X - Partial Differential Equations and Boundary Value Problems

Abstract

Partial differential equations are classified as to order and linearity in the same way as ordinary differential equations. The order of an equation is the order of the highest-order partial derivatives of the unknown function that appear in the equation. A boundary value problem possessing a unique solution that depends on the prescribed values in the boundary conditions is said to be a well-posed problem. Boundary conditions that yield a well-posed problem with a hyperbolic equation do not in general yield a well-posed problem with an equation of elliptic type. For some partial differential equations, it is possible to find expressions that represent all solutions, that is, represent the general solution. Such expressions contain arbitrary functions instead of arbitrary constants, as in the case of ordinary differential equations. The expression for the Laplacian of a function in spherical coordinates can be derived in the same way as was done for cylindrical coordinates, although the algebra is more complicated. In problems where the number of independent variables is greater than two, the method of separation of variables leads to the notion of a multiple Fourier series.