An efficient collocation method based on Legendre and Romanovski polynomials for solving Riesz distributed fractional differential equations

In this article, we provide effective numerical solutions with high accuracy and exponential convergence for Riesz distributed fractional differential equations. In this paper, in order to solve numerically initial–boundary value problems of RDFDEs in one and two dimensional, we propose and explore a novel collocation approach in two successive steps. The first stage handles the spatial discretization (one and two dimensional spaces), and primarily relies on the shifted Legendre Gauss–Lobatto collocation method. The spatial derivatives that show up in the RDFDEs and the approximate solution are evaluated using an expansion of shifted Legendre polynomials. After that, we reduce the equation and associated conditions to a system of fractional differential equations (SFDEs) for these coefficients. The second step is to propose a Romanovski Gauss–Radau collocation approach for temporal discretization, to reduce such system into a system of algebraic equations which is far easier to be solved. We effectively solved one and two-dimensional RDFDEs using the suggested collocation strategy in both spatial and temporal discretizations, and provided examples to numerically verify the spectral effectiveness and accuracy of the suggested algorithm.