Classical and non-classical symmetries and analytical solutions of the system of fractional HGF differential equations

Authors
Mir Sajjad Hashemi
Ali Hahi-Badali
Farzaneh Alizadeh
University of Bonab
Abstract
In this paper, we consider the classical and non-classical Lie symmetries of fractional HGF differential equations. Indeed, Lie symmetries are utilized to reduce the fractional nonlinear PDEs into the fractional nonlinear ODEs. Finally, exact solutions of the corresponding equations are extracted.
Keywords
Non-classical symmetry
Lie groups
Exact solution
System of fractional HGF differential equations
[1] G. Bluman, S. Anco, Symmetry and integration methods for differential equations, Vol. 154, Springer Science and Business Media, 2008.

[2] N. H. Ibragimov, CRC handbook of Lie group analysis of differential equations, Vol. 3, CRC press, 1995.

[3] P. J. Olver, Applications of Lie groups to differential equations, Vol. 107, Springer Science & Business Media, 2000.

[4] M. S. Hashemi, D. Baleanu, Lie Symmetry Analysis of Fractional Differential Equations, Chapman and Hall/CRC, 2020.

[5] K. Oldham, J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.

[6] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.

[7] A. Kilbas, Theory and applications of fractional differential equations.

[8] R. Cherniha, V. Davydovych, A hunter-gatherer-farmer population model: Lie symmetries, exact solution and their interpretation,

[9] V. V. Uchaikin, Fractional derivatives for physicists and engineers, Vol. 2, Springer, 2013.

[10] J. F. Gómez-Aguilar, A. Atangana, V. F. Morales-Delgado, Electrical circuits rc, lc, and rl described by atangana-baleanu fractional derivatives, International Journal of Circuit Theory and Applications 45 (11) (2017) 1514–1533.

[11] M. Heydari, A. Atangana, A cardinal approach for nonlinear variableorder time fractional schrüodinger equation deffned by atanganabaleanu-caputo derivative, Chaos, Solitons & Fractals 128 (2019) 339–348.

[12] A. Atangana, Modelling the spread of covid-19 with new fractalfractional operators: Can the lockdown save mankind before vaccination, Chaos, Solitons & Fractals 136 (2020) 109860.

[13] R. L. Magin, Fractional calculus in bioengineering, Vol. 2, Begell House Redding, 2006.

[14] R. Hilfer, et al., Applications of fractional calculus in physics, Vol. 35, World scientiffic Singapore, 2000.

[15] D. Baleanu, J. A. T. Machado, A. C. Luo, Fractional dynamics and control, Springer Science & Business Media, 2011.

[16] G. W. Bluman, J. D. Cole, The general similarity solution of the heat equation, Journal of Mathematics and Mechanics 18 (11) (1969) 1025–1042.
Mathematical Research
Volume 9, Issue 1 - Serial Number 24
Spring 2023
Pages 264-283