apply the considered method to the given Problem 3 corresponding to a scale level . Plot the exact and approximate solution as in Figure 5. We see from Figure 5 that the adapted techniques work excellently and there is a close agreement between approximate and exact solutions. Further, we have computed maximum absolute error for Problem 3 in Table 5 at scale level . The maximum absolute error corresponding to the scale level has been provided in Figure 6, which is below . The absolute error can be further decreased by enlarging scale level with various values of .
CPU time for test Problem 3 is provided in Table 6.

Table 5: Maximum absolute error in and of Problem 3 at corresponding to various values of .

Table 6: Maximum absolute error and CPU times in and of Problem 3 at corresponding to various values of .

CPU time

CPU time

Figure 5: Evaluation between exact and approximate solutions for scale level of test Problem 3.

Figure 6: Maximum absolute error in and at for different values of for Problem 3.

6. Conclusion

In this article we have developed an efficient numerical technique by using shifted Legendre polynomials. A new operational matrix of mixed partial derivative is introduced to solve a multiterm coupled system of FPDEs. We have also tested our method by some test problems. The comparison between the exact and approximate solutions of the problems has been provided. We observed that the maximum absolute error is very small even at very small scale level. This minimum value of the absolute error indicates that the method is highly accurate and can be applied to a large numbers of FPDEs.

Conflicts of Interest

The authors declare that no conflicts of interest exist regarding this manuscript.

Authors’ Contributions

Both authors equally contributed to this paper and approved the final version.

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (11571378).

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