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Revista Colombiana de Matemáticas

versão impressa ISSN 0034-7426

Rev.colomb.mat. vol.54 no.1 Bogotá jan./jun. 2020 

Original articles

New analytical method for solving nonlinear time-fractional reaction-diffusion-convection problems

Nuevo método analítico para resolver problemas no lineales fraccionados por tiempo reacción-difusión-convección

Ali Khalouta1  * 

Abdelouahab Kadem1 

1Ferhat Abbas Sétif University, Sétif, Algeria


In this paper, we propose a new analytical method called generalized Taylor fractional series method (GTFSM) for solving nonlinear timefractional reaction-diffusion-convection initial value problems. Our obtained results are given in the form of a new theorem. The advantage of the proposed method compared with the existing methods is, that method solves the nonlinear problems without using linearization and any other restriction. The accuracy and efficiency of the method is tested by means of two numerical examples. Obtained results interpret that the proposed method is very effective and simple for solving different types of nonlinear fractional problems.

Keywords: Nonlinear time-fractional reaction-diffusion-convection problems; Caputo fractional derivative; generalized Taylor fractional series method


En este artículo, proponemos un nuevo método analítico denominado método generalizado de la serie fraccional de Taylor (MGSFT) para resolver problemas de valor inicial no lineales fraccionales en el tiempo de reacción-difusión-convección. Nuestros resultados obtenidos se dan en la forma de un nuevo teorema. La ventaja del método propuesto en comparación con los métodos existentes es que ese método resuelve los problemas no lineales sin utilizar la linealización y cualquier otra restricción. La precisión y la eficiencia del método se prueban mediante dos ejemplos numéricos. Los resultados obtenidos interpretan que el método propuesto es muy eficaz y simple para resolver diferentes tipos de problemas fraccionarios no lineales.

Palabras clave: Problemas no lineales fraccionados en el tiempo de reacción-difusión-convección; derivado fraccional de Caputo; método de serie fraccional de Taylor generalizado

Full text available only in PDF format.


The authors would like to thank Professor Hernando Gaitán Orjuela (Adjunct Editor) and Professor Francisco José Marcellán (Managing Editor) as well as the anonymous referees who have made valuable and careful comments, which improved the paper considerably.


[1] M. Hamdi Cherif, K. Belghaba, and D. Ziane, Homotopy perturbation method for solving the fractional fisher’s equation, International Journal of Analysis and Applications 10 (2016), no. 1, 9-16. [ Links ]

[2] D. Das and R. K. Bera, Generalized differential transform method for nonlinear inhomogeneous time fractional partial differential equation, International Journal of Sciences and Applied Research 4 (2017), 71-77. [ Links ]

[3] K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications 265 (2002), 229-248. [ Links ]

[4] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, New York, 1993. [ Links ]

[5] H. Molaei, Solutions of the nonlinear reaction diffusion convection problems using analytical method, Revista Publicando 5 (2018), no. 15, 109-119. [ Links ]

[6] Z. Odibat and S. Momani, The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers and Mathematics with Applications 58 (2009), 2199-2208. [ Links ]

[7] K. B. Oldham and J. Spanier, The fractional calculus, Academic Press, New York, 1974. [ Links ]

[8] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999. [ Links ]

[9] S. G. Samko, A.A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, Yverdon, 1993. [ Links ]

[10] A. Shidfar, A. Babaei, A. Molabahrami, and M. Alinejadmofradi, Approximate analytical solutions of nonlinear reaction-diffusion-convection problem, Mathematical and Computer Modeling 53 (2011), 261-268. [ Links ]

[11] A. M. Shuku, Adomian decomposition method for certain space-time fractional partial differential equations, IOSR Journal of Mathematics 11 (2015), no. 1, 55-65. [ Links ]

[12] X. B. Yin, S. Kumar, and D. Kumar, A modified homotopy analysis method for solution of fractional wave equations, Advances in Mechanical Engineering 7 (2015), no. 12, 1-8. [ Links ]

Received: January 06, 2019; Accepted: July 13, 2019

*Correspondencia: Ali Khalouta, Laboratory of Fundamental and Numerical Mathematics, Department of Mathematics, Faculty of Sciences, Ferhat Abbas Sétif University 1, 19000 Sétif, Algeria. Correo electrónico: DOI:

2010 Mathematics Subject Classification. 35R11, 26A33, 74G10, 34K28.

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