Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations

Abstract We show, using invariant subspace method, how to derive exact solutions to the time fractional Korteweg-de Vries (KdV) equation, potential KdV equation with absorption term, KdV-Burgers equation and a time fractional partial differential equation with quadratic nonlinearity. Also we extend the invariant subspace method to nonlinear time fractional differential-difference equations and derive exact solutions of the time fractional discrete KdV and Toda lattice equations.

[1]  R. Sahadevan,et al.  On solutions of two coupled fractional time derivative Hirota equations , 2014 .

[2]  Stanislav Spichak,et al.  On the Poincare-Invariant Second-Order Partial Equations for a Spinor Field , 1996 .

[3]  I. Podlubny.  Fractional differential equations , 1998 .

[4]  R. Sahadevan,et al.  Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations , 2012 .

[5]  Morikazu Toda,et al.  Theory Of Nonlinear Lattices , 1981 .

[6]  Teodor M. Atanackovic,et al.  Similarity solutions to nonlinear heat conduction and Burgers/Korteweg-deVries fractional equations , 2008 .

[7]  Victor A. Galaktionov,et al.  Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[8]  R. K. Gazizov,et al.  Symmetry properties of fractional diffusion equations , 2009 .

[9]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[10]  S. A. El-Wakil,et al.  Time-fractional KdV equation for plasma of two different temperature electrons and stationary ion , 2011 .

[11]  R. K. Gazizov,et al.  Construction of exact solutions for fractional order differential equations by the invariant subspace method , 2013, Comput. Math. Appl..

[12]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[13]  S. A. El-Wakil,et al.  Ion-acoustic waves in unmagnetized collisionless weakly relativistic plasma of warm-ion and isothermal-electron using time-fractional KdV equation , 2012 .

[14]  Mark Kac,et al.  On an Explicitly Soluble System of Nonlinear Differential Equations Related to Certain Toda Lattices , 1975 .

[15]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[16]  S. A. El-Wakil,et al.  Time-fractional KdV equation for electron-acoustic waves in plasma of cold electron and two different temperature isothermal ions , 2011 .

[17]  A general theory of Lie symmetries for fractional differential equations , 2014 .

[18]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[19]  Anatoly A. Kilbas,et al.  On solution of integral equation of Abel-Volterra type , 1995, Differential and Integral Equations.

[20]  S. R. Svirshchevskii,et al.  Lie-Bäcklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations , 1995 .

[21]  G. Bluman,et al.  Symmetry and Integration Methods for Differential Equations , 2002 .

[22]  F. Mainardi,et al.  Fractional models of anomalous relaxation based on the Kilbas and Saigo function , 2014, Meccanica.

[23]  Evelyn Buckwar,et al.  Invariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Transformations , 1998 .

[24]  MA Wen-Xiu.  A refined invariant subspace method and applications to evolution equations , 2012 .

[25]  P. Artale Harris,et al.  Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method , 2013, 1306.1942.