Theory

Introduction

Next we will study the wave equation and heat equation. The solution procedures will involve Fourier series, but also some of the basic ideas about boundary value problems.

The idea of a boundary value problem (BVP) is a counterpoint to the idea of an initial value problem (IVP). The BVP is to find solutions to the differential equation that pass through specified points $(x_0,y_0),(x_1,y_1),\dots$ (if their graphs are considered), or which take specified values at specified points (if their inputs only are considered in space). The ‘$x$’ variable is used again here because it is often a spatial variable. These points are called ‘boundary values’ because of the fact that in real applications, these values are fixed known quantities taken at the boundaries of a system.

Both BVPs and IVPs are problems of determining which (if any) members of a complete family of solutions satisfy the given conditions. The BVP or IVP becomes a problem of algebra as soon as the complete family of solutions is known.

Boundary value problems are generally much more complicated than initial value problems. Given some ‘complete family of solutions,’ it is often a tricky question whether any of these solutions take on the boundary values. On the other hand, our standard theorems of existence and uniqueness often apply to yield (at least local) solutions of initial value problems. It may be hard to see why the IVP is easier than the BVP just by looking at a complete family of solutions, however.

BVP = IVP for first-order ODE

Notice! BVP = IVP in the case of first-order ODEs.
Both conditions are simply the choice of a single input-output pair $(x_0,y_0)$ on the solution curve.

Eigenfunctions

We have seen previously that linear differential operators determine linear differential equations of the form $L[f]=g$ for any driving function $g$.

Given a linear operator $L$, an eigenfunction of $L$ is a function $f$ (not identically zero) satisfying $L[f]=\lambda f$ for some scalar $\lambda$.

For example, the operator $L=\frac{d}{dx}$ has eigenfunction $f=e^{\lambda x}$ that satisfies $L[f]=\lambda f$ with eigenvalue $\lambda$. For another example, the operator $L=\frac{d^2}{d{x}^2}$ has eigenfunctions $f=A\sin (\omega x+\varphi )$ that satisfy $L[f]=-{\omega }^{2}f$ with eigenvalue $-{\omega }^2$.

Linearity and the concept of eigenfunction are applicable to boundary value problems when the specified boundary values are all zero. (Otherwise the sum of solutions would not be a solution.) So, for example, the BVP given by $L[f]=\lambda f$ and $y(0)=0=y(x_1)$ determines a vector space of solutions. All solutions can be written as linear combinations of eigenfunctions. Infinite series of eigenfunctions also give solutions, provided they converge.

Eigenfunction solutions have a special importance in the basic theory of PDEs because many important physical PDEs specify an eigenvalue problem in each variable. This means the solutions must be eigenfunctions, but the eigenvalues themselves are free – they are not determined by the problem setup. The complete family of solutions therefore comprises the linear combinations of eigenfunctions of the BVP.