A Method For Solving Multiple Solutions Of Neumann Boundary Value Problems For A Class Of Partial Differential Equations

In this master thesis,we discuss the multiple solution problem for a class of partial differential equations with Neumann boundary condition.This paper includes two parts: in the first part,we research the multiple solution of the Schr ¨odinger equation on a square,its forms is where x₀ is the center of domain ? = [-1,1] × [-1,1],p > 1,? > 0,? ? R,? ? R and r ? 0 are given parameters.We first compute the multiple non-trivial solutions of the equation?0.1?on a square by using symmetry-breaking bifurcation theory and pseudospectral methods.Then,starting from the non-trivial solution branches of the corresponding nonlinear problem,we take ?,? or r in the equation?0.1?as bifurcation parameters respectively and further obtain the symmetric positive solution branch of the equation?0.1?numerically by the continuation method.During continuation,we find the potential symmetry-breaking bifurcation points and propose the extended systems,which can detect symmetry-breaking bifurcation points on the branch of the symmetric positive solutions accurately.We compute the multiple positive solutions with various symmetries of the equation?0.1?by the branch switching method based on the Liapunov-Schmidt reduction.These solutions are concerned by scientists.Finally,the bifurcation diagrams are constructed,showing the symmetry-breaking bifurcation positive solutions of the equation?0.1?on a square.In the second part,we research the multiple solution of the Concave-convex equation on a square,its form is where x₀ is the center of domain ? = [-1,1] × [-1,1],0 < q < 1 < p,? ? R,? ? R,? ? R and r ? 0 are given parameters.Based on the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory,we compute and visualize multiple solutions of the Concave-convex equation on a square by using Jacobi spectral method.Starting from the non-trivial solution branches of the corresponding nonlinear bifurcation problem,we take ?,? or r in the equation?0.2?as bifurcation parameters respectively and obtain multiple solutions of Concave-convex equation with various symmetries numerically.During continuation,we find the potential symmetry-breaking bifurcation points and propose the extended systems,which can detect symmetry-breaking bifurcation points on the branch of the symmetric positive solutions.The symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.The bifurcation diagrams are constructed,showing the symmetry-breaking bifurcation positive solutions of the equation?0.2?on a square.Numerical results demonstrate the effectiveness of these approaches.The final section is for some concluding discussions.