Improved Partial Newton-correction Method And Its Applications In Finding Multiple Solutions Of Nonlinear Partial Differential Equations

Based on partial Newton-correction method(i.e.,PNC method),we develop a new numerical method(i.e.,improved partial Newton method)for solving multiple unstable solutions of nonlinear PDEs,and prove the local convergence of the new algorithm.First of all,by modifying or supplementing the definition of the original augmented transform,the singularity in the original augmented transform is removed.Then,by virtue of the separability of Nehari manifold N and zero solution,we establish and prove a global separation theorem and its generalization(i.e.,a local separation result)on some submanifold M of N.Analysis shows that the separation theorem is very important for the original method and the new algorithm to find multiple unstable so-lutions;meanwhile the separation conditions required by the theorem can be strictly verified.When available,certain symmetry of those nonlinear PDEs is used so that the new method can find their multiple symmetric solutions more rapidly.For the new aug-mented transform,we show that some unknown solutions of certain nonlinear PDEs are closely related to their zero kernel space.In the proof,we remove the so-called standard convergence assumption required by the original method.In addition,the local conver-gence result of the new method is given meanwhile the key inequality therein is also numerically verified.Finally,for the square,circular,dumbbell-shaped and annular domains,we obtain multiple solutions and their functional energy(if the correspond-ing problem is variational)of five types of nonlinear PDEs through different numerical experiments.Comparison between our results and the previous results demonstrate the feasibility and effectiveness of the proposed method.