The Research Of Numerical Solution And Bifurcation For The Gilson-Pickering Equation

As is well-known，the partial differential equation is a relatively wide rang ofsubject, it contains many aspects of the Principles of Mathematical Analysis, in the18th century, some scholars began to combined with the content of mechanics andphysical to study partial differential equations. Among them, the equation of thevibrating string and the equation of heat conduction and the harmonic equation arethe earliest partial differential equation that they are studied. This paper tries tostudy the Gilson-Pickering equation that is a partial differential equation, the thesishas 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. Andthere 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 thenumerical solution, so through some methods we can reduce orders of theGilson-Pickering equation and make it into the ordinary differential equation.Then, this part mainly introduces two numerical methods, namely theRunge-Kutta method and the parallel algorithm. Using the Runge-Kutta methodsolves the special case of the Gilson-Pickering equation which is based on theRunge-Kutta method for the numerical solution of ordinary differential equations.Through the research of the parallel algorithm, the numerical solution model of theGilson-Pickering equation is studied, and using the parallel algorithm solves thespecial case of the Gilson-Pickering equation.At last, The main content of the third chapter is the image branch of theGilson-Pickering equation, this chapter first introduces the bifurcation theory, andintroduces the corresponding bifurcation when it meet the conditions. In this partthere are a lot of forms of the bifurcation, such as Flod bifurcation, Flip (doublecycle) bifurcation, Hopf bifurcation, etc.; and then we study the bifurcationphenomena when the parameters of the equation take different values, and give thecorresponding image which are based on some theoretical basises.