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 modelling 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 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. Numerical results for one- and two-dimensional examples are presented to highlight the convergence rate and the flexibility of this approach. The proposed method yields an exponential rate of convergence when the solution is smooth and allows great flexibility to handle multi-dimensional temporally-distributed order fractional Rayleigh–Stokes problems. Our results confirm that nonlocal numerical methods are best suited to discretize distributed order fractional differential equations as they naturally take the global behavior of the solution into account.