# 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:
 [1] A. M. O. Anwar, F. Jarad, D. Baleanu and F. Ayaz, Fractional Caputo heat equation within the double Laplace transform, Romanian Journal of Physics, 58 (2013), 15-22. [2] A. Atangana, A note on the triple Laplace transform and its applications to some kind of third-order differential equation, Abstr. Appl. Anal., 2013 (2013), Art. ID 769102, 10 pp. doi: 10.1155/2013/769102. [3] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. [4] A. Atangana, On the new fractional derivative and application to nonlinear Fishers reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. [5] A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractionl derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1-7. [6] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus: Models and Numerical Methods, World Science, 2012. doi: 10.1142/9789814355216. [7] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC press, 2004. doi: 10.1201/9781420035148. [8] T. A. Estrin and T. J. Higgins, The solution of boundary value problems by multiple Laplace transformations, Journal of the Franklin Institute, 252 (1951), 153-167. doi: 10.1016/0016-0032(51)90950-7. [9] F. Gao and X. J. Yang, Fractional Maxwell fluid with fractional derivative without singular kernel, Therm. Sci., 20 (2016), 871-877. doi: 10.2298/TSCI16S3871G. [10] H. Jafari, A. Kadem, D. Baleanu and T. Yalmaz, Solutions of the fractional Davey-Stewartson equations with variational iteration method, Rom. Rep. Phy., 64 (2017), 337-346. [11] F. Jarad, T. Abdeljawad, E. Gndogdu and D. Baleanu, On the Mittag-Leffler stability of q-fractional nonlinear dynamical systems, P. Romanian Acad. A, 12 (2011), 309-314. [12] Y. Khan, J. Diblik, N. Faraz and Z. Smarda, An efficient new perturbative Laplace method for space-time fractional telegraph equations, Adv. Differ. Equ-NY, 2012 (2012), 9pp. doi: 10.1186/1687-1847-2012-204. [13] T. Khan, K. Shah, A. Khan and R. A. Khan, Solution of fractional order heat equation via triple Laplace transform in 2 dimensions, Math. Meth. Appl. Sci., 41 (2018), 818-825. doi: 10.1002/mma.4646. [14] A. A. Kilbas, O. I. Marichev and S. G. Samko, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993. [15] A. Kilicman and H. E. Gadain, On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institute, 347 (2010), 848-862. doi: 10.1016/j.jfranklin.2010.03.008. [16] D. Kumar, J. Singh and S. Kumar, Analytic and approximate solutions of space-time fractional telegraph equations via Laplace transform, Walailak J. Sci. and Tech., 11 (2014), 711-728. [17] S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model., 38 (2014), 3154-3163. doi: 10.1016/j.apm.2013.11.035. [18] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 2677-2682. doi: 10.1016/j.na.2007.08.042. [19] R. Metzler, W. Schick, H. G. Kilian and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phy., 103 (1995), 7180-7186. doi: 10.1063/1.470346. [20] R. C. Mittal and R. Bhatia, A numerical study of two dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method, Appl. Math. Comput., 244 (2014), 976-997. doi: 10.1016/j.amc.2014.07.060. [21] M. K. Owolabi and A. Atangana, Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system, Adv. Appl. Math. Mech., 9 (2017), 1438-1460. doi: 10.4208/aamm.OA-2016-0115. [22] K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo Fabrizio derivative in Riemann Liouville sense, Chaos Solitons & Fractals, 99 (2017), 171-179. doi: 10.1016/j.chaos.2017.04.008. [23] K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus., 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9. [24] K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, Eur. Phys. J. Plus., 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x. [25] K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119-127. doi: 10.1016/j.chaos.2018.04.019. [26] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [27] J. Unsworth and F. J. Duarte, Heat diffusion in a solid sphere and Fourier theory: an elementary practical example, Am. J. Phys., 47 (1979), 981-983. doi: 10.1119/1.11601. [28] A. M. Yang, Y. Han, J. Li J and W. X. Liu, On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel, Therm. Sci., 20 (2016), 717-721. doi: 10.2298/TSCI16S3717Y.

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##### References:
 [1] A. M. O. Anwar, F. Jarad, D. Baleanu and F. Ayaz, Fractional Caputo heat equation within the double Laplace transform, Romanian Journal of Physics, 58 (2013), 15-22. [2] A. Atangana, A note on the triple Laplace transform and its applications to some kind of third-order differential equation, Abstr. Appl. Anal., 2013 (2013), Art. ID 769102, 10 pp. doi: 10.1155/2013/769102. [3] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. [4] A. Atangana, On the new fractional derivative and application to nonlinear Fishers reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. [5] A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractionl derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1-7. [6] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus: Models and Numerical Methods, World Science, 2012. doi: 10.1142/9789814355216. [7] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC press, 2004. doi: 10.1201/9781420035148. [8] T. A. Estrin and T. J. Higgins, The solution of boundary value problems by multiple Laplace transformations, Journal of the Franklin Institute, 252 (1951), 153-167. doi: 10.1016/0016-0032(51)90950-7. [9] F. Gao and X. J. Yang, Fractional Maxwell fluid with fractional derivative without singular kernel, Therm. Sci., 20 (2016), 871-877. doi: 10.2298/TSCI16S3871G. [10] H. Jafari, A. Kadem, D. Baleanu and T. Yalmaz, Solutions of the fractional Davey-Stewartson equations with variational iteration method, Rom. Rep. Phy., 64 (2017), 337-346. [11] F. Jarad, T. Abdeljawad, E. Gndogdu and D. Baleanu, On the Mittag-Leffler stability of q-fractional nonlinear dynamical systems, P. Romanian Acad. A, 12 (2011), 309-314. [12] Y. Khan, J. Diblik, N. Faraz and Z. Smarda, An efficient new perturbative Laplace method for space-time fractional telegraph equations, Adv. Differ. Equ-NY, 2012 (2012), 9pp. doi: 10.1186/1687-1847-2012-204. [13] T. Khan, K. Shah, A. Khan and R. A. Khan, Solution of fractional order heat equation via triple Laplace transform in 2 dimensions, Math. Meth. Appl. Sci., 41 (2018), 818-825. doi: 10.1002/mma.4646. [14] A. A. Kilbas, O. I. Marichev and S. G. Samko, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993. [15] A. Kilicman and H. E. Gadain, On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institute, 347 (2010), 848-862. doi: 10.1016/j.jfranklin.2010.03.008. [16] D. Kumar, J. Singh and S. Kumar, Analytic and approximate solutions of space-time fractional telegraph equations via Laplace transform, Walailak J. Sci. and Tech., 11 (2014), 711-728. [17] S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model., 38 (2014), 3154-3163. doi: 10.1016/j.apm.2013.11.035. [18] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods and Applications, 69 (2008), 2677-2682. doi: 10.1016/j.na.2007.08.042. [19] R. Metzler, W. Schick, H. G. Kilian and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phy., 103 (1995), 7180-7186. doi: 10.1063/1.470346. [20] R. C. Mittal and R. Bhatia, A numerical study of two dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method, Appl. Math. Comput., 244 (2014), 976-997. doi: 10.1016/j.amc.2014.07.060. [21] M. K. Owolabi and A. Atangana, Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system, Adv. Appl. Math. Mech., 9 (2017), 1438-1460. doi: 10.4208/aamm.OA-2016-0115. [22] K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo Fabrizio derivative in Riemann Liouville sense, Chaos Solitons & Fractals, 99 (2017), 171-179. doi: 10.1016/j.chaos.2017.04.008. [23] K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus., 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9. [24] K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, Eur. Phys. J. Plus., 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x. [25] K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119-127. doi: 10.1016/j.chaos.2018.04.019. [26] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [27] J. Unsworth and F. J. Duarte, Heat diffusion in a solid sphere and Fourier theory: an elementary practical example, Am. J. Phys., 47 (1979), 981-983. doi: 10.1119/1.11601. [28] A. M. Yang, Y. Han, J. Li J and W. X. Liu, On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel, Therm. Sci., 20 (2016), 717-721. doi: 10.2298/TSCI16S3717Y.
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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