An explicit‐implicit numerical scheme for time fractional boundary layer flows. (16th March 2022) - Nawaz, Yasir; Arif, Muhammad Shoaib; Abodayeh, Kamaleldin
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. Abstract : Anexplicit‐implicit three‐stage fractional numerical scheme has been proposed for solving time‐dependent partial differential equations with integer and noninteger tempered derivatives. Some core characteristic like stability and consistency of the proposed scheme are presented using Von Neumann's stability criteria. Several numerical examples for the fractional boundary layer flows were constructed. The scheme was implemented to solve these considered models and show the feasibility and high efficiency of the proposed scheme. The scheme showed faster convergence than two of the conventional existing numerical schemes. Scientific research into everyday physical phenomena became possible due to the advent of fractional calculus, which opened up new research opportunities. If the analytical solution of the Stokes model is considered via a fractional derivative, the exact solution is rare. The fractional constitutive relationship model is found to be significantly more important than the traditional constitutive relationship model for several fluids ranging from elastic to viscous materials. The fractional derivative produces extremely promising results when dealing with more complicated dynamics.