Numerical Solutions Of Lane-Emden Equation By Bifurcation Method

Using the Liapunov-Schmidt method and symmetry-breaking bifurcation theory, wecompute and visualize multiple solutions of Lane-Emden equation on the plane of R2 with ahomogeneous Dirichlet boundary condition, which plays an important role in stellar structureand evolution theory. The reward will be twofold: ?rstly the computation will be simpli?ed,secondly the numerical solutions can easily be classi?ed according to their symmetry. Onthe other hand, we also obtain some interesting symmetry results from our numerical exper-iments. The domains we consider include unit square, disk and other complex geometries.An outline of the paper is as follows. In Chapter 1, the historical background to the prob-lem has been introduced. Next we give our theoretical analysis of the problem in Chapter2. In Section 2.1, we apply Lyapunov-Schmidt method to show that the nontrivial solu-tion branches of (1.2) bifurcating from the bifurcation points turn left. In Section 2.2 thesymmetries of solutions of (1.1) and (1.2) are discussed in detail, then we can reduce thecomputation as much as possible and subsequently solve numerical solutions with differ-ently symmetric property. Finally, in Chapter 3, we give the numerical results. In Section3.1, several effective numerical methods to solve multiple solutions of (1.1) for differentdomains are devised. In Section 3.2, we plot some multiple solutions of (1.1) for 2 ≤p ≤5.