An Implicit Numerical Approach for 2D Rayleigh Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

Anam Naz,Ashraf Elfasakhany,Ahmed A Al-Ghamdi,Umair Ali,Khadiga Ahmed Ismail,Abdullah G Al-Sehemi

doi:10.3390/fractalfract5040283

Anam Naz, Ashraf Elfasakhany + Show 4 more

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https://doi.org/10.3390/fractalfract5040283

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Abstract

In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme can be applied to the fractional RSP-HGSGF. Numerical experiments will be conducted, to show that the scheme is stress-free to implement, and the outcomes reveal the ideal execution of the suggested technique. The Fourier series will be used to examine the proposed scheme stability and convergence. The technique is stable, and the approximation solution converges to the exact result. To demonstrate the applicability and viability of the suggested strategy, a numerical demonstration will be provided.

Highlights

Fractional calculus is associated with the study and application of arbitrary order derivatives and integrals
The discussion between Leibniz and L’ Hospital at the end of the seventeenth century had the first discussion about fractional calculus
Many books on fractional calculus and its applications have been authored by scholars such as Ross and Miller [1], Spanier and Oldham [2], Podlubny [3] is the most well-known book in the field of fractional calculus, and Samko et al [4] explains the underlying theory of fractional calculus, as well as its applications and solutions

Summary

Introduction

Fractional calculus is associated with the study and application of arbitrary order derivatives and integrals. The presentations of fractional-order calculus in many areas of science and engineering, including geometric phenomena, have aroused great interest in this field. The discussion between Leibniz and L’ Hospital at the end of the seventeenth century had the first discussion about fractional calculus. The Erdelyi, Abel, Riemann, Laplace, Heaviside, Levy, Liouville, Riesz, Grünwald, Letnikov, and Fourier are the of the great mathematician who worked on it and contributed. The use of fractional-order integrals and derivatives plays a significant part in the solution of some chemical problems, and this field has received more attention since 1968. Many books on fractional calculus and its applications have been authored by scholars such as Ross and Miller [1], Spanier and Oldham [2], Podlubny [3] is the most well-known book in the field of fractional calculus, and Samko et al [4] explains the underlying theory of fractional calculus, as well as its applications and solutions

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