Two shifted Jacobi-Gauss collocation schemes for solving two-dimensional variable-order fractional Rayleigh-Stokes problem

Because of the non-local properties of fractional operators, obtaining the analytical solutions of partial differential equations with fractional variable order is more challenging. Efficiently solving these equations naturally becomes an urgent topic. This paper reports an efficient numerical solution of the Rayleigh-Stokes (R-S) problem with variable-order fractional derivative for a heated generalized second grade fluid. The shifted Jacobi polynomials are employed as basis functions for the approximate solution of the aforementioned problem in a bounded domain, and the variable-order derivative is given by the means of Riemann-Liouville sense. The proposed method is a combination of the shifted Jacobi-Gauss collocation (SJ-G-C) approach for the spatial discretization and the shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) approach for temporal discretization. The aforementioned problem is then reduced to a problem that consists in a system of easily solvable algebraic equations. Finally, numerical problems are presented to show the effectiveness of the proposed numerical method.