The Existence And Uniqueness Of Periodic Solution Of Duffing Equations With Numerical Solution Methods

In the thesis, based on knowing about the su?cient conditions for the exis-tence and uniqueness of periodic solution and numerical methods of di?erentialequation at home and abroad, we consider the existence and numerical solutionmethods of the delay Du?ng di?erential equation. Then, we brie?y introducethe su?cient conditions and numerical solution methods for several kinds ofsimilar equations. Finally we discuss methods of solving the standard Du?ngequation.This thesis consists of three chapters. The main contents are as follows:In chapter 1, we mainly introduce the development of Du?ng equation andreview some significant research.In chapter 2, we first introduce the following delay Du?ng di?erentialequationax" (t) + bx(t) + g(x(t -Ï„)) = f(t).Here a and b are constants, g(x) and f(t) are functions defined in R with f(t)T-periodic. We introduce the background of delay Du?ng equation and discussthe su?cient conditions for the existence and uniqueness of periodic solution.Then we derive an e?cient numerical method for solving delay Du?ng equa-tion. Numerical examples are given to illustrate the e?ciency of our algorithms.Finally we brie?y discuss su?cient conditions for the existence and uniquenessof periodic solution of the following equations:ax" (t) + cx' (t) + bx(t) + g(x(t -Ï„)) = f(t),ax" (t) + cx' (t) + bx(t) + g(x(t -Ï„(t))) = f(t),ax" (t) + f[x'(t)] + bx(t) + g(x(t -Ï„(t))) = p(t).In chapter 3, we discuss a kind of standard Du?ng equation:u (t) + grad(G(u(t))) = p(t) And discuss the numerical solution method of the equation. We extend thenumerical method discussed in the second chapter. Numerical examples of two-dimensional and three-dimensional are given to illustrate the e?ciency of thealgorithm.