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June  2020, 13(3): 755-768. doi: 10.3934/dcdss.2020042

Extension of triple Laplace transform for solving fractional differential equations

Amir Khan 1,, , Asaf Khan 1, , Tahir Khan 2, and  Gul Zaman 2,

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.

Keywords: Fractional integral, fractional derivative, fractional telegraph equation, triple Laplace transform.
Mathematics Subject Classification: 34A08, 35R11.
Citation: Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 755-768. 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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus: Models and Numerical Methods, World Science, 2012. doi: 10.1142/9789814355216.  Google Scholar

[7] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC press, 2004.  doi: 10.1201/9781420035148.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. A. Kilbas, O. I. Marichev and S. G. Samko, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus: Models and Numerical Methods, World Science, 2012. doi: 10.1142/9789814355216.  Google Scholar

[7] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC press, 2004.  doi: 10.1201/9781420035148.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. A. Kilbas, O. I. Marichev and S. G. Samko, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The plot shows comparison between AB (lower surface) and Caputo (upper surface) for $u(x,y,t)$ at fixed $y = 0.5$
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Figure 2.  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 $y = 0.5$
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Figure 3.  The plot shows comparison between AB (lower surface) and Caputo (upper surface) for $u(x,y,t)$ at fixed $t = 0.5$
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Figure 4.  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 $t = 0.5$
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Figure 5.  The plot shows comparison between AB (upper surface) and Caputo (lower surface) for $u(x,y,t)$ at fixed x = -1.5.
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Figure 6.  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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