文章主要研究了同伦摄动法在求解非线性偏微分方程中的应用问题。简要介绍了同伦摄动法，该法的基本思想是通过行波变换并结合同伦摄动理论，把求解某些非线性偏微分方程的问题转化为求解常微分方程的初值问题，最后得出近似解。文中求解了非线性平流方程和Fisher方程。结果表明，这种方法简单而有效，显示同伦摄动法具有一些显著特点，例如可以任意选取初始猜测解、不依赖非线性方程中的小参数等等，同时可以简化复杂的求解过程，它的二阶近似解就相当精确。同伦摄动方法是一种很普遍的解决非线性问题的方法。%This paper mainly studied the homotopy perturbation method in solving the application problems of nonlinear partial differential equations. This paper first briefly introduces the homotopy perturbation method, the basic idea of the method is to transform the nonlinear partial differential equations into ordinary differential equations by combining the traveling wave transformation with the homotopy perturbation theory, and the approximate solution will be obtained. In this paper,we solve the nonlinear advection equation and the Fisher equation.The results show that this method is simple and effective,Also shows the homotopy perturbation method has some significant features, such as an arbitrary choice of initial guess solutions , the independence of the small parameters in nonlinear equations, etc., and at the same time, this method can also simplify the complex solving process, Its second-order approximate solution is quite accurate, the homotopy perturbation method is a very common method for solving nonlinear problems.
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