Study on lie symmetry analysis to differential equations

Abstract

In this work, travelling wave and a family of invariant solutions of nonlinear filtration partial differential equation are found. In general, by using Lie symmetry transformations, it is possible to reduce the partial differential equations into ordinary differential equations, if the symmetries admitted by target equations allow to determine the Lie point transformations. In the case of nonlinear filtration equation, we are looking for travelling wave solutions which are invariant under a particular group of Lie symmetry. By using the invariance surface condition of nonlinear filtration equation we get a much simpler form of partial differential equation to solve than the original one. After that applying the method of characteristics to obtain fundamental differential invariant such that the travelling wave solutions of nonlinear filtration equation can be found. In case of a solution of any partial differential equation that is invariant under a group of Lie symmetry it is possible to find new solution from invariant solution by using the other Lie symmetries of given PDE. Since travelling wave solutions are invariant, therefore with the help of above strategy new solutions of nonlinear filtration equation are found by utilizing the remaining symmetries of filtration equation from travelling wave solutions.