12 - Partial Differential Equations

Abstract

Many physical and mathematical situations are described by ordinary differential equations and others are described by partial differential equations. This chapter discusses that one way to solve some partial differential equations is the method of separation of variables. Partial differential equations cannot be studied without an introduction of some of the terminology associated with the topic, that is, linear second-order partial differential equation. A method that can be used to solve linear partial differential equations is called separation of variables. The goal of the method of separation of variables is to transform the partial differential equation into a system of ordinary differential equations by separating the variables. One of the more important differential equations is the heat equation. In one spatial dimension, the solution of the heat equation represents the temperature in a thin rod or wire of length. As the rate at which heat flows through the rod depends on the material that makes up the rod, the constant that is related to the thermal diffusivity of the material is included in the heat equation.