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

• Published in: Applied numerical mathematics Vol. 161; pp. 244 - 274
• Main Authors: Srivastava, Nikhil, Singh, Aman, Kumar, Yashveer, Singh, Vineet Kumar
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2021
• Subjects: Matrix transform method; Operational matrix; Optimal error bound; Riesz-space fractional advection-dispersion equation; Riesz-space fractional diffusion equation; Unconditional stability; Operational matrix; Unconditional stability; Matrix transform method; Optimal error bound; Riesz-space fractional advection-dispersion equation; Riesz-space fractional diffusion equation
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2020.10.032

•New numerical schemes are proposed for solving the Riesz-space fractional partial differential equation.•The matrix transform method in space and operational matrix method in the time domain is applied.•Optimal error bound and stability analysis are investigated for the schemes.•Numerical stability of the schemes is verified by applying some noise.•The effectiveness and accuracy of the numerical schemes are discussed. 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 L2-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.