# 2: Second Order Partial Differential Equations


"Either mathematics is too big for the human mind or the human mind is more than a machine." - Kurt Gödel (1906-1978)

• 2.1: Introduction
In this chapter we will introduce several generic second order linear partial differential equations and see how such equations lead naturally to the study of boundary value problems for ordinary differential equations. These generic differential equation occur in one to three spatial dimensions and are all linear differential equations.
• 2.2: Derivation of Generic 1D Equations
The wave equation for a one dimensional string is derived based upon simply looking at Newton’s Second Law of Motion for a piece of the string plus a few simple assumptions, such as small amplitude oscillations and constant density.
• 2.3: Boundary Value Problems
For an initial value problem one has to solve a differential equation subject to conditions on the unknown function and its derivatives at one value of the independent variable. Typically, initial value problems involve time dependent functions and boundary value problems are spatial. With an initial value problem one knows how a system evolves in terms of the differential equation and the state of the system at some fixed time; one seeks to determine the state of the system at a later time.
• 2.4: Separation of Variables
Solving many of the linear partial differential equations presented in the first section can be reduced to solving ordinary differential equations. We will demonstrate this by solving the initial-boundary value problem for the heat equation. We will employ a method typically used in studying linear partial differential equations, called the Method of Separation of Variables.
• 2.5: Laplace’s Equation in 2D
Another generic partial differential equation is Laplace’s equation, ∇²u=0 . Laplace’s equation arises in many applications. Solutions of Laplace’s equation are called harmonic functions.
• 2.6: Classification of Second Order PDEs
We have studied several examples of partial differential equations, the heat equation, the wave equation, and Laplace’s equation. These equations are examples of parabolic, hyperbolic, and elliptic equations, respectively. Given a general second order linear partial differential equation, how can we tell what type it is? This is known as the classification of second order PDEs.
• 2.7: d’Alembert’s Solution of the Wave Equation
A general solution of the one-dimensional wave equation can be found. This solution was first Jean-Baptiste le Rond d’Alembert (1717- 1783) and is referred to as d’Alembert’s formula. In this section we will derive d’Alembert’s formula and then use it to arrive at solutions to the wave equation on infinite, semi-infinite, and finite intervals.
• 2.8: Problems

Thumbnail: Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. (CC BY-SA 3.0; via Wikipedia)

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