Branching of Solutions to Some Nonlinear Eigenvalue Problems

Abstract

The multiplicity of solutions to nonlinear eigenvalue problems is analyzed by a method originally proposed by Hammerstein, and the results are given in tables which display the relation between the number of solutions for eigenvalues in the neighborhood of a critical eigenvalue and the properties of the nonlinear function and its derivatives. Several general types of nonlinear functions are considered, and a simple method of estimating the critical eigenvalue for each type is presented. Since these functions describe physical phenomena, a stability analysis is given for the cases of multiple solutions in order to determine which solution represents the observed physical state. The results are applied to the following nonlinear eigenvalue problems: (i) nonlinear heat generation and the temperature distribution in conducting solids; (ii) temperature distribution in a heat‐conducting gas undergoing chemical reactions, leading to a thermal explosion; (iii) nonlinear effects of temperature‐dependent viscosity on the temperature distribution of a fluid flowing in a pipe; (iv) neutron flux distribution in a reactor for temperature‐dependent cross sections.