Efficient numerical algorithms for Riesz-space fractional partial differential equations based on finite difference/operational matrix

In this paper, we construct two efficient numerical schemes by combining the finite difference method and operational matrix method (OMM) to solve Riesz-space fractional diffusion equation (RFDE) and Riesz-space fractional advection-dispersion equation (RFADE) with initial and Dirichlet boundary conditions. We applied matrix transform method (MTM) for discretization of Riesz-space fractional derivative and OMM based on shifted Legendre polynomials (SLP) and shifted Chebyshev polynomial (SCP) of second kind for approximating the time derivatives. The proposed schemes transform the RFDE and RFADE into the system of linear algebraic equations. For a better understanding of the methods, numerical algorithms are also provided for the considered problems. Furthermore, optimal error bound for the numerical solution is derived, and theoretical unconditional stability has been proved with respect to L-2-norm. The stability of the schemes is also verified numerically. The schemes are observed to be of second-order accurate in space. The effectiveness and accuracy of the schemes are tested by taking two numerical examples of RFDE and RFADE and found to be in good agreement with the exact solutions. It is observed that the numerical schemes are simple, easy to implement, yield high accurate results with both the basis functions. Moreover, the CPU time taken by the schemes with SLP basis is very less as compared to schemes with SCP basis. (c) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

This paper proposes two efficient numerical schemes for solving Riesz-space fractional diffusion equation and Riesz-space fractional advection-dispersion equation. These schemes transform the equations into systems of linear algebraic equations and achieve second-order accuracy in space. Numerical results show that the schemes are simple, easy to implement, yield high accuracy, and require less CPU time when based on SLP basis.