# 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
##### References:

show all references

##### References:
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 $t = 1$
 [1] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 709-722. doi: 10.3934/dcdss.2020039 [2] Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615 [3] Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001 [4] Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975 [5] Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657 [6] Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025 [7] Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031 [8] Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 741-754. doi: 10.3934/dcdss.2020041 [9] Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 723-739. doi: 10.3934/dcdss.2020040 [10] Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems & Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27 [11] Ekta Mittal, Sunil Joshi. Note on a $k$-generalised fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 797-804. doi: 10.3934/dcdss.2020045 [12] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 975-993. doi: 10.3934/dcdss.2020057 [13] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 937-956. doi: 10.3934/dcdss.2020055 [14] Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039 [15] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [16] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248 [17] Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032 [18] Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204 [19] Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058 [20] Alexandre Boritchev. Decaying turbulence for the fractional subcritical Burgers equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2229-2249. doi: 10.3934/dcds.2018092

2018 Impact Factor: 0.545