ON APPROXIMATE SOLUTIONS FOR TIME-FRACTIONAL DIFFUSION EQUATION
1Department of Mathematics, College of Science, Qassim University, Saudi Arabia
ABSTRACT
In the last decades differential equations involving fractional derivatives and integrals have been studied by many researchers. Due to their ability to model more adequately some phenomena, fractional partial differential equations have been used in numerous areas such as finance, hydrology, porous media, engineering and control systems, etc. Numerical schemes based on rotated finite difference approximation have been proven to work well in solving standard diffusion equations. However, the formulation of these strategies on time fractional diffusion counterpart is still at its infancy. A well-designed preconditioning for these types of problems reduces the number of iterations to reach convergence. In this research work, we have derived new preconditioned fractional rotated finite difference method for solving 2D time-fractional diffusion equation. Numerical experiments are conducted to examine the effectiveness of the proposed method.
Keywords:Preconditioned rotated method Time-fractional diffusion Equation.
ARTICLE HISTORY: Received:23 August 2018Revised:28 September 2018 Accepted:5 November 2018Published:30 November 2018.
Contribution/ Originality:This study contributes in the existing literature about the foundation of fast iterative schemes from the preconditioned methods for solving the time-fractional diffusion equation. It is one of the few studies which combine a suitable pre-conditioner matrix with the rotated iterative scheme as a way to further improve the convergence rate of the method in solving the 2D time-fractional diffusion equation.
1. INTRODUCTION
The importance of this study lies in the various applications of fractional partial differential equations (FPDE's) in finance, physics, image processing and engineering [1 , 2
]. It is well known that FPDE's is a generalized of the classical partial differential equations (PDE's). As a result of that there is no general method that can be used in solving FPDE's same as classical PDE's. Approximation methods such as finite difference methods have played important role for solving FPDE's in the last few years [3
, 4
]. It is noteworthy to observe that the finite difference schemes derived from skewed (rotated) difference operators have been extensively investigated over the years for solving FPDE's. These iterative methods have been shown to be much faster than the methods based on the standard five-point formula which is due to the formers’ overall lower computational complexities (Saeed and Ali [5
]; Ali and Saeed [6
]; Saeed and Ali [7
]. In Saeed [8
] the preconditioned rotated finite difference method applied successfully for solving fractional elliptic partial differential equations and the reveal results was very encouraging.
This work involves an investigation on the utilization of the new preconditioned fractional rotated finite difference method for solving 2D Time-Fractional Diffusion Equations. An outline of this paper is as follows. In Section 2, the proposed accelerated version of fractional rotated five point’s approximation method will be formulated. The numerical results will be presented to show the efficiency of the new proposed methods in Section 3. Finally, Conclusion and Future work are given in Section 4.
2. FORMULATION OF THE PROPOSED PRECONDITIONED ITERATIVE METHOD
Consider the following time fractional diffusion equation
3. NUMERICAL RESULTS
In this section, we present numerical results for the proposed method applied to two particular examples. The first problem as the following [16]:
4 CONCLUSION AND FUTURE WORK
In this study, we have introduced new preconditioned iterative methods based on fractional rotated finite difference method for solving 2D time-fractional diffusion equation. From observation of all experimental results, it can be conclude that the proposed P2FRFD method requires less time and iterations number when compared to FRFD and P1FRFD methods with same levels of precision. Therefore, the proposed scheme P2FRFD may be a good alternative to solve this type of equations and many other numerical problems. Numerical results strongly suggest that the efficiency of the proposed preconditioning methods. The convergence analysis of the present iterative method regarding solutions for 2D time-fractional diffusion equation is currently under study. Furthermore, the idea of this proposed method can be extended to group iterative solver which will be reported separately in the future.
| Funding: The author gratefully acknowledges Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1123-cos-2016-12-s) during the academic year 1438 AH/2017AD). |
| Competing Interests: The author declares that there are no conflicts of interests regarding the publication of this paper. |
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