# American Institute of Mathematical Sciences

## Extension of triple Laplace transform for solving fractional differential equations

 1 Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan 2 Department of Mathematics, University of Malakand, Chakadara, Lower Dir, Khyber Pakhtunkhwa, Pakistan

* Corresponding author: Amir Khan

Received  June 2018 Revised  July 2018 Published  March 2019

In this article, we extend the concept of triple Laplace transform to the solution of fractional order partial differential equations by using Caputo fractional derivative. The concerned transform is applicable to solve many classes of partial differential equations with fractional order derivatives and integrals. As a consequence, fractional order telegraph equation in two dimensions is investigated in detail and the solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. The same problem is also solved by taking into account the Atangana-Baleanu fractional derivative. Numerical plots are provided for the comparison of Caputo and Atangana-Baleanu fractional derivatives.

Citation: Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020042
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The plot shows comparison between AB (lower surface) and Caputo (upper surface) for $u(x,y,t)$ at fixed $y = 0.5$
The plot shows comparison between AB (dotted) and Caputo (solid) by considering solution profile of $u(x,y,t)$ at fixed $x = 0.5$ and $y = 0.5$
The plot shows comparison between AB (lower surface) and Caputo (upper surface) for $u(x,y,t)$ at fixed $t = 0.5$
The plot shows comparison between AB (red/bottom curve) and Caputo (blue/top curve) by considering solution profile of $u(x,y,t)$ at fixed $y = 1$ and $t = 0.5$
The plot shows comparison between AB (upper surface) and Caputo (lower surface) for $u(x,y,t)$ at fixed x = -1.5.
The plot shows comparison between AB (dotted curve) and Caputo (solid curve) by considering solution profile of $u(x,y,t)$ at fixed $x = -1.5$ and $t = 1$
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