Nonlinear Partial Differential Equations

Abstract

In the previous chapters, we discussed the solution of linear partial differential equations. Special focus was given to the solution of internal heat transfer problems in duct flows. However, in most technical applications, problems are often described by nonlinear partial differential equations. For a lot of these applications, the equations have to be solved by numerical methods. In contrast to the large amount of literature dealing with the solution of linear partial differential equations, much less literature exists on the solution of nonlinear partial differential equations. One of the major difficulties arising in the solution of nonlinear problems is that we are no longer able to use the powerful superposition method for constructing solutions as for linear problems. Sometimes, the equations under consideration may be linearised by using perturbation methods. An example on how to use this sort of method is shown in Chap. 4 for the solution of eigenvalue problems. In the present chapter, we do not discuss this solution approach. The interested reader is referred to the books of van Dyke (1964), Kevorkian and Cole (1981), Simmonds and Mann (1986), Aziz and Na (1984) and Schneider (1978) for many interesting applications of the perturbation method to fluid dynamics and heat transfer problems. In the following sections, we intend to provide a short overview on some selected solution methods for nonlinear partial differential equations occurring in heat transfer and fluid flow problems. The solution approaches discussed here include, for example, the method of separation of variables, the Kirchhoff transformation, and special solutions of the energy equation.