Chapter 8 - Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables

Abstract

This chapter solves the Laplace's equation, the wave equation, and the heat equation in polar or cylindrical coordinates. In Cartesian coordinates, the ordinary differential equations (ODEs) that arise are simple to solve. The chapter shows that in cylindrical and spherical coordinates not all the ODEs are as agreeable. The solutions to these more difficult ODEs go by the names of Bessel functions and Legendre polynomials. In cylindrical coordinates only Bessel functions are needed. The chapter presents an example where Bessel functions arise. The chapter defines a Bessel equation, demonstrates one solution to such equations, and then makes a transformation that enables a person to solve a Bessel-like equation.