Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

Hafez, Ramy M.; Hammad, Magda; Doha, E. H.

doi:10.1007/s00366-020-01180-y

Cited by 6 publications

(3 citation statements)

References 49 publications

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“…Computing the discrete inner product of (41) with 𝜂, then taking the real part of the resulting equation, we obtain…”

Section: Convergence Analysis Of the Semi-discrete Formmentioning

confidence: 99%

“…Wang [34] proposed a fast and efficient spectral-Galerkin method for the nonlinear complex Ginzburg-Landau equation involving the fractional Laplacian in R d . The space-fractional Schrödinger equation as special form of fractional Ginzburg-Landau equation was considered extensively in literature [35][36][37][38][39][40][41][42]. However, to the best of our knowledge, the L2 -1 𝜎 /Galerkin spectral method, which is an important method to solve fractional partial differential equations [43][44][45][46][47][48][49][50][51][52], has not been considered for the nonlinear time-space fractional Ginzburg-Landau equation.…”

Section: Introductionmentioning

confidence: 99%

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High‐order finite difference/spectral‐Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation

Zaky

Hendy

Macías‐Díaz

2020

Numerical Methods Partial

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This paper proposes and analyzes a high‐order difference/Galerkin spectral scheme for the time–space fractional Ginzburg–Landau equation. For the time discretization, the L2 ‐ 1σ differentiation formula is used to approximate the Caputo fractional derivative. While for the space discretization, the Legendre–Galerkin spectral method is used to approximate the Riesz fractional derivative. It is shown that the scheme is efficiently applied with spectral accuracy in space and second‐order in time. The error estimates of the solution are established by applying a fractional Grönwall inequality and its discrete form. In addition, a detailed implementation of the numerical algorithm is provided. Furthermore, numerical experiments are presented to confirm the theoretical claims. As an application of the proposed method, the effect of fractional‐order parameters on the pattern formation of time–space fractional Ginzburg–Landau equation is discussed.

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“…Computing the discrete inner product of (41) with 𝜂, then taking the real part of the resulting equation, we obtain…”

Section: Convergence Analysis Of the Semi-discrete Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High‐order finite difference/spectral‐Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation

Zaky

Hendy

Macías‐Díaz

2020

Numerical Methods Partial

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“…The essential step in all spectral approaches is to express the solution as a finite series of distinct functions. There are many different kinds of spectral approaches, such as collocation [22], tau [23], Galerkin [24], and Petrov-Galerkin [25]. After that, the coefficients will be selected to reduce the absolute error.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Amin,

Abdelkawy,

Solouma

et al. 2023

Fractal Fract

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One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.

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Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

et al. 2023

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The time-fractional advection–diffusion reaction equation (TFADRE) is a fundamental mathematical model because of its key role in describing various processes such as oil reservoir simulations, COVID-19 transmission, mass and energy transport, and global weather production. One of the prominent issues with time fractional differential equations is the design of efficient and stable computational schemes for fast and accurate numerical simulations. We construct in this paper, a simple and yet efficient modified fractional explicit group method (MFEGM) for solving the two-dimensional TFADRE with suitable initial and boundary conditions. The proposed method is established using a difference scheme based on L1 discretization in temporal direction and central difference approximations with double spacing in spatial direction. For comparison purposes, the Crank–Nicolson finite difference method (CNFDM) is proposed. The stability and convergence of the presented methods are theoretically proved and numerically affirmed. We illustrate the computational efficiency of the MFEGM by comparing it to the CNFDM for four numerical examples including fractional diffusion and fractional advection–diffusion models. The numerical results show that the MFEGM is capable of reducing iteration count and CPU timing effectively compared to the CNFDM, making it well-suited to time fractional diffusion equations.

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Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

Cited by 6 publications

References 49 publications

High‐order finite difference/spectral‐Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation

High‐order finite difference/spectral‐Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

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