Nonlinear Laplace/Poisson Equation

Abstract

In the previous chapter, we presented the boundary element and Green element formulations for the linear version of the steady 1-D. second-order differential equation given by eq. (2.1) in which the parameter K assumed a constant value. We used the example of flow in a confined aquifer of uniform thickness to derive the linear form of the differential equation. Here we relax that condition and consider cases where K is a function of the spatial variable x (heterogeneous case), and a function of the dependent variable h (nonlinear case). Nonlinear heterogeneous problems are frequently encountered in many engineering applications — heat flux through a material whose thermal properties are nonlinear, infiltration into unsaturated soils, elastic deformation of a bar of varying section subject to axial loads, etc. We elect to use a heat transfer example to derive the nonlinear equation of the form of eq. (2.1).