An Explicit‐Implicit Numerical Scheme for Time Fractional Boundary Layer Flows

Abstract

This contribution is concerned with constructing a fractional explicit-implicit numerical scheme for solving time-dependent partial differential equations. The proposed scheme has the advantage over some existing explicit in providing better stability region. But it has one of its limitations of being conditionally stable, even having one implicit stage. For spatial discretization, a fourth-order compact scheme is considered. The stability and convergence of the proposed scheme for respectively the scalar parabolic equation and system of parabolic equations are given. For the sake of application of the scheme, fractional models of flow between parallel plates and mixed convection flow of Stokes' problems under the effects of viscous dissipation and thermal radiation are constructed. The proposed scheme for the classical model is also compared with built-in Matlab solver pdepe for solving parabolic and elliptic equations and existing numerical schemes. It is found that Matlab solver pdepe is failed to find the solution of the considered flow problem with larger values of Eckert number or coefficient of the nonlinear term. But, the proposed scheme successfully finds the solution for classical and fractional models and shows faster convergence than the existing scheme. We provide illustrative computer simulations to show the principal computational features of this approach.