An efficient numerical algorithm for solving the two-dimensional fractional cable equation

Alias

et al. 2023

Sci Rep

In this article, we developed a new higher-order implicit finite difference iterative scheme (FDIS) for the solution of the two dimension (2-D) time fractional Cable equation (FCE). In the new proposed FDIS, the time fractional and space derivatives are discretized using the Caputo fractional derivative and fourth-order implicit scheme, respectively. Moreover, the proposed scheme theoretical analysis (convergence and stability) is also discussed using the Fourier analysis method. Finally, some numerical test problems are presented to show the effectiveness of the proposed method.

Section: Numerical Experiementsmentioning

confidence: 78%

A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation

Alias

et al. 2023

Sci Rep

Math Methods in App Sciences

“…In 2016, Liu et al 10 have proposed two grid finite element method for solving the FC equation. In 2018, Li et al 11 have investigated the Fourier analysis method for the numerical solution, stability, and convergence analysis of the FC equation. In 2019, Sweilam and Al‐Mekhlafi 12 have proposed a nonstandard implicit compact finite difference method for the numerical solution and stability analysis of FC equation.…”

Section: Introductionmentioning

confidence: 99%

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

Mittal

Balyan

Panda

et al. 2021

In recent years, major attention has been devoted to fractional‐order partial differential equations since they seem to be more efficient in modeling complex processes. Fractional‐order partial differential equations are considered as generalizations of classical integer‐order partial differential equations. In many areas of electrophysiology and in modeling neuronal dynamics, the cable equation plays a major role. In this article, the authors constructed to approximate the numerical solutions of the two‐dimensional fractional cable equation using the time‐space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions. Moreover, we use fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi–Gauss–Lobatto (JGL) points. The fractional derivative is defined in the modified Riemann–Liouville fractional derivatives formula at JGL points. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The stability analysis of the proposed method, uniqueness of the solution, and error estimate are also provided. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

“…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.

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.