7.1 Introduction

In many branches of mathematics, we do classify the objects of study. For example, in linear algebra, we classify the square matrices into symmetric matrices, orthogonal matrices and others. Similarly, we may group a particular set of partial differential equations (PDE) whose solutions have common qualitative and/or quantitative properties. Among the important equations of mathematical physics, the three main equations, namely the wave equation, Laplace equation and the heat equation, are classified as follows:.

Wave Equation: This equation is a prototype of hyperbolic equations. In one dimension, this equation models many real-world problems, such as small transversal vibrations of a string, longitudinal vibrations of a rod, electrical oscillations in a wire, torsional oscillations of shafts and oscillations in gases.

In two dimensions, the wave equation models vibrations of a membrane, and in three dimensions, it models the propagation of sound waves, light waves and electromagnetic waves in a medium.

Heat Equation: It models the heat diffusion in a medium and is a prototype of parabolic equations. It also describes the diffusion of a chemical substance in a medium. As such, it is also called diffusion equation.

Laplace Equation: A steady-state process, that is, a physical process that does not change with time, is generally modelled by a Laplace or Poisson equation and is classified as an elliptic equation. A physical process that is in equilibrium state is also described by an elliptic equation.

Another reason for the classification of PDE is more mathematical in nature by the consideration of a comparison with ordinary differential equation (ODE). Typically, an ODE is associated with a Cauchy problem or an IVP.