A hybrid method based on the classical/piecewise Chebyshev cardinal functions for multi-dimensional fractional Rayleigh–Stokes equations

Abstract

This study presents a numerical hybrid strategy for deriving approximate solutions to the one- and two-dimensional fractional Rayleigh–Stokes equations involving the Caputo derivative. This scheme mutually utilizes the classical and piecewise Chebyshev cardinal functions as basis functions. To this end, the operational matrices of the ordinary integral and fractional derivative of the piecewise Chebyshev cardinal functions, along with the ordinary and partial derivatives of the one- and two-variable Chebyshev cardinal functions, are derived. To create the desired approach by considering a hybrid expansion of the solution of the problem using the Chebyshev cardinal functions (for the spatial variable) and piecewise Chebyshev cardinal functions (for the temporal variable), and employing the aforementioned operational matrices, solving the problem under consideration turns into solving an algebraic system of linear equations. The convergence analysis of the established method is examined both theoretically and numerically. The accuracy and validity of the developed scheme are examined by solving several numerical examples.