Numerical Solution of the Time Fractional Reaction-advection-diffusion Equation in Porous Media

Document Type : Research Paper


1 Department of Mathematical Sciences. Indian Institute of Technology (BHU), Varanasi, 221005, India

2 CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca Morelos, Mexico


In this work, we obtained the numerical solution for the system of nonlinear time-fractional order advection-reaction-diffusion equation using the homotopy perturbation method using Laplace transform method with fractional order derivatives in Liouville-Caputo sense. The solution obtained is very useful and significant to analyze many physical phenomenons. The present technique demonstrates the coupling of homotopy perturbation method and the Laplace transform technique using He’s polynomials, which can be applied to numerous coupled systems of nonlinear fractional models to find the approximate numerical solutions. The salient features of the present work is the graphical presentations of the numerical solution of the concerned nonlinear coupled equation for several particular cases and showcasing the effect of reaction terms on the nature of solute concentration of the considered mathematical model for different particular cases. To validate the reliability, efficiency and accuracy of the proposed efficient scheme, a comparison of numerical solutions and exact solution are reported for Burgers’ coupled equations and other particular cases of concerned nonlinear coupled systems. Here we find high consistency and compatibility between exact and numerical solution to a high accuracy. Presentation of absolute errors for given examples are reported in tabulated and graphical forms that ensure the convergence rate of the numerical scheme.


Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] H.G. Sun, Z. Li, Y. Zhang, W. Chen, Fractional and fractal derivative models for transient anomalous diffusion: Model comparison. Chaos, Solitons & Fractals, 102, 2017, 346–353.
[2] S. Zubair, N.I. Chaudhary, Z.A. Khan, W. Wang, Momentum fractional lms for power signal parameter estimation. Signal Processing, 142, 2018, 441–449.
[3] A. Ahlgren, R. Wirestam, F. Ståhlberg, L. Knutsson, Automatic brain segmentation using fractional signal modeling of a multiple flip angle, spoiled gradient-recalled echo acquisition. Magnetic Resonance Materials in Physics, Biology and Medicine, 27(6), 2014, 551–565.
[4] D. Kumar, J. Singh, S. Kumar, A fractional model of navier–stokes equation arising in unsteady flow of a viscous fluid. Journal of the Association of Arab Universities for Basic and Applied Sciences, 17(1), 2015, 14–19.
[5] J. Han, S. Migórski, H. Zeng,  Weak solvability of a fractional viscoelastic frictionless contact problem. Applied Mathematics and Computation, 303, 2017, 1–18.
[6] D.A. Murio. Implicit finite difference approximation for time fractional diffusion equations. Computers & Mathematics with Applications, 56(4), 2008, 1138–1145.
[7] H. Wang, K. Wang, T. Sircar, A direct O (N log2 N) finite difference method for fractional diffusion equations. Journal of Computational Physics, 229(21), 2010, 8095–8104.
[8] P. Darania, A. Ebadian, A method for the numerical solution of the integro-differential equations. Applied Mathematics and Computation, 188(1), 2007, 657–668.
[9] S.S. Ray, R.K. Bera, Solution of an extraordinary differential equation by adomian decomposition method. Journal of Applied Mathematics, 2004(4), 2004, 331–338.
[10] Y. Li, N. Sun, Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Computers & Mathematics with Applications, 62(3), 2001, 1046–1054.
[11] H. Jafari, S.A. Yousefi, M.A. Firoozjaee, S. Momani, C.M. Khalique, Application of legendre wavelets for solving fractional differential equations. Computers & Mathematics with Applications, 62(3), 2011, 1038–1045.
[12] S. Kumar, P. Pandey, S. Das, Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction–diffusion and galilei invariant advection–diffusion equations. Computational and Applied Mathematics, 38(4), 2019, 162.
[13] Y. Li, Solving a nonlinear fractional differential equation using chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation, 15(9), 2010, 2284–2292.
[14] M. Tavassoli Kajani, A. Hadi Vencheh, M. Ghasemi, The chebyshev wavelets operational matrix of integration and product operation matrix. International Journal of Computer Mathematics, 86(7), 2009 , 1118–1125.
[15] N.H. Sweilam, A.M. Nagy, A.A El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted chebyshev polynomials of the third kind. Journal of King Saud University-Science, 28(1), 2016, 41–47.
[16] H. Saeedi, M. Mohseni Moghadam, N. Mollahasani, G.N. Chuev, A quasi wavelet method for solving nonlinear fredholm integro-differential equations of fractional order. Communications in Nonlinear Science and Numerical Simulation, 16(3), 2011, 1154–1163.
[17] Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied Mathematics and Computation, 216(8), 2010, 2276–2285.
[18] V.F. Morales-Delgado, J.F. Gómez-Aguilar, K.M. Saad, M. Altaf Khan, P. Agarwal. Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach. Physica A: Statistical Mechanics and its Applications, 523, 2019, 48–65.
[19] P. Pandey, S. Kumar, S. Das. Approximate analytical solution of coupled fractional order reaction-advection-diffusion equations. The European Physical Journal Plus, 134, 2019, 364.
[20] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation. Journal of Mathematical Analysis and Applications, 351(1), 2009, 218–223.
[21] J.D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics, 9(3), 1951, 225–236.
[22] J.M. Burgers, A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics, 1, 1948, 171–199.W
[23] W. Hereman, A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations. Mathematics and Computers in Simulation, 43(1), 1997, 13–27.
[24] M. Thangarajan, Groundwater models and their role in assessment and management of groundwater resources and pollution. Groundwater, 2007, 189–236.
[25] J.F. Gómez-Aguilar, M. Miranda-Hernández, M.G. López-López, V.M. Alvarado-Martínez, D. Baleanu, Modeling and simulation of the fractional space-time diffusion equation. Communications in Nonlinear Science and Numerical Simulation, 30(1-3), 2016, 115–127.
[26] J. Bear, A. Verruijt, Modeling groundwater flow and pollution, Vol. 2. Springer Science & Business Media, 2012.
[27] J.J. Fried, Groundwater pollution mathematical modelling: improvement or stagnation?. Studies in Environmental Science, 17, 1981, 807–822.
[28] A.A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations. Journal of Applied Mathematics, 1(4), 2001, 141–155.
[29] A. Ghorbani, Beyond adomian polynomials: He polynomials. Chaos, Solitons & Fractals, 39(3), 2009, 1486–1492.
[30] A. Ghorbani, J. Saberi-Nadjafi, He’s homotopy perturbation method for calculating adomian polynomials. International Journal of Nonlinear Sciences and Numerical Simulation, 8(2), 2007, 229–232.
[31] D.D. Ganji, A. Sadighi, Application of he’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations. International Journal of Nonlinear Sciences and Numerical Simulation, 7(4), 2006, 411–418.
[32] L. Cveticanin, Homotopy–perturbation method for pure nonlinear differential equation. Chaos, Solitons & Fractals, 30(5), 2006, 1221–1230.
[33] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier Science Limited, 2016.
[34] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Elsevier, 1998.
[35] Z.J. Fu, W. Chen, H.T. Yang, Boundary particle method for laplace transformed time fractional diffusion equations. Journal of Computational Physics, 235, 2013, 52–66.
[36] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 135(1), 2003, 73–79.
[37] J. Nee, J. Duan, Limit set of trajectories of the coupled viscous burgers’ equations. Applied Mathematics Letters, 11(1), 1998, 57–61.
[38] R.C. Mittal, H. Kaur, V. Mishra, Haar wavelet-based numerical investigation of coupled viscous burgers’ equation. International Journal of Computer Mathematics, 92(8), 2015, 1643–1659.
[39] Ö. Oruç, F. Bulut, A. Esen, Chebyshev wavelet method for numerical solutions of coupled burgers’ equation. Hacettepe Journal of Mathematics and Statistics, 48(1), 2019, 1–16.
[40] A. Ali, K. Shah, Y. Li, R. Khan, Numerical treatment of coupled system of fractional order partial differential equations. Journal of Mathematics and Computer Science, 19, 2019, 74–85.
[41] A. Jannelli, M. Ruggieri, M.P. Speciale, Exact and numerical solutions of time-fractional advection–diffusion equation with a nonlinear source term by means of the Lie symmetries, Nonlinear Dynamics, 92, 2018, 543.
[42] A. Jannelli, M. Ruggieri, M.P. Speciale, Analytical and numerical solutions of fractional type advection- diffusion equation, AIP Conference Proceedings, 1863(1), 2017, 530005.
[43] R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Symmetry properties of fractional diffusion equations, Physica Scripta, T136, 2009, 014016.
[44] R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Group Invariant solutions of Fractional Differential Equations. Nonlinear Science and Complexity, 2011, 51–59.
[45] A. Jannelli, M. Ruggieri, M.P. Speciale, Numerical solutions of space fractional advection-diffusion equation, with source term. AIP Conference Proceedings, 2116, 2019, 280007.
[46] P. Zhuang, F. Liu, Implicit difference approximation for the time fractional diffusion equation. Journal of Applied Mathematics and Computing, 22, 2006, 879.
[47] A. Jannelli, M. Ruggieri, M.P. Speciale, Analytical and numerical solutions of time and space fractional advection-diffusion reaction equation. Communications in Nonlinear Science and Numerical Simulations, 70, 2019, 89-101.
[48] F. Zeng, C. Li, F. Liu, I. Turner, The use of finite difference/element approaches for solving the time-fractional sub-diffusion equation. SIAM Journal on Scientific Computing, 35, 2013, 2976-3000.