| This paper is focused on the bifurcation of the numerical solution of the generalized Burgers equation ut ? u xx ?λu + uux=0in one-dimensional space with periodic boundary condition. The Burgers equation is a nonlinear partial differential equation of the second order. It can be used not only in fluid dynamics and boundary layer behavior, but also a simplified model for mass transport. So the study of the Burgers equation of numerical solution can be used to enrich the numerical analysis theory, and has much practical value.This paper is divided into three parts. Firstly, we introduce the background and the practical significance of the problem we are considering, then review some existing results and the numerical analysis methods.The second part is the theoretical work, divided into two aspects. The first is focused on the generalized Burgers equation, and the numerical solution method is given, after which we get a high dimension of the difference equation. Due to the high number of dimensions of the equation, the iterative matrix is a large sparse matrix. According to the characteristics of the matrix, the analysis method is used strictly to prove the existence of the bifurcation. It is shown that the numerical solution of the generalized Burgers equation undergoes a fold bifurcation whenλ= 0.Furthermore, Runge-Kutta method which is stable of high with high precision is used to solve the generalized Burgers equation, and more complex difference equation of high dimension is got, then the existence of the bifurcation is analyzed. We show that whenλ= 0the numerical solution of the generalized Burgers equation undergoes a fold bifurcation.In the second place, we investigate a Reaction-Diffusion system:We introduce the equation undergoes a Hopf bifurcation as the parameterλchanges.The third part is the numerical experiment. A lot of numerical experiments is did to verify that the theoretical results have been of the accuracy of the results. |
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