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# COMPUTATIONAL METHODS FOR NONLINEAR ELLIPTIC EIGENVALUE PROBLEMS

#### Abstract

A basic problem is that of finding nontrivial solutions of the following nonlinear eigenvalue problem: $-\Delta$u = $\lambda$F$\prime$ (u) in D, u = 0 on $\partial$D, where D is a domain in R$\sp{\rm N}$, N = 1,2,3, and F$\prime$(u) is non-monotone. Problems of this kind arise, for example, in plasma physics, fluid dynamics, and astrophysics.^ In the first part of this thesis, an equivalent variational formulation is used to obtain an iterative procedure for solving the problem with a general function F(u). The global convergence of this procedure is established, i.e., convergence from any initial guess. The method is applied to a test problem with F(u) = $-$cos u.^ In the last part of this thesis, a problem of internal solitary waves in stratified fluids is studied. Attention is paid to establishing the range of validity of existing asymptotic theories. New large amplitude solutions are obtained numerically. ^

Mathematics

#### Recommended Citation

SHIRLEY BARBARA POMERANZ, "COMPUTATIONAL METHODS FOR NONLINEAR ELLIPTIC EIGENVALUE PROBLEMS" (January 1, 1987). Doctoral Dissertations Available from Proquest. Paper AAI8727097.
http://scholarworks.umass.edu/dissertations/AAI8727097

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