# Chapter 11. Partial Differential Equations

11.1 Introduction

11.2 Poisson's Equation

11.3 Laplace's Equation

11.4 Heat Equation

11.5 Wave Equation

11.6 Visual Solution: Code11

11.7 Summary

Numerical Exercises

Programming Challenges

# INTRODUCTION

Partial differential equation (PDE) is an equation that has one or more partial derivatives as independent variables in its terms. Some typical examples are

**Equation 11.1. **

The *order* of a partial differential equation is defined as the highest partial derivative of the terms in the equation. Therefore, the first example above is the first-order PDE, whereas the second is the second-order PDE. The *degree* of a partial differential equation is defined as the power of the highest derivative term in the equation. It can be verified from the definition that the two equations above have the degrees of three and one, respectively.

In general, a partial differential equation of order *n* having *m* variables *x _{i}* for

*i*= 1, 2,...,

*m*is expressed as

**Equation 11.1. **

In the above equation,

In this chapter, we will concentrate on numerical problems involving second-order partial differential equations only. A second-order partial differential equation with variables *x*_{1}, *x*_{2}, and ...

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