TABLE OF CONTENTS With this chapter we begin a systematic discussion of partial differential equations (PDEs). Equations of this type arise when several independent variables are involved in a differential equation. In typical physical problems the independent variables are one or more spatial variables and time. Examples of PDEs include (a) the heat equation, which describes evolution of spatial temperature distribution in a conducting body, and (b) the kinematic wave equation, which describes processes such as adsorption of a component from a flowing fluid.
Some fundamental aspects of PDEs are discussed in this chapter. These equations are first classified with regard to whether they are linear, quasilinear, or nonlinear. Following this, characteristic curves of first-order PDEs, which provide a parametric description of the solutions, are discussed in detail. Finally, second-order PDEs are treated and a general classification of commonly encountered two-dimensional equations is presented.
These fundamental aspects are utilized in chapters 6 and 7, which also deal with PDEs. Chapter 6 is devoted to first-order equations, while generalized Fourier transform methods for solving second-order equations are discussed in chapter 7.
In general, an m-dimensional nth-order PDE can be represented as a function of m independent variables ( x 1, x 2, , x m), one dependent variable (the unknown z), and all the partial derivatives of z with respect to the independent variables up to order n:
where F is an arbitrary function and the...
