Expansion Method for Nonlinear Boundary-Value Problems

Abstract

A homogeneous nonlinear boundary-value problem, which reduces to the Helmholtz equation when the nonlinearity is removed, is solved by an expansion method using as a basis the eigenfunctions of the linear Helmholtz equation. The nonlinear differential equation is reduced to a nonlinear algebraic system in the expansion coefficients, which can be easily solved with any desired degree of accuracy. It is found that with only three terms in the eigenfunction expansion, a satisfactory agreement with the exact numerical solution of the problem is obtained, even for strongly nonlinear cases. Solutions are presented both in one-dimensional slab and cylindrical geometries. It is also shown that the method can be applied to inhomogeneous problems.