American Institute of Mathematical Sciences

Advanced
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, 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 $y = 0.5$
Figure Options

Download full-size image

Download as PowerPoint slide

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 $y = 0.5$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The plot shows comparison between AB (lower surface) and Caputo (upper surface) for $u(x,y,t)$ at fixed $t = 0.5$
Figure Options

Download full-size image

Download as PowerPoint slide

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 $t = 0.5$
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 $t = 1$
Figure Options

Download full-size image

Download as PowerPoint slide

[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

Tools

Metrics

  • PDF downloads (85)
  • HTML views (664)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences