Numerical simulation of time partial fractional diffusion model by Laplace transform

Amjad Ali; Iyad Suwan; Thabet Abdeljawad; Abdullah; Amjad Ali; Iyad Suwan; Thabet Abdeljawad; Abdullah

doi:10.3934/math.2022159

AIMS Mathematics

2022, Volume 7, Issue 2: 2878-2890. doi: 10.3934/math.2022159

Previous Article Next Article

Research article Special Issues

Numerical simulation of time partial fractional diffusion model by Laplace transform

1.
Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics and Statistics, Arab American University, Zababdeh, Palestine
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia
4.
Department of Medical Research, China Medical Universit, 40402 Taichung, Taiwan

Received: 03 September 2021 Accepted: 08 November 2021 Published: 22 November 2021
MSC : Primary: 37A25; Secondary: 34D20, 37M01

In the present work, the authors developed the scheme for time Fractional Partial Diffusion Differential Equation (FPDDE). The considered class of FPDDE describes the flow of fluid from the higher density region to the region of lower density, macroscopically it is associated with the gradient of concentration. FPDDE is used in different branches of science for the modeling and better description of those processes that involve flow of substances. The authors introduced the novel concept of fractional derivatives in term of both time and space independent variables in the proposed FPDDE. We provided the approximate solution for the underlying generalized non-linear time PFDDE in the sense of Caputo differential operator via Laplace transform combined with Adomian decomposition method known as Laplace Adomian Decomposition Method (LADM). Furthermore, we established the general scheme for the considered model in the form of infinite series by aforementioned techniques. The consequent results obtained by the proposed technique ensure that LADM is an effective and accurate technique to handle nonlinear partial differential equations as compared to the other available numerical techniques. At the end of this paper, the obtained numerical solution is visualized graphically by Matlab to describe the dynamics of desired solution.
- partial fraction diffusion equations,
- Laplace transform,
- numerical approximation,
- Matlab
Citation: Amjad Ali, Iyad Suwan, Thabet Abdeljawad, Abdullah. Numerical simulation of time partial fractional diffusion model by Laplace transform[J]. AIMS Mathematics, 2022, 7(2): 2878-2890. doi: 10.3934/math.2022159

Related Papers:

Abstract

In the present work, the authors developed the scheme for time Fractional Partial Diffusion Differential Equation (FPDDE). The considered class of FPDDE describes the flow of fluid from the higher density region to the region of lower density, macroscopically it is associated with the gradient of concentration. FPDDE is used in different branches of science for the modeling and better description of those processes that involve flow of substances. The authors introduced the novel concept of fractional derivatives in term of both time and space independent variables in the proposed FPDDE. We provided the approximate solution for the underlying generalized non-linear time PFDDE in the sense of Caputo differential operator via Laplace transform combined with Adomian decomposition method known as Laplace Adomian Decomposition Method (LADM). Furthermore, we established the general scheme for the considered model in the form of infinite series by aforementioned techniques. The consequent results obtained by the proposed technique ensure that LADM is an effective and accurate technique to handle nonlinear partial differential equations as compared to the other available numerical techniques. At the end of this paper, the obtained numerical solution is visualized graphically by Matlab to describe the dynamics of desired solution.

References

[1]	L. Perko, Differential equations and dynamical systems, New York: Springer, 2008.
[2]	T. Toni, M. P. H. Stumpf, Simulation-based model selection for dynamical systems in systems and population biology, Bioinformatics, 26 (2010), 104–110. doi: 10.1093/bioinformatics/btp619. doi: 10.1093/bioinformatics/btp619
[3]	M. W. Hirsch, S. Smale, R. L. Devaney, Differential equations, dynamical systems and an introduction to chaos, Elsevier, USA, 2012. doi: 10.1016/B978-0-12-382010-5.00015-4.
[4]	K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 133 (2018). doi: 10.1140/epjp/i2018-11863-9.
[5]	A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, UK, 1995. doi: 10.1017/CBO9780511809187.
[6]	L. Perko, Differential equations and dynamical systems, 2 Eds., New York: Springer, 1996. doi: 10.1007/978-1-4684-0249-0.
[7]	S. Saravi, M. Saravi, A Short survey in application of ordinary differential equations on cancer research, American J. Comp. Appl. Math., 10 (2020), 1–5. doi:10.5923/j.ajcam.20201001.01. doi: 10.5923/j.ajcam.20201001.01
[8]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000. doi: 10.1142/9789812817747_0001.
[9]	B. Ghanbari, Chaotic behaviors of the prevalence of an infectious disease in a prey and predator system using fractional derivatives, 44 (2021), 9998–10013. doi: 10.1002/mma.7386.
[10]	C. Chicone, Ordinary differential equations with applications, New York: Springer, 2006. doi: 10.1007/0-387-35794-7.
[11]	K. Wang, K. J. Wang, C. H. He, Physical insight of local fractional calculus and its application to fractional kdv-burgers-kuramoto equation, Fractals, 27 (2019). doi: 10.1142/S0218348X19501226.
[12]	D. Baleanu, A. Fernandez, A. Akgül, On a fractional operator combining proportional and classical differintegrals, Mathematics, 8 (2020), 360. doi: 10.3390/math8030360. doi: 10.3390/math8030360
[13]	E. Al Awawdah, The adomian decomposition method for solving partial differential equations, Palestine: Birzeit University, 2016.
[14]	S. S. Ray, A. Atangana, S. C. Noutchie, M. Kurulay, N. Bildik, A. Kilicman, Fractional calculus and its applications in aplied mathematics and other sciences, Math. Probl. Eng., 2014 (2014), Article ID 849395, 2 pages. doi: 10.1155/2014/849395.
[15]	A. Ali, B. Samet, K. Shah, R. A. Khan, Existence and stability of solution to a toppled system of differential equations of non-integer order, Bound. Value Probl., 2017 (2017), 16. doi: 10.1186/s13661-017-0749-1. doi: 10.1186/s13661-017-0749-1
[16]	M. K. Ishteva, Properties and application of the Caputo fractional operator, Department of Mathematics, Universitat Karlsruhe (TH), 2005.
[17]	B. Ghanbari, A fractional system of delay differential equation with nonsingular kernels in modeling hand-foot-mouth disease, Adv. Differ. Equ., 2020 (2020), 536. doi: 10.1186/s13662-020-02993-3. doi: 10.1186/s13662-020-02993-3
[18]	S. Djilali, B. Ghanbari, The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative, Adv. Differ. Equ., 2021 (2021), 20. doi: 10.1186/s13662-020-03177-9. doi: 10.1186/s13662-020-03177-9
[19]	B. Ghanbari, A. Atangana, Some new edge detecting techniques based on fractional derivatives with non-local and non-singular kernels, Adv. Differ. Equ., 2020 (2020), 435. doi: 10.1186/s13662-020-02890-9. doi: 10.1186/s13662-020-02890-9
[20]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equation, Amsterdam: Elsevier, 2006.
[21]	A. ALi, K. Shah, R. Ali Khan, Numerical treatment for traveling wave solution of fractional Whitham-Broer-Kaup equation, Alex. Eng. J., 57 (2018), 1991–1998. doi: 10.1016/j.aej.2017.04.012. doi: 10.1016/j.aej.2017.04.012
[22]	A. Elsaid, M. S. Abdel Latif, M. Maneea, Similarity solution for multiterm time fractional diffusion equation, Adv. Math. Phys., 2016 (2016), Article ID 7304659, 7 pages. doi: 10.1155/2016/7304659.
[23]	S. Kumar, A. Yildirim, Y. Khan, L. Wei, A fractional model of the diffusion equation and its analytical solution using Laplace transform, Sci. Iran., 19 (2012), 1117–1123. doi: 10.1016/j.scient.2012.06.016. doi: 10.1016/j.scient.2012.06.016
[24]	S. Das, Analytical solution of a fractional diffusion equation by variational iteration method, Comput. Math. Appl., 57 (2009), 483–487. doi: 10.1016/j.camwa.2008.09.045. doi: 10.1016/j.camwa.2008.09.045
[25]	B. R. Sontakke, A. S. Shelke, Approximate scheme for time fractional diffusion equation and its application, Global J. Pure Appl. Math., 13 (2017), 4333–4345.
[26]	D. K. Maurya, R. Singh, Y. K. Rajoria, Analytical solution of new approach to reaction diffusion equation by NHPM, Int. J. Res. Eng. IT So. Sci., 9 (2019), 197–207.
[27]	J. J. Zhao, J. Y. Xiao, Y. Xu, A finite element method for the multiterm space Ries advection-diffusion equations in finite domain, Abstr. Appl. Anal., 2013 (2013). doi: 10.1155/2013/868035.
[28]	V. D. Gejje, S. Bhalekar, Solving fraction diffusion-wave equations using a new iterative method, Fract. Calc. Appl. Anal., 11 (2008), 193–202.
[29]	N. A. Shah, S. Saleem, A. Akgül, K. Nonlaopon, J. D. Chung, Numerical analysis of time-fractional diffusion equations via a novel approach, J. Funct. Space, 2021 (2021), Article ID 9945364, 12 pages. doi: 10.1155/2021/9945364. doi: 10.1155/2021/9945364
[30]	X. J. Yang, J. A. Machado, D. Baleanuet, A new numerical technique for local fractional diffusion equation in fractal heat transfer, J. Nonlinear Sci. Appl., 9 (2016), 5621–5628. doi: 10.22436/jnsa.009.10.09. doi: 10.22436/jnsa.009.10.09
[31]	L. D'Amore, R. Campagna, V. Mele, A. Murli, Algorithm 946: ReLIADiff–A C++ software package for real Laplace transform inversion based on Algorithmic differentiation, ACM T. Math. Software, 40 (2014). doi: 10.1145/2616971.
[32]	L. D'Amore, V. Mele, R. Campagna, Quality assurance of Gaver's formula for multi-precision Laplace transform inversion in real case, Inverse Probl. Sci. En., 26 (2018), 553–580. doi: 10.1080/17415977.2017.1322963. doi: 10.1080/17415977.2017.1322963
[33]	A. Ali, A. Zeb, V. E. Turk, R. A. Khan, Numerical solution of fractional order immunology and AIDS via Laplace transform adomian decomposition method, J. Frac. Calc. Appl., 10 (2019), 242–252.
[34]	E. Kreyszig, Advanced engineering mathematics, New York: John Wiley & Sons, 2011.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)