Abstract
This chapter deals with partial differential equations (PDEs). It surveys the most important PDEs with the Dirichlet or Neumann boundary conditions: the Laplace, Poisson, wave, and diffusion equations. The method of separation of variables is discussed for Cartesian, cylindrical, and spherical coordinate systems. Two situations are identified: One with the solution a unique linear combination of basic solutions (normal modes), and one with many individual solutions, each associated with an eigenvalue. Examples of both problem types are given. The angular solutions to central-force PDEs are examined in detail. When in a specific form they are identified as spherical harmonics. The properties of the spherical harmonics are surveyed; they are classified using the standard angular-momentum symbols. Brief discussions deal with Green’s-function methods for inhomogeneous problems and integral transform methods for converting PDEs into ODEs. All topics of this chapter are treated using symbolic computation.
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