Solve The Nonlinear Equation Based On Homotopy Method

Nonlinear equation solution is one of the very important problems in physics and engineering. There are many methods for the solution of weakly nonlinear equation, such as perturbation method, KBM method, L-P perturbation method, Adomian decomposition method, etc. But in practice it is common to see some strongly nonlinear equation which can't be solved by the traditional perturbation method.Appling homotopy perturbation method to solve Nonlinear equation is not required to make small parameters exist in the equation in form, but based on homotopy topology theory to introduce an imbedding parameter to make up homotopy continuation and expresses the solutions in the form of power series of q, the method of which has been applied to the Bifurcation nonlinear problems, initial value problem of nonlinear and integral equation.Homotopy analysis method is to introduce imbedding parameter q∈[0,1] as well, get Zero order deformation equation and High order deformation equation by constituting homotopy function, constitute infinite number by changing the original nonlinear equations into High order deformation equation, and take the sum of the former sub-problem's solution to approach the access to exact solution.This method doesn't rely on small parameter at all, and can control and adjust the convergence of the solution, choose the initial assumable solution, assistant to linear operator, which is widely applied to solve all kinds of nonlinear equations.This thesis applies perturbation method to solve a class of nonlinear vibration equation and compares with L-P perturbation method and Krylov expansion method in the aspect of result, and meanwhile solves Bratu-type equation, the solution from this method is more close to the exact solution than that of Iterative perturbation method. This article applies homotopy analysis to solve Stefan equation, makes error comparison between the analytical solution of the first steps and the exact solution, solves wave equation and modifies Burgers equation, which illustrates the applicability and superiority.