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.

Comment on the paper “an explicit‐implicit numerical scheme for time fractional boundary layer flows, International Journal for Numerical Methods in Fluids, 2022, 94:920–940”

“…( 68) is not compatible with the term D 𝛼 t 𝜙 in eq. ( 78) in Nawaz et al 1 Papers with the same errors have been criticized by the present author in Pantokratoras 2 and in Pantokratoras. 3…”

“…In a Physics equation all terms must have the same units and the equation ( 5) is correct only for 𝛼 = 1. However the results in figures 7,8,9,10 and 11 in Nawaz et al 1 correspond to 𝛼 = 0.90 and in figs 12, 13 and 14 in Nawaz et al 1 the results correspond to 𝛼 = 0.97 and these results are wrong.…”

“…( 67) is not compatible with the term D 𝛼 t 𝜃 in eq. ( 77) in Nawaz et al 1 The term 𝜕C 𝜕t * in eq. ( 68) is not compatible with the term D 𝛼 t 𝜙 in eq.…”

A Computational Approach to a Mathematical Model of Climate Change Using Heat Sources and Diffusion

“…The numerical approaches that have been provided are versatile enough to be applied to a variety of time-dependent problems whose spatial derivative terms can be of either integer or 𝑞𝑡ℎ order. In addition, the proposed numerical approach has the potential to be utilized in epidemiological illness models as well as other types of issues in fractional calculus [33,34]. The limitation of most of the proposed schemes is their conditional stability.…”

Finite difference schemes for MHD mixed convective Darcy–forchheimer flow of Non-Newtonian fluid over oscillatory sheet: A computational study

“…An iterative scheme was also adopted to solve discretized or difference equations obtained by proposing a scheme on the considered system of ODEs. Following this research, different applications for the current approach may be developed [48][49][50][51][52]. The main concluding points can be expressed as.…”