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.
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 [
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) [
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
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 Taucollocations method is required. By doing so, the proposed system (
The manuscript is organized as follows: Section
The MittagLeffler function subjected to one parameter is recalled by
The fractional integral of RiemannLiouville type of order
For a given function
Hence, it follows that
The solution of FDE
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 RiemannLiouville derivatives have certain disadvantages when trying to model realworld phenomena with fractional differential equations. For further details, see [
The wellknown Legendre polynomials described over
Consider a sufficiently smooth function
Extending the notion of the operational matrices of fractional order integration and differentiation from one dimension [
The operational matrix corresponding to fractional integration of order
Consider
The fractional order derivative of
Take
The fractional derivative with order
To prove this theorem it can be easily followed from the proof of Theorem
The fractional derivative of
Take
Thanks to the orthogonality conditions and convolution theorem of Laplace transform, evaluating the integrals (
Thanks to the operational matrices established in previous section, we now in position to obtain numerical solutions of the proposed coupled system (
With the help of operational matrices, we consider the approximations as
Simplifying (
Here, we remark that we have used a machine type DESKTOPV8125H8 with processor intel(R) Core (TM) i53210M CPU
This section is concerning to the numerical test problems and their visualization.
Let the coupled system of FPDEs given as
Further in Table
Absolute error at various values of



















































Absolute error and CPU times at various values of

CPU time 

CPU time 















































Evaluation of numerical and exact solutions
Absolute error in
To support the aforesaid established results, we consider the following problem:
Maximum absolute error at



















































Absolute error and CPU times at various values of

CPU time 

CPU time 















































Evaluation of exact and numerical solutions
The maximum absolute error in
Consider another coupled system of FPDEs as
CPU time for test Problem
Maximum absolute error in



















































Maximum absolute error and CPU times in

CPU time 

CPU time 















































Evaluation between exact and approximate solutions
Maximum absolute error in
In this article we have developed an efficient numerical technique by using shifted Legendre polynomials. A new operational matrix of mixed partial derivative
The authors declare that no conflicts of interest exist regarding this manuscript.
Both authors equally contributed to this paper and approved the final version.
This work has been supported by the National Natural Science Foundation of China (11571378).