Numerical Calculation Method For Multiple Solution Problems Of A Class Of Partial Differential Systems

The nonlinear partial differential equations are extensively applied to science and engi-neering,and studying their numerical methods attracts more and more attention from scholars.Inspired by the existing algorithms,this thesis mainly discusses a high-precision algorithm for numerical computation of multiple solutions to a class of partial differential systems.The form of the partial differential systems is as follows:(?) where Ω=[-1,1]×[-1,1],γ,λ,θ,κ are given parameters,and satisfy δ>0,γ≤λ.When a(x)=0,the differential systems have variational structure.There are many mature algorithms for numerical multiple solutions.When a(x)≠ 0,the systems don’t have variational structure and the existing algorithms related to energy are no longer applicable.When κ=-1,they are called definite systems.when κ=1,they are called indefinite systems.Firstly,the partial Newton-correction method are introduced.We present a new singular transformation for systems.Then its mathematical validation and a flow chart of partial Newton-correction algorithm.Then we establish a Legendre-Gauss-Lobatto pseudo spectral format for solving multiple solutions of systems,and numerical experiments are carried out to verify the accuracy of the numerical scheme.The partial Newton correction algorithm is used to solve the multiple solutions of the partial differential systems with variational structure.Next,we provide the solving process and the numerical results for the definite and the indefinite systems.The numerical results show that the algorithm not only can calculate the solutions by the existing methods,but also can compute the new solutions.Finally,we give the solving process of partial Newton-correction method and the numerical results for the definite and indefinite systems without variational structure,though they are difficult to be solved by existing algorithms.The numerical results demonstrate the effectiveness of these methods.Our methods can transform the multiple solutions of nonlinear partial differential systems into two linear partial differential equations,which greatly simplifying the computation.The suggested methods can effectively overcome the difficulty of initial value selection encountered in other algorithms.