Approximate Solutions And Symmetries Of Several Nonlinear Differential Equations

Nonlinear phenomena appear in various fields of modern scienceand technology and their mathematical models are usually described bynonlinear equations. Therefore，the study of the method for solvingnonlinear equations is important in theoretical and practical.As we all know, to solve nonlinear partial differential equations byuse of the homotopy perturbation method has many advantages, forexample which has a high precision, clearly in ideas， simple incalculation, fast and effective, etc. The homotopy perturbation methodhas been used to solve nonlinear equations with constant coefficients andvariable coefficients which related to time variables t in soliton theory.But so far, the variable coefficient nonlinear equations related to thespatial variables x is less studied than the variable coefficients equationsrelated to time variables t by the homotopy perturbation method.Therefore， this dissertation focuses the study of the variablecoefficient nonlinear equations in which the coefficients related to timevariables and spatial variables by use of the homotopy perturbationmethod and attempt to give the approximate analytical solution of thevariable coefficients KdV-MKdV equation, the error comparison ofapproximate solution with the exact solution and improve the scope ofapplication of homotopy perturbation method.Second，the research and construction of the exact solution of nonlinear partial differential equations have a better understanding andgrasp for the nonlinear systems. Therefore，solving nonlinear partialdifferential equations and get their exact solution has a very importanttheoretical and practical values. Lie group method is a powerful tool forstudy differential equations and symmetry is one of the basic concept ofLie group theory. Lie symmetry method can give soliton solution ofequations and linear reduction of the original nonlinear partial differentialequations, so the construction of symmetry is very important in solitontheory. In the second chapter we study the symmetry of a nonlinearpartial differential equations and the reduction of this equation is obtainedby applying the symmetry.