Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations

Hendy, Ahmed S.; Zaky, Mahmoud A.

doi:10.1007/s00366-020-01095-8

Cited by 35 publications

(19 citation statements)

References 37 publications

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“…For time-dependent partial differential equations, if the spectral scheme is used in spatial, the difference scheme is usually adopted in time. However, when the exact solution is very smooth, the accuracy of the approximate solution would be limited by the finite difference scheme in time [35][36][37][38][39]. The aim of this section is to introduce spectral Galerkin methods based on Romanovski-Jacobi polynomials for the approximation of solutions of the Cauchy problem for the linear hyperbolic equations in one and two space dimensions.…”

Section: Applications To First-order Hyperbolic Equationsmentioning

confidence: 99%

On Romanovski–Jacobi polynomials and their related approximation results

Abo‐Gabal

Zaky

Hafez

et al. 2020

Numerical Methods Partial

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The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F-distribution over the positive real line. We introduce some basic properties of the Romanovski-Jacobi polynomials, the Romanovski-Jacobi-Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski-Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes. KEYWORDS differentiation matrices, finite orthogonal polynomials, Gauss-type quadrature, Romanovski-Jacobi polynomials, spectral methods 1 INTRODUCTION The different families of classical orthogonal polynomials are nowadays part of the basic mathematical machinery of numerous physical, engineering, and mathematical algorithms and methodologies [1-3]. The classical hypergeometric polynomials of Hermite, Laguerre, and Jacobi are infinite types

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Section: Applications To First-order Hyperbolic Equationsmentioning

confidence: 99%

On Romanovski–Jacobi polynomials and their related approximation results

Abo‐Gabal

Zaky

Hafez

et al. 2020

Numerical Methods Partial

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“…Similarly, an application of that inequality to discuss the convergence and stability for fractional in time and space fractional schrödinger equation with smooth solutions was proposed in [25]. An efficient finite difference/spectral method to solve a coupled system of nonlinear multi-term time-space fractional diffusion equations is introduced in [13]. Based on the L1 formula on nonuniform meshes for time stepping and the Legendre-Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed.…”

Section: Introductionmentioning

confidence: 99%

A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

Hendy

Zaky

Staelen

2021

Applied Numerical Mathematics

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“…A high-order efficient difference/Galerkin spectral approach was proposed in [35] for solving the time-space fractional Ginzburg-Landau equation. Hendy and Zaky [36] proposed a finite difference/spectral method based on the L1 formula on nonuniform meshes for time stepping and the Legendre-Galerkin spectral approach for solving a coupled system of nonlinear multiterm timespace fractional diffusion equations.…”

Section: Introductionmentioning

confidence: 99%

Interpolating Stabilized Element Free Galerkin Method for Neutral Delay Fractional Damped Diffusion-Wave Equation

Abbaszadeh

Dehghan

Zaky

et al. 2021

Journal of Function Spaces

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A numerical solution for neutral delay fractional order partial differential equations involving the Caputo fractional derivative is constructed. In line with this goal, the drift term and the time Caputo fractional derivative are discretized by a finite difference approximation. The energy method is used to investigate the rate of convergence and unconditional stability of the temporal discretization. The interpolation of moving Kriging technique is then used to approximate the space derivative, yielding a meshless numerical formulation. We conclude with some numerical experiments that validate the theoretical findings.

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Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations

Cited by 35 publications

References 37 publications

On Romanovski–Jacobi polynomials and their related approximation results

On Romanovski–Jacobi polynomials and their related approximation results

A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

Interpolating Stabilized Element Free Galerkin Method for Neutral Delay Fractional Damped Diffusion-Wave Equation

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