Numerical Solution Method For A Class Of Partial Differential Equations With Multiple Fixed Points

In this master thesis,inspired by several numerical methods for solving multiple solutions of nonlinear partial differential equations(PDEs),we discusses multiple fixed points of a class of nonlinear PDEs.The form of the PDE is(?)where u?U=W01,p(?),?pu(x)=div(|?u(x)|p2?u(x)),?=[-1,1]×[-1,1]and p>1,r?0,?,? ?R and l>0 are given parameters.Firstly,a partial Newton-correction method is introduced to find multiple fixed points of non-linear PDE.A new augmented singular transformation is developed to form a barrier surrounding previously found fixed points so that an algorithm search from outside cannot pass the barrier and penetrate into the inside to reach a previously found fixed points.Then its mathematical valida-tion and a flow chart of the partial Newton-correction method are given.Finally,we establish Legendre-Gauss-Lobatto pseudo spectral scheme.When p=2,the equation is Laplace equation.By constructing the augmented singular trans-formation,the original problem is transformed into the problem of solving the augmented equa-tion.The augmented equation is discretized by Legendre-Gauss-lobatto pseudo spectral method.Using partial Newton-correction method,we got the multiple fixed points of the boundary value problems of Henon equation,Schrodinger equation and nonlinear Laplace equation without vari-ational structure successively.When p?2,the operator ?p is nonlinear,which adds a lot of complexity for our comput-ing.By constructing a more general augmented singular transformation,the original problem is also transformed into a fixed points problem for solving the augmented equation.The augment-ed equation is discretized by Legendre-Gauss-lobatto pseudo spectral method.The multiple fixed points for p-Henon equation,p-Schrodinger equation and nonlinear p-Laplace equation without variational structure are solved successively by the partial Newton-correction method.Numerical results demonstrate the effectiveness of these approaches.Our methods can transform the non-linear fixed point problem into two linear partial differential equations,which greatly simplifying the computation.The suggested algorithms can overcome effectively the difficulty for searching the initial guess encountered in some other popular algorithms.