Symmetry properties and explicit solutions of the nonlinear time fractional KdV equation

• Published in: Boundary value problems Vol. 2013; no. 1
• Main Authors: Wang, Gangwei, Xu, Tianzhou
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 08-11-2013
• Subjects: Analysis; Approximations and Expansions; Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Erdélyi-Kober operators; exact solutions; modified Riemann-Liouville derivative; Lie symmetry analysis; fractional KdV equation
• ISSN: 1687-2770; 1687-2770
• DOI: 10.1186/1687-2770-2013-232

The time fractional KdV equation in the sense of the Riemann-Liouville derivatives is considered. The symmetry properties of the time fractional KdV equation is investigated by using the Lie group analysis method. On the basis of the point symmetry, the vector fields of the time fractional KdV equation are presented. And then, the symmetry reductions are constructed. By right of the obtained Lie point symmetries, it is shown that this equation could transform into a nonlinear ordinary differential equation of fractional order with the new independent variable
ξ
=
x
t
−
α
/
3
. The derivative is an Erdélyi-Kober derivative depending on a parameter
α
. At last, by means of the sub-equation method, some exact and explicit solutions of the time fractional KdV equation are constructed.
MSC:
22E70, 26A33.