Convergence analysis of tau scheme for the fractional reaction-diffusion equation

Rashidinia, Jalil; Mohmedi, Elham

doi:10.1140/epjp/i2018-12200-2

Cited by 17 publications

(17 citation statements)

References 46 publications

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“…This indicates that the order of convergence of present method is two orders of magnitude higher than that of the method in [16]. Example We consider the following TFRD equation [16, 22].

_{0} D_{t}^{α} u false(x, t false) = u_{italicxx} false(x, t false) - u false(x, t false) + f false(x, t false), false(x, t false) \in false[0, 2 false] \times false[0, 1 false],

subject to the IC

u false(x, 0 false) = 0,

and BCs

u false(0, t false) = 0, u false(1, t false) = 0,

where

f false(x, t false) = \frac{2 t^{2 - α} x false(2 - x false)}{Γ false(3 - α false)} + t^{2} x false(2 - x false) + 2 t^{2}

.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…In [21], Wang et al developed an efficient parallel algorithm based on an implicit finite-difference method to solve the fractional reaction-diffusion model (1)- (3). In [22], Rashidinia and Mohammadi described an approximation technique to solve the problem (1)- (3). This method is based on a combination of Legendre tau spectral method and generalized shifted Legendre operational matrix.…”

Section: Introductionmentioning

confidence: 99%

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A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Roul

Goura

2020

Numerical Methods Partial

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This paper is concerned with the development of a high order numerical technique for solving time‐fractional reaction–diffusion equation. The fractional derivative in the governing equation is described in the Caputo sense and a collocation method based on quintic B‐spline basis function is used to discretize the space variable. The stability and convergence analysis of the method are investigated, and it is shown that the proposed method converges to the exact solution of the problem with order of convergence O(Δt2 − α + Δx4), where α is the order of fractional derivative (0 < α < 1). Three test problems are considered to illustrate the efficiency and accuracy of present numerical scheme and to verify the theoretical result. It is shown that the order of fractional derivative has a profound effect on the solution of time‐fractional reaction–diffusion equation.

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“…This indicates that the order of convergence of present method is two orders of magnitude higher than that of the method in [16]. Example We consider the following TFRD equation [16, 22].

_{0} D_{t}^{α} u false(x, t false) = u_{italicxx} false(x, t false) - u false(x, t false) + f false(x, t false), false(x, t false) \in false[0, 2 false] \times false[0, 1 false],

subject to the IC

u false(x, 0 false) = 0,

and BCs

u false(0, t false) = 0, u false(1, t false) = 0,

where

f false(x, t false) = \frac{2 t^{2 - α} x false(2 - x false)}{Γ false(3 - α false)} + t^{2} x false(2 - x false) + 2 t^{2}

.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Roul

Goura

2020

Numerical Methods Partial

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“…Sungu and Demir [21] derived the hybrid generalized differential method and finite difference method (FDM) for solving the TFRD model numerically. Several numerical techniques for the TFRD model are seen in literature; such as explicit FDM [22], H 1 -Galerkin mixed finite element method [23], implicit FDM [24], the explicit-implicit and implicit-explicit method [10], Legendre tau spectral method [25]. Ersoy and Dag [26] solved the FRDM using the exponential cubic B-spline technique.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

et al. 2020

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The time–fractional reaction–diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo–Fabrizio operator is applied for the discretization of time levels while the extended cubic B-spline (ECBS) function is applied for the space direction. The ECBS function preserves geometrical invariability, convex hull and symmetry property. Unconditional stability and convergence analysis are also proved. The projected numerical method is tested on two numerical examples. The theoretical and numerical results demonstrate that the order of convergence of 2 in time and space directions.

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“…Most fractional differential equations do not have exact analytical solutions, so for tackling such approximation and numerical schemes must be applied. There are many researchers which have been interested to develop novel numerical methods for fractional partial differential equations (Ren and Wang 2017;Xing and Yan 2018;Zhang and Yang 2018;Sakar et al 2018;Mirzaee and Samadyar 2018) such as explicit finite difference (Shen et al 2011;Sousa 2012;Zhang and Yang 2018;Costa and Pereira 2018), implicit finite difference (Burrage et al 2012;Karatay et al 2011;Sunarto et al 2014), compact finite difference (Cui 2012;Wang and Ren 2019;Wang 2015), finite element (Ford et al 2011;Jiang and Ma 2011;Li and Yang 2017), spline (Arshed 2017;Siddiqi and Arshed 2015;Qiao and Xu 2018), Fourier analysis (Li et al 2018), radial basis functions (Golbabai et al 2019;Ahmadi et al 2017;Dehghan et al 2016;Hosseini et al 2016;Ghehsareh et al 2018), wavelets (Heydari et al 2015;Kargar and Saeedi 2017;Soltani Sarvestani et al 2019), sinc radial basis function (Permoon et al 2016), local radial basis function-generated finite difference (Nikan et al 2020b;Nikan et al 2020a) and spectral methods (Rashidinia and Mohmedi 2018;Yang et al 2018;Zaky 2018a, b;Aghdam et al 2020)…”

Section: Introductionmentioning

confidence: 99%

Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations

Rashidinia

Mohmedi

2020

Comp. Appl. Math.

Self Cite

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We develop a numerical scheme for finding the approximate solution for one-and twodimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.

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Convergence analysis of tau scheme for the fractional reaction-diffusion equation

Cited by 17 publications

References 46 publications

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations

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