关于gilson-pickering方程数值解及分支分析-numerical solution and branch analysis of gilson - pickering equation.docx

AbstractAs is well-known，the partial differential equation is a relatively wide rang of subject, it contains many aspects of the Principles of Mathematical Analysis , in the 18th century, some scholars began to combined with the content of mechanics and physical to study partial differential equations. Among them, the equation of the vibrating string and the equation of heat conduction and the harmonic equation are the earliest partial differential equation that they are studied. This paper tries to study the Gilson-Pickering equation that is a partial differential equation, the thesis has three chapters, each one mostly contents as follows:First ， there is an introduction that introduces the background and thedevelopment of the domestic and abroad of the Gilson-Pickering equation. And there are some examples when the parameters of the equation take different values, and the application of these equations in practice are introduced; at the same time, because the order of the Gilson-Pickering equation is high, it is difficult to get the numerical solution, so through some methods we can reduce orders of the Gilson-Pickering equation and make it into the ordinary differential equation.Then, this part mainly introduces two numerical methods, namely the Runge-Kutta method and the parallel algorithm. Using the Runge-Kutta method solves the special case of the Gilson-Pickering equation which is based on the Runge-Kutta method for the numerical solution of ordinary differential equations. Through the research of the parallel algorithm, the numerical solution model of the Gilson-Pickering equation is studied, and using the parallel algorithm solves the special case of the Gilson-Pickering equation.At last, The main content of the third chapter is the image branch of the Gilson-Pickering equation, this chapter first introduces the bifurcation theory, and introduces the corresponding bifurcation when it meet the conditions. In this part there are a lot of forms of th