The Study Of A New Method For Solving Multiple Solutions Of Semilinear Partial Differential Equations

Along with the more widespread applications of the semilinear partial dif-ferential equations in the fields of scientific research and engineering design, the scientists have paid more and more attentions to the research of the related nu-merical methods about them. Illumined by Mountain Pass Algorithm (MPA), High Linking Algorithm (HLA), Search Extension Method (SEM), Local Min-max Method (LMM) and so on, this paper studies an efficient algorithm for solving multiple solutions of the semilinear partial differential boundary prob-lems.Motivated by the spanned subspace S in [20]-[21], we introduce a trans-formation which is singular on the boundary of the subspace S by using the augmented idea, such that when choosing an initial guess out of the subspace S, the solution sequence from the algorithm can’t pass the singular lines to converge to a solution in the subspace, thus it extends the Newton Method to get more different solutions. In this paper by inducting a variable t to construct a more general singular nonlinear augmented transformation G, the semilinear partial differential problem F is translated to solve the augmented equation G. The key of the augmented Newton Method is to construct the augmented transformation so that the barrier structure changes, which is also the brightest spot differing from the methods proposed before.Numerical experiments for Henon equation with γ=6and f(x,u)=u3in a square domain are presented in this paper. The numerical results show that with the same initial guess uO of the augmented transformation G, the augmented Newton Method can get multiple different solutions with the same symmetries, which also implies that in some sense it relieves the dependence of the initial guess, and we find some general solutions in the numerical exper-iments. It’s more interesting to find that, comparing with the deflation matrix method (DMM), this method reaches the same goal that it avoid converging to founded solutions repeatedly by different means, but without constructing the especial deflation matrix which is defined case by case depending on people’s knowledge of a problem. The combination of the founded solutions is not a new solution, but the founded solutions have influence on the convergent solution, so we can consider the properties of the founded solutions in the experiment and set suitable initial coefficients tO,tlO,..., and tkOsuch that the initial guess is in the neighborhood of the solution u*and speed up the convergent rate with litter iteration. The augmented Newton Method has the merits of easy implementation and general applications to all kinds of equations and equation sets.