As we know nearly all physical, chemical, and biological processes in nature can be described or modeled by dint of a differential equation or a system of differential equations, an integral equation or an integro-differential equation. The differential equations can be ordinary or partial, linear or nonlinear. So, we concentrate our attention in problem that can be presented in terms of a differential equation with fractional derivative. Our research in this work is to use symmetry transformation method and its analysis to search exact solutions to nonlinear fractional partial differential equations.
The Lie group analysis or in other word the symmetry analysis method is extended to deal with a timefractional order derivative nonlinear generalized KdV equation. Our research in this work aims to use transformation methods and their analysis to search for exact solutions to the nonlinear generalized KdV differential equation. It is shown that this equation can be reduced to an equation with Erdelyi-Kober fractional derivative. In this study, we research the initial and boundary conditions, considering them invariant, and so we get two ordinary initial value problems, i.e. two Cauchy problems. Conservation laws for the given equation are also investigated. In this work, we introduce symmetry analysis and find conservation laws for the nonlinear generalized time-fractional KdV equation by the Lie groups method using fractional derivatives in the Riemann-Liouville sense.
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