Lie symmetry analysis for the solution of first-order linear and nonlinear fractional differential equations


  • Mousa Ilie Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
  • Jafar Biazar Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O.Box.41335-1914, Guilan, Rasht, Iran
  • Zainab Ayati Department of Engineering sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran





Linear and Nonlinear Fractional Equations, Lie Symmetry Method, Conformable Fractional Derivative, Bernoulli Fractional Equation, Ric-cati Fractional Equation.


Obtaining analytical or numerical solution of fractional differential equations is one of the troublesome and challenging issues among mathematicians and engineers, specifically in recent years. The purpose of this paper is to solve linear and nonlinear fractional differential equations such as first order linear fractional equation, Bernoulli, and Riccati fractional equations by using Lie Symmetry method, based on conformable fractional derivative. For each equation, some numerical examples are presented to illustrate the proposed approach.  


[1] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014) 65-70.

[2] T. Abdeljawad, on conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015) 57-66.

[3] Daniel J. Arrigo, Symmetry analysis of differential equations an introduction, 1nd Ed. John Wiley & Sons, Inc. 2015.

[4] P. E. Hyden, Symmetry Methods for Differential Equations (A Beginner’s Guide), Cambridge Texts in Applied Mathematics, 2000.

[5] Peter J. Olver, Applications of Lie groups to differential equations, 2nd Ed. Springer-Verlag, 1993.

[6] A. Ouhadan, E. H. EL. kinani, Exact solutions of time fractional Kolomogorov equation by using lie symmetry analysis, Journal of Fractional Calculus and Applications, 5 (1) (2014) 97-104.

[7] A. Elsaid, M. S. Abdel Latif, M. Maneea, Similarity solutions for solving Riesz fractional partial differential equations, Progress in Fractional Differentiation and Applications, 2 (4) (2016) 293-298.

[8] Y. Zhanglie, Symmetry analysis to general time-fractional Korteweg-De Vries equations, Fractional Differential Calculus, 5 (2) (2015) 125-133.

[9] A. M. Yang, Y. Z. Zhang, C. Cattani, G. N. Xie, M. M. Rashidi, Y. J. Zhou, X. J. Yang, Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets, Abstract and Applied Analysis, Volume 2014 (2014), Article ID 372741, 6 pages.

[10] S. Kumar, D. Kumar, S. Abbasbandy, M. M. Rashidi, Analytical solution of fractional Navier–Stokes equation by using modified Laplace decomposition method, Ain Shams Engineering Journal, 5 (2) (2014) 569-574.

[11] H. Khalil, R. Khan, M. M. Rashidi, Brenstien polynomials and its application to fractional differential equation, Computational methods for differential equations, 3 (1) (2015) 14-35.

[12] J. Singh, M. M. Rashidi, D. Kumar, R. Swroop, A fractional model of a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions, Nonlinear Engineering 5 (4) (2016) 277-285.

[13] M. Gaur, K. Singh, Symmetry analysis of time-fractional potential Burgers' equation, Mathematical Communications, 22 (2017) 1–11.

[14] M. Gaur, K. Singh, Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation, International Journal of Differential Equations, Article ID 4270724, 8 pages, (2016).

[15] M. Ilie, J. Biazar, Z. Ayati, General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative, International Journal of Applied Mathematical Research, 6(2) (2017) 49-51.

[16] M. Ilie, J. Biazar, Z. Ayati, Application of the Lie Symmetry Analysis for second-order fractional differential equations, Iranian Journal of Optimization, 9(2) (2017) 79-83.

[17] M. Ilie, J. Biazar, Z. Ayati, Analytical solutions for conformable fractional Bratu-type equations, International Journal of Applied Mathematical Research, 7 (1) (2018) 15-19.

[18] M. Ilie, J. Biazar, Z. Ayati, Optimal Homotopy Asymptotic Method for first-order conformable fractional differential equations, Journal of Fractional Calculus and Applications, 10 (1) (2019) 33-45.

[19] M. Ilie, J. Biazar, Z. Ayati, The first integral method for solving some conformable fractional differential equations, Optical and Quantum Electronics, 50 (2) (2018),

[20] M. Ilie, J. Biazar, Z. Ayati, Resonant solitons to the nonlinear Schrödinger equation with different forms of nonlinearities, Optik, 164 (2018) 201-209.

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