Numerical Solution of Caputo-Fabrizio Time Fractional Distributed Order Reaction-diffusion Equation via Quasi Wavelet based Numerical Method

Document Type: Research Paper


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

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


In this paper, we derive a novel numerical method to find out the numerical solution of fractional partial differential equations (PDEs) involving Caputo-Fabrizio (C-F) fractional derivatives. We first find out the approximation formula of C-F derivative of function tk. We approximate the C-F derivative in time with the help of the Legendre spectral method and approximation formula of tk. The unknown function and their derivatives in spatial direction are approximated with the quasi wavelet-based numerical method. We apply this newly derived method to solve the nonlinear distributed order reaction-diffusion in which time-fractional derivative is of C-F type. The accuracy and validity of the proposed method is exhibited by giving a solution to some numerical examples. The obtained numerical results are compared with the analytical results and conclude that our proposed numerical method achieves accurate results. On the other hand, the method is easy to apply on higher-order fractional partial differential equations and variable-order fractional partial differential equations.


[1] R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27(3), 1983, 201-210.

[2] A. Kilbas, H. Srivastava, J.J.Trujillo, Theory and Applications of the Fractional Differential Equations, Vol. 204, Elsevier (North-Holland), Amsterdam, 2006.

[3] I. Podlubny, Fractional differential equations, to methods of their solution and some of their applications, Fractional Differential Equations: An Introduction to Fractional Derivatives, Academic Press, San Diego, CA, 1998.

[4] B. Karaagac, Analysis of the cable equation with non-local and non-singular kernel fractional derivative. The European Physical Journal Plus, 133(2), 2018, 1-15.

[5] A. Atangana, J.F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena. The European Physical Journal Plus, 133(4), 2018, 1-21.

[6] B. Karaagac, A study on fractional Klein Gordon equation with non-local and non-singular kernel. Chaos, Solitons & Fractals, 126, 2019, 218-229.

[7] K.M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana–Baleanu fractional derivative. Chaos, Solitons & Fractals, 115, 2018, 127-134.

[8] K.M. Owolabi, Numerical patterns in system of integer and non-integer order derivatives. Chaos, Solitons & Fractals, 115, 2018, 143-153.

[9] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties. Physica A: Statistical Mechanics and its Applications, 505, 2018, 688-706.

[10] K.M. Owolabi, Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives. Chaos, Solitons & Fractals, 115, 2018, 160-169.

[11] K.M. Owolabi, Z. Hammouch, Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative. Physica A: Statistical Mechanics and its Applications, 523, 2019, 1072-1090.

[12] A. Atangana, T. Mekkaoui, Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus. Chaos, Solitons & Fractals, 128, 2019, 366-381.

[13] K.M. Owolabi, A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(2), 2019, 1-13.

[14] A. Atangana, S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal–fractional operators. Chaos, Solitons & Fractals, 123, 2019, 320-337.

[15] K.M. Owolabi, Z. Hammouch, Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative. Physica A: Statistical Mechanics and its Applications, 523, 2019, 1072-1090.

[16] A. Atangana, Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors. The European Physical Journal Plus, 134(9), 2019, 1-12.

[17] L. Suarez, A. Shokooh,  An eigenvector expansion method for the solution of motion containing fractional derivatives. Journal of Applied Mechanics, 64(3), 1997, 629-635

[18] P. Darania, A. Ebadian, A method for the numerical solution of the integrodifferential equations. Applied Mathematics and Computation, 188, 2007, 657-668.

[19] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPS. Communications in Nonlinear Science and Numerical Simulation, 14(3), 2009, 674-684.

[20] K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1-4), 2002, 3-22.

[21] Y. Li, N. Sun, Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Computers & Mathematics with Applications, 62(3), 2011, 1046-1054.

[22] H. Jafari, S. Yousefi, M. Firoozjaee, S. Momani, C.M. Khalique, Application of Legendre wavelets for solving fractional differential equations. Computers & Mathematics with Applications, 62(3), 2011, 1038-1045.

[23] L. Yuanlu, Solving a nonlinear fractional differential equation using Chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation, 15(9), 2010, 2284-2292.

[24] 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.

[25] Z. Odibat, On Legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations. Journal of Computational and Applied Mathematics, 235(9), 2011, 2956-2968.

[26] B. Gürbüz, M. Sezer, Laguerre polynomial solutions of a class of initial and boundary value problems arising in science and engineering fields. Acta Physica Polonica A, 130(1), 2016, 194-197.

[27] S. Araci, Novel identities for q-Genocchi numbers and polynomials. Journal of Function Spaces and Applications, 2012, 2012, 1-12.

[28] H. Zhang, X. Yang, D. Xu, A high-order numerical method for solving the 2d fourth-order reaction-diffusion equation. Numerical Algorithms, 80(3), 2019, 849-877.

[29] P. Couteron, O. Lejeune, Periodic spotted patterns in semi-arid vegetation explained by a propagation-inhibition model. Journal of Ecology, 89(4), 2001, 616-628.

[30] S. Kondo, R. Asai, A reaction-diffusion wave on the skin of the marine angelfish pomacanthus. Nature, 376(6543), 1995, 1-7.

[31] J.D. Murray, A pre-pattern formation mechanism for animal coat markings. Journal of Theoretical Biology, 88(1), 1981, 161-199.

[32] S. Kondo, How animals get their skin patterns: fish pigment pattern as a live turing wave, in: Systems Biology, Springer, 2009, 37-46.

[33] J.R. Loh, A. Isah, C. Phang, Y.T. Toh, On the new properties of Caputo-Fabrizio operator and its application in deriving shifted Legendre operational matrix. Applied Numerical Mathematics, 132, 2018, 138-153.

[34] G. Walter, J. Blum, Probability density estimation using delta sequences. The Annals of Statistics, 7(2), 1979, 328-340.

[35] G. Wei, Discrete singular convolution for the solution of the Fokker-Planck equation. The Journal of Chemical Physics, 110(18), 1999, 8930-8942.

[36] W. De-cheng, W. Guo-Wei. The study of quasi wavelets based numerical method applied to Burgers' equations. Applied mathematics and Mechanics, 21(10), 2000, 1099-1110.

[37] X. Yang, D. Xu, H. Zhang, Quasi-wavelet based numerical method for fourth-order partial integro-differential equations with a weakly singular kernel. International Journal of Computer Mathematics, 88(15), 2011, 3236-3254.