Partial differential equations

To apply the ideas of this chapter to partial differential equations, we need to view them as ordinary differential equations in a function space. We explain how to do this using the example of the heat equation. Consider the equation

 

u_t=u_xx,

for $x\in (0,1)$, t>0, with boundary conditions

u(0,t)=u(1,t)=0

and some initial condition

u(x,0)=u₀(x).

The abstract approach to this problem is to recast it as an ordinary differential equation in an infinite-dimensional function space. For this purpose, one defines a space X which consist of functions of x defined on the interval (0,1). We then regard u as a function u(t), which, for every given t, takes values in X. That is, u(t) is still a function of x; to avoid ambiguous notation, we shall use square brackets to denote the x-dependence: u(x,t)=u(t)[x]. The infinite-dimensional space X replaces ${\rm I\!R}^n$ in the previous sections. There are many possible choices for this function space; the one most commonly employed is the space of square integrable functions: X=L²(0,1).

The abstract version of the heat equation is  

$$\dot u=Au,\quad u(0)=u_0$$ (23)

where the operator A maps the function u[x] to the function u''[x]. In contrast to the case of ordinary differential equations, however, A is not defined on the whole space X. First of all, u''[x] does not exist, at least not as an element of X, unless we assume some smoothness of u. The appropriate space is H²(0,1), defined as the space of all functions on (0,1) which have a second derivative which lies in L²(0,1) (we have to refer to texts on functional analysis or partial differential equations for a precise definition of what this means, see e.g. [12]). Even more importantly, the boundary conditions are viewed as a restriction on the domain of the operator A. This may appear strange at first; we certainly do not have to require that u satisfy any boundary conditions to define u''[x]. The reason for incorporating the boundary conditions as a restriction on the domain of A is the objective of defining a meaningful eigenvalue problem $Au=\sigma u$.If we interpret this equation simply as $u''=\sigma u$, then solutions always exist, and every $\sigma$ would be an eigenvalue. This is not what we want. On the other hand, the eigenvalue problem  

$$u''=\sigma u,\quad u[0]=u[1]=0$$ (24)

has the eigenvalues $\sigma_n=-n^2\pi^2$, $n\in{\rm I\!N}$ and the associated eigenfunctions $u_n[x]=\sin(n\pi x)$. We therefore define

Au[x]=u''[x],

where A is defined on the domain  

$$D(A)=\{u\in H^2(0,1)\,\vert\,u[0]=u[1]=0\}.$$ (26)

Using the Fourier expansion

$$u(x,t)=\sum_{n\in{\rm I\!N}} u_n(t)\sin(n\pi x),$$ (27)

we can rewrite the heat equation in the form

$$\dot u_n=-n^2\pi^2u_n.$$ (28)

In this form, the abstract system $\dot u=Au$ appears explicitly as an infinite system of ODEs, and we can read off the eigenvalues $-n^2\pi^2$, and it is clear the system is stable.

Of course having simple closed form expressions for the eigenvalues and eigenfunctions is exceptional. Indeed, there are partial differential equations for which eigenvalues do not even exist! This makes it necessary to examine at a more fundamental level which systems $\dot u=Au$ admit, in some sense, a solution $u=\exp(At)u(0)$ and what can be said about the stability of these systems. A thorough discussion of this issue goes well beyond the scope of this course, but we shall review some of the main issues in the next section.