Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel
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Date
2020
Authors
Kumar, Sachin
Baleanu, Dumitru
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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 t(k) 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.
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Cable Equation, Fractional Derivative With Mittag-Leffler Kernel, Legendre Polynomial, Operational Matrix, Two-Dimensional Fractional PDE
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Citation
Kumar, Sachin; Baleanu, Dumitru (2020). "Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel",Mathematical Methods in the Applied Sciences, Vol. 43, No. 15, pp. 8348-8362.
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Mathematical Methods in the Applied Sciences
Volume
43
Issue
15
Start Page
8348
End Page
8362