Eigenfunction expansion method for multiple solutions of fourth-order ordinary differential equations with cubic polynomial nonlinearity

Abstract

Multiple solutions of fourth-order ordinary differential equations (ODEs) with cubic polynomial nonlinearity are presented in this paper. The ODEs are discretized by the eigenfunction expansion method. Discretization error estimates are derived. We construct an efficient polynomial homotopy to find all solutions for the system of polynomial equations on the coarse level by recursion. Two kinds of filters are suggested for removing possible spurious solutions of the discretized system of polynomial equations. Numerical experiments are included to verify the error estimates and efficiency of the proposed homotopy.