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

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.

Download Full-text

A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

Advances in Difference Equations ◽

10.1186/s13662-020-03061-6 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Muhammad Asim Khan ◽

Norhashidah Hj. Mohd Ali ◽

Nur Nadiah Abd Hamid

Keyword(s):

Finite Difference ◽

Stokes Problem ◽

Stability And Convergence ◽

Second Grade ◽

Group Method ◽

Two Dimensional ◽

Second Grade Fluid ◽

The Matrix ◽

Grade Fluid ◽

Generalized Second Grade Fluid

Abstract In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

Download Full-text

Advanced Implicit Meshless Approaches for the Rayleigh–Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

International Journal of Computational Methods ◽

10.1142/s0219876218500329 ◽

2018 ◽

Vol 15 (05) ◽

pp. 1850032 ◽

Cited By ~ 1

Author(s):

Xiaolei Bi ◽

Shanjun Mu ◽

Qingxia Liu ◽

Quanzhen Liu ◽

Baoquan Liu ◽

...

Keyword(s):

Fractional Derivative ◽

Bounded Domain ◽

Stokes Problem ◽

Second Order ◽

Second Grade ◽

Second Grade Fluid ◽

Discrete Scheme ◽

First Order ◽

Grade Fluid ◽

Generalized Second Grade Fluid

To solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain is important in the research for diffusion processes. In this paper, novel implicit meshless approaches based on the moving least squares (MLS) approximation for spatial discretization and two different time discrete schemes, which are the first-order semi-discrete scheme and the second-order semi-discrete scheme for time, are developed for the numerical simulation of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain. Based on these two time discretization schemes, the newly developed meshless approaches will have the first-order and the second-order accuracy in time, respectively. The stability and convergence of the implicit MLS meshless approaches are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approaches. It has found that the newly developed meshless approaches are accurate and convergent for fractional partial differential equations (FPDEs). Most importantly, the meshless approaches are robust for arbitrarily distributed nodes and complex domains.

Download Full-text

The Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative model

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2005.09.007 ◽

2006 ◽

Vol 7 (5) ◽

pp. 1072-1080 ◽

Cited By ~ 83

Author(s):

Fang Shen ◽

Wenchang Tan ◽

Yaohua Zhao ◽

Takashi Masuoka

Keyword(s):

Fractional Derivative ◽

Stokes Problem ◽

Second Grade ◽

Second Grade Fluid ◽

Fractional Derivative Model ◽

Grade Fluid ◽

Generalized Second Grade Fluid

Download Full-text

On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on ℝ N

Advances in Nonlinear Analysis ◽

10.1515/anona-2021-0211 ◽

2021 ◽

Vol 11 (1) ◽

pp. 580-597

Author(s):

Jia Wei He ◽

Yong Zhou ◽

Li Peng ◽

Bashir Ahmad

Keyword(s):

Fractional Derivative ◽

Blow Up ◽

Stokes Problem ◽

Second Grade ◽

Laplacian Operator ◽

Well Posedness ◽

Second Grade Fluid ◽

Liouville Fractional Derivative ◽

Grade Fluid ◽

Generalized Second Grade Fluid

Abstract We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝ N , which is derived from a non-Newtonain fluid for a generalized second grade fluid with Riemann-Liouville fractional derivative. We show that a solution operator involving the Laplacian operator is very effective to discuss the proposed problem. In this paper, we are concerned with the global/local well-posedness of the problem, the approaches rely on the Gagliardo-Nirenberg inequalities, operator theory, standard fixed point technique and harmonic analysis methods. We also present several results on the continuation, a blow-up alternative with a blow-up rate and the integrability in Lebesgue spaces.

Download Full-text