The General Series Expansion Method Of Theory And Application

The solution of nonlinear differential equation is one of the kernel problems. Professor Liao Shijun presented the homotopy analysis method in 1992, which is different from traditional analytic methods.Different from the traditional analysis method,this method provides a convenient approach to control and adjust the convergence region and rate of the series solution of nonlinear problems. And it also gives us more freedom to choose different base functions to express the solution of nonlinear problem.The homotopy analysis method also contains some traditional methods, such as Adomian decomposition method, Lyapunov artificial small parameter method andδexpanding method.Professor Liao and his team,domestic and foreign scholars did a lot of work in the homotopy analysis method, but the mathematic meanings of the auxiliary parameter h in the method were not studied clearly.The range of the value of h and the convergence region of series solution were solved by numerical methods. Professor Liao proved the generalized Taylor theorem in 2003. He believed that this theorem could explain the mathematic meanings of h , and the validity and rationality of the homotopy analysis method was proved from the aspects of symbolic logic.Professor Liu proved the generalized Taylor theorem was only the Taylor series espansion at the other point t₀ of the ordinary Taylor's theorem. He established a connection between t 0 and h and explained the mathematic meanings of h . He also revealed the essence of the generalized Taylor theorem.On the basis, he brought forward the general series expansion method, used it to solve some examples and gained a better result than the homotopy analysis method.The difficulty of general series expansion method lies in how to calculate the coefficient when we select different base function and different initial point. Professor Liu studied general series expansion method When (t - t₀)ⁿ and ( e^-2t- t₀)ⁿ were selected as basic functions.By using general series expansion method ,this paper used the more complicated base function {(t - t₀)^m e^-m|m=0,1,2,…n=1,2,…}, to studied the differential equation V′(t) = 1-V²(t), V(0)= 0. We could obtain a better result than the homotopy analysis method.