4Solving Partial Differential Equations

4.1 Introduction

Many problems in mathematical finance result in partial differential equations (PDEs) that need to be solved. The best way to solve these equations is to find the exact solution using analytical methods. This is however a hard task and only the simplest PDEs have exact analytical formulas for the solution. In this chapter, we talk about some PDEs that provide exact solution. We will also present transformation methods that may produce exact analytical solutions. When applied in practice these are in fact approximations as well, but they are theoretical ones as opposed to the numerical approximations presented in later chapters.

We begin the chapter with a discussion of the three main types of PDEs and some important properties.

4.2 Useful Definitions and Types of PDEs

PDEs describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. PDEs are used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, and several others.