Jacobi Spectral Galerkin Method for Distributed-Order Fractional Rayleigh–Stokes Problem for a Generalized Second Grade Fluid

Distributed-order fractional differential operators provide a powerful tool for mathematical modeling of multiscale multiphysics processes, where the differential orders are distributed over a range of values rather than being just a fixed fraction. In this work, we consider the Rayleigh-Stokes problem for a generalized second-grade fluid which involves the distributed-order fractional derivative in time. We develop a spectral Galerkin method for this model by employing Jacobi polynomials as temporal and spatial basis/test functions. The suggested approach is based on a novel distributed order fractional differentiation matrix for Jacobi polynomials. Numerical results for one- and two-dimensional examples are presented illustrating the performance of the algorithm. The results show that our scheme can achieve the spectral accuracy for the problem under consideration with smooth solution and allows a great flexibility to deal with multi-dimensional temporally-distributed order fractional Rayleigh-Stokes problems as the global behavior of the solution is taken into account.