Exact solution of nonlinear time-fractional reaction-diffusion-convection equation via a new coupling method

Document Type : Research Paper


Laboratory of Fundamental and Numerical Mathematics, Department of Mathematics Faculty of Sciences, Ferhat Abbas Sétif University 1, 19000 Sétif, Algeria


The main aim of this work is to find the exact solution in the form of the Mittag-Leller function for the nonlinear time-fractional reaction-diffusion-convection equation via a new coupling method namely, the Aboodh variational iteration method (AVIM). The proposed method is a coupling of the Aboodh transform method with the variational iteration method and the fractional derivative defined with the Liouville-Caputo operator. Three different numerical applications are given to demonstrate the validity and applicability of the proposed method and compare it to existing methods. The results shown through figures and tables demonstrate the accuracy of our method. It is concluded here that the proposed method is very efficient, simple and can be applied to other nonlinear problems arising in science and engineering.


[1] K.S. Aboodh, The new integral transform ”Aboodh transform”, Glob. J. Pure Appl. Math. 9 (2013), no. 1, 35–43.
[2] L.K. Alzakia and H.K. Jassima, The approximate analytical solutions of nonlinear fractional ordinary differential equations, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 2, 527–535.
[3] M. Arshad, D. Lu and J. Wang, (N+1)-dimensional fractional reduced differential transform method for fractional order
partial differential equations, Commun. Nonlinear Sci. Numer. Simul. 48 (2017), 509–519.
[4] M. Hamarsheh, A.I. Ismail and Z. Odibat, An analytic solution for fractional order Riccati equations by using optimal homotopy asymptotic method, Appl. Math. Sci. 10 (2016), no. 23, 1131–1150.
[5] M. Khader and M.H. DarAssi, Residual power series method for solving nonlinear reaction-diffusion-convection problems, Bol. Soc. Parana. Mat. 39 (2021), no. 3, 177–188.
[6] A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Application of Fractional Differential equations, Elsevier, North-Holland, 2006.
[7] S. Kumar, A. Kumar, S. Abbas, M. Al Qurashi and D. Baleanu, A modified analytical approach with existence and uniqueness for fractional Cauchy reaction-diffusion equations, Adv. Diff. Equ. 2020 (2020), 28.
[8] S. Momani and A. Yildirim, Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He’s homotopy perturbation method, Int. J. Comput. Math. 87 (2010), no. 5, 1057–1065.
[9] R. Saadeh, M. Alaroud, M. Al-Smadi, R.R Ahmad, and U.K.S. Din, Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order, Symmetry 11 (2019), no. 12, 1431.
[10] B.R. Sontakke, A.S. Shelke and A.S. Shaikh, Solution of non-linear fractional differential equations by variational iteration method and applications, Far East J. Math. Sci. 110 (2019), no. 1, 113–129.
[11] M. Suleman, D. Lu, J.H, He, U. Farooq, S. Noeiaghdam and F.A Chandio, Elzaki projected differential transform method for fractional order system of linear and nonlinear fractional partial differential equation, Fractals 26 (2018), no. 3, 1850041–1093.
[12] P. Veeresha, D.G. Prakasha and J. Singh. Solution for fractional forced KdV equation using fractional natural decomposition method, AIMS Math. 5 (2020), no. 2, 798–810.
Volume 13, Issue 2
July 2022
Pages 333-344
  • Receive Date: 01 May 2021
  • Revise Date: 09 June 2021
  • Accept Date: 12 June 2021