Linear Boundary Value Problems

Abstract

We begin this chapter by showing that solutions of second-order linear boundary value problems exist (and analogously for the higher order) if and only if corresponding systems of algebraic equations of second (higher) order have solutions. Then, we shall use the method of separation of variables also known as the Fourier method (after Jean Baptiste Joseph Fourier (1768–1830)) to solve some basic partial differential equations. An important feature of this method is that it leads to finding the solutions of boundary value problems for ordinary differential equations. We shall also consider problems in infinite domains which can be effectively solved by finding the Fourier transform, or the Fourier sine or cosine transform of the unknown function.