Section 1.2 Partial Differential Equations
A main part of applied mathematics are differential equations. Ordinary differential equations (ODEs) include a finite number of differential equations of a single independent variable, and the analysis of those is covered in courses on ODEs, boundary value problems, and dynamical systems. The defining feature of ODEs is the fact that they can be formulated as a finite dimensional systems, often in \(\RR^n\) for appropriate dimension \(n\text{.}\) Partial differential equations (PDEs), however, can be seen as infinite dimensional dynamical systems, a concept that I will make precise in this book. PDEs are used for problems in physics, mechanics, engineering, chemistry, biology and many other disciplines. They are the work-horses of applied mathematics and they form a topic of central importance in our field. Compared to ODEs, PDEs require a new language. The state space of a PDE is a Banach space, the PDE itself can be seen as a combination of operators between Banach spaces, and solutions often arise as weak or weak\(^*\) limits in those Banach spaces. All these concepts need to be learned, and that is what we do in this book. As an example, consider a standard reaction-diffusion equation
\begin{equation*}
u_t = d \Delta u + f(u)
\end{equation*}
for an unknown function \(u(x,t)\) that depends on space \(x\) and time \(t\text{.}\) The constant \(d>0\) denotes the diffusion coefficient, \(u_t=\frac{\partial}{\partial t} u(x,t)\) denotes the partial time derivative, \(\Delta = \frac{\partial^2}{\partial x_1^2} + \dots + \frac{\partial^2}{\partial x_n^2}\) denotes the Laplace operator, and \(f(u)\) is a given function that describes growth or decay. If we introduce a linear operator
\begin{equation*}
Au = d \Delta u,
\end{equation*}
we write the reaction diffusion equation as
\begin{equation}
u_t = Au + f(u), \label{ode}\tag{1.2.1}
\end{equation}
which now looks like an ODE (ordinary differential equation). In fact, if it were an ODE, we could use matrix exponentials and the variation of constant formula like
\begin{equation}
u(t) = e^{At} u_0 + \int_0^t e^{A(t-s)} f(u(s)) ds, \label{variationofconstant}\tag{1.2.2}
\end{equation}
where \(e^{At}\) is a matrix exponential. But now, \(A=d\Delta\) is a differential operator, and not a matrix. The question arises whether we can define a matrix exponential for unbounded operators. The answer is YES, using semigroup theory, which we cover in the final Chapter Chapter 9. If \(u(x,t)\) is a solution of the "ODE" (1.2.1), then for each \(t\text{,}\) \(u(x,t)\) is a function of \(x\text{.}\) Hence it lies in a function space. But in which function space? We will discuss appropriate choices for a large variety of function spaces, such as Banach spaces, Hilbert spaces, Sobolev spaces, and their dual spaces later in Chapters Chapter 3, Chapter 5, and Chapter 6. In this scenario \(A\) becomes a linear operator between those spaces, hence we need to consider operator theory, as we do in Chapter Chapter 4.
Example 1.2.1.
\begin{align}
u_t \amp=\amp d u_{xx},\tag{1.2.3}\\
u(t,0) \amp=\amp u(t,L) =0, \nonumber\label{Dirichletexample}\tag{1.2.4}\\
u(0,x) \amp=\amp u_0(x), \nonumber\tag{1.2.5}
\end{align}
with a given initial condition \(u_0(x)\text{.}\) Using separation of constants and the superposition principle \cite{HillenPDE}, we find a solution as Fourier-sine series
\begin{equation*}
u(x,t) = \sum_{n=1}^\infty a_n e^{-\lambda_n d t} \sin\left(\frac{n\pi x}{L}\right),
\end{equation*}
where the coefficients \(a_n\) are the Fourier-sine coefficients of the initial condition
\begin{equation*}
u_0(x)=\sum_{n=1}^\infty a_n \sin\left(\frac{n\pi x}{L}\right).
\end{equation*}
The family of functions
\begin{equation*}
\mathcal{S}:=\left\{\sin\left(\frac{n\pi x}{L}\right), n\in \NN\right\}
\end{equation*}
forms an orthogonal set with inner product
\begin{equation*}
\langle\phi_n,\phi_m\rangle = \int_0^L\phi_n(x) \phi_m(x) dx .
\end{equation*}
\(\mathcal{S}\) can be seen as a basis of the function space \(L^2(0,L)\) of square integrable functions on \([0,L]\text{.}\) Since \(\mathcal{S}\) is infinite, we say that \(L^2(0,L)\) is infinite dimensional. In this sense we understand ((1.2.4)) as an infinite dimensional ODE. Also note that the Dirichlet boundary conditions \(u(t,0) = u(t,L) =0\) became part of the basis functions \(\mathcal{S}\text{,}\) since the sine functions satisfy these boundary conditions. This suggest that other boundary conditions, such as Neumann boundary conditions, for example, might lead to a different basis functions. In fact, much of PDE theory is concerned about the identification of the "right" basis and the "right" function space. We come back to this question when we talk about domains of definition of unbounded operators in Chapter Chapter 4.