Numerical solution of two‐dimensional time fractional cable equation with Mittag‐Leffler kernel

Abstract

The main motive of this article is to study the recently developed Atangana‐Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag‐Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two‐dimensional cable equation in which time‐fractional derivative is of Mittag‐Leffler kernel type. First, we derive an approximation formula of the fractional‐order ABC derivative of a function tk using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two‐dimensional model of the time‐fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two‐dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag‐Leffler fractional derivative.

Numerical solution of ABC space–time fractional distributed order reaction–diffusion equation

“…The Riemann-Liouville fractional order derivative of a function 𝜉(𝜃) is defined by the following equation [12]…”

“…These FDEs have so many application in various fields as groundwater contamination problem, biology, medical science [6], chemistry and physics. Some biological and physical models like as predator-prey dynamical model [7], two interacting species [8], heat equation [9], SIR epidemic model [10], heat mass transfer [11] and tumor model [12]. FDEs have many applications in porous media [13].…”

On a time fractional diffusion with nonlocal in time conditions

“…By replacing many differential operators of fractional order with different type of PDEs of integer order, we formulate various types of boundary value problems with fractional order. Let us refer to many papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

“…In 2020, Zheng et al 14 have proposed a finite difference method in time direction and Legendre spectral method in space direction for the numerical solutions of two‐dimensional distributed‐order time‐FC equation. Kumar and Baleanu 15 have proposed the ABC operational matrix of shifted Legendre method for the numerical solutions of a two‐dimensional time FC equation with Mittag–Leffler kernel. Kumar et al 16 have solved a nonlinear fractional predator–prey biological model of two species using Bernstein wavelet and Euler methods.…”