Solve The Nonlinear Equation Based On Modified Homotopy Method

ABSTRACT: With the development of modern science and technology invarious fields, the nonlinear problems appear, and they become the commonproblems which are usual in the modern science and technology in naturalscience and engineering. The problems are described by nonlinear equations.So solving the nonlinear equation is always hot topic studied by scholars. In1998, with the inspiration of homotopy analysis method, HeJihuan combinedperturbation method and homotopy method and put forward homotopyperturbation method(HPM). When homotopy perturbation method is used tosolve nonlinear equation, it does not depend on the small parameter chosen,but an equation with embedding parameter q∈[0,1]is structured by thehomotopy technology. The solution is showed in the way of the power seriesform about q, and the embedding parameter is regarded as a small parameterto deal with. The method has been applied in generalized eigenvalue problem,the initial and boundary value problem about partial differential equation,integral equation and differential equation and so on. When homotopyanalysis method is applied to solve nonlinear equation, a embedded variablesq is introduced, and homotopy function is structured through the homotopymethod. Make homotopy as zero, get zero order deformation equation andhigh order deformation equation, and turn the original nonlinear equationsinto infinite linear problems. At last, take the sum of the first few linearproblems to approximate the exact solution. This method, as well as thehomotopy perturbation method, doesn’t rely on small parameters, and theinitial guess solution and auxiliary linear operator can be chosen freely.Thismakes that the homotopy analysis method is widely applied to solve all kindsof different nonlinear equation problems. This paper applies the homotopy perturbation method to solve nonlineardifferential and integral equations, and at the same time, comparerespectively the results solved by improved homotopy perturbation methodimproved, the modified homotopy perturbation method and Adomiandecomposition method. Through the comparison, it shows that the homotopyperturbation method is simpler than the relative variational iteration method.In addition, aim at the solution of the generalized regularization equation, wecompare the solution got by the homotopy perturbation method with exactsolution and it reflects that the homotopy perturbation method can make thesolution closer to the exact solution. With the help of the homotopy analysismethod in solving Klein-Gordon equation, the comparison about numbervalues and example analysis shows the homotopy analysis method is ofeffectiveness and applicability.