International Journal of Nonlinear Analysis and Applications
Mixed fractional partial differential equations by the base method
Document Type : Research Paper
Authors
Department of Mathematics, Ministry of Education, Thi-Qar, Iraq
Abstract
In this paper, the invariant subspace method is generalized and improved and is then used to have an exact solution for a wide class of the linear/ non-linear mixed fractional partial differential equations $(FPDEs)$; with constant, non-constant coefficients. Some examples are given here to illustrate the efficiency of this method.
Keywords
[1] S. Choudhary and V. Daftardar-Gejji, Invariant subspace method: A tool for solving fractional partial differential
equations, Frac. Calc. Appl. Anal. 2 (2017) 477–493.
[2] M. Eslami, B. F. Vajargah, M. Mirzazadeh and A. Biswas, Applications of first integral method to fractional
partial differential equations, Indian J. Phys. 2 (2014) 177–184.
[3] V. A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proc. R. Soc. Edinb. Sect. A Math. 2 (1995) 225–246.
[4] V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of nonlinear Partial Differential Equations in Mechanics and Physics, Chapman and Hall/CRC, London, 2007.
[5] R.K. Gazizov and A. A. Kasatkin, Construction of exact solutions for fractional order differential equations by
invariant subspace method, Comput. Math. Appl. 5 (2013) 576–584.
[6] P. Artale Harris and R.Garra, Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace
method, arXiv preprint arXiv:1410.8085, (2013) 1–8.
[7] P. Artale Harris and R.Garra, Nonlinear time-fractional dispersive equations, arXiv preprint arXiv:1410.8085,
(2014) 1–14.
[8] J. Jiang, Y. Feng and S. Li, Exact solutions to the fractional differential equations with mixed partial derivatives,
Axioms, 1 (2018) 1–10.
[9] A. Ouhadan and E.H. El Kinani, Invariant subspace method and fractional modified Kuramoto-Sivashinsky equation, arXiv preprint arXiv:1503.08789, (2015) 1–12.
[10] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential
Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, New York, 1998.
[11] R. Sahadevan and Thangarasu, Invariant subspace method and exact solutions of certain nonlinear time fractional
partial differential equations, Fract. Calc. Appl. Anal. 18 (2015) 146–162.
[12] R. Sahadevan and P. Prakash, Exact solution of certain time fractional nonlinear partial differential equations,
Nonlinear Dyn. 85 (2016) 659–673.
[13] S.R. Svirshchevskii and R. Sergey, Invariant linear spaces and exact solutions of nonlinear evolution equations,
J. Nonlinear Math. Phys. 3 (1996) 146–169.
[14] K.V. Zhukovsky, Operational solution for some types of second order differential equations and for relevant physical
problems, J. Math. Anal. 446 (2017) 628–647.
equations, Frac. Calc. Appl. Anal. 2 (2017) 477–493.
[2] M. Eslami, B. F. Vajargah, M. Mirzazadeh and A. Biswas, Applications of first integral method to fractional
partial differential equations, Indian J. Phys. 2 (2014) 177–184.
[3] V. A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proc. R. Soc. Edinb. Sect. A Math. 2 (1995) 225–246.
[4] V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of nonlinear Partial Differential Equations in Mechanics and Physics, Chapman and Hall/CRC, London, 2007.
[5] R.K. Gazizov and A. A. Kasatkin, Construction of exact solutions for fractional order differential equations by
invariant subspace method, Comput. Math. Appl. 5 (2013) 576–584.
[6] P. Artale Harris and R.Garra, Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace
method, arXiv preprint arXiv:1410.8085, (2013) 1–8.
[7] P. Artale Harris and R.Garra, Nonlinear time-fractional dispersive equations, arXiv preprint arXiv:1410.8085,
(2014) 1–14.
[8] J. Jiang, Y. Feng and S. Li, Exact solutions to the fractional differential equations with mixed partial derivatives,
Axioms, 1 (2018) 1–10.
[9] A. Ouhadan and E.H. El Kinani, Invariant subspace method and fractional modified Kuramoto-Sivashinsky equation, arXiv preprint arXiv:1503.08789, (2015) 1–12.
[10] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential
Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, New York, 1998.
[11] R. Sahadevan and Thangarasu, Invariant subspace method and exact solutions of certain nonlinear time fractional
partial differential equations, Fract. Calc. Appl. Anal. 18 (2015) 146–162.
[12] R. Sahadevan and P. Prakash, Exact solution of certain time fractional nonlinear partial differential equations,
Nonlinear Dyn. 85 (2016) 659–673.
[13] S.R. Svirshchevskii and R. Sergey, Invariant linear spaces and exact solutions of nonlinear evolution equations,
J. Nonlinear Math. Phys. 3 (1996) 146–169.
[14] K.V. Zhukovsky, Operational solution for some types of second order differential equations and for relevant physical
problems, J. Math. Anal. 446 (2017) 628–647.
)
Volume 12, Issue 2
November 2021Pages 1687-1697
- Receive Date: 03 January 2021
- Revise Date: 08 March 2021
- Accept Date: 22 April 2021