Lie Symmetric Analysis Of Time - Fractional KdV Equation And Coupled KdV Equations

The fractional differential equation is an important branch of modern mathematics.In the real world, because many phenomena cannot be described by using the integerorder differential equations, the fractional differential equations have been paid moreand more attentions and becoming great importance in mathematical world in recentyears. The fractional differential equations have been researched and developeddeeply. The Lie symmetry analysis is an effective tool to study partial differentialequations; it can be applied to get the symmetry and invariance of systems. Asymmetry corresponds to a conservation law. The invariance property can be used totest the accuracy of the numerical results. In this dissertation, we shall apply Liesymmetry method to study time fractional KdV equation and coupled KdV equations,and obtain the following results:Firstly, applying Lie symmetry analysis method to study time fractional KdVequation, we obtain infinitesimal symmetry generator and self-containedcommutation table, and give the corresponding symmetry group for symmetrygenerator. Then, by using the invariance, we obtain symmetry reductions and somegroup invariant solutions of time fractional KdV equation.Secondly, we extend Lie symmetry analysis method to the time fractional coupledequations, and study three time fractional coupled KdV equations with physicalbackground. We obtain infinitesimal symmetry generator and self-containedcommutation table, and give the corresponding symmetry group for symmetrygenerator. Then, by using the invariance, we obtain symmetry reductions and somegroup invariant solutions of three time-fractional coupled KdV equations.Finally, we summary the conclusions of this dissertation, and put forward some ofnew problems to be studied in the future time.