Transitioning to Second-order Linear PDEs
Up until now, we have been focusing on ODEs which involve ordinary derivatives only. However, one major incentive of ODEs, particularly about the special functions introduced in the last chapter, is to build up the solution of Partial Differential Equations (PDEs). As the name suggests, they now involve partial derivatives of multiple independent variables (e.g. \(x\), \(t\), sometimes we also use \(y\) as an independent variable, and the dependent variable will usually be denoted by \(u\)). To connect ODEs and PDEs, we shall concentrate on second-order linear PDEs. Second-order here means that the highest order of (mixed) partial derivatives is two, such as \( \partial^2 u/\partial x^2, \partial^2 u/\partial t^2, \partial^2 u/\partial x\partial t \). For simplicity, we often use the notation of \( u_{xx}, u_{tt}, u_{xt} \) instead. Linearity has a similar meaning as in the ODE counterpart: \( u \) and its partial derivatives cannot appear as a product together or have a power different from \( 1 \), and they are not placed in a non-linear function.
In general, second-order linear PDEs (in two independent variables) have the form of
\begin{align}
Au_{xx} + Bu_{xt} + Cu_{tt} + Du_{x} + Eu_{t} + Fu = g(x,t) \tag{1}
\end{align}
where \( A,B,C,D,E,F \) may be functions of \(x, t\), but for most usages they will be constants, and \(g(x,t)\) is the source term. If \(g(x,t) \equiv 0\) then (1) is called homogeneous.
Three Major Types of PDE
To categorize second-order linear PDEs, we can put the first three terms of (1) into the matrix/quadratic form as
\begin{align}
A\frac{\partial^2}{\partial x^2} + B\frac{\partial^2}{\partial x\partial t} + C\frac{\partial^2}{\partial t^2} = \begin{bmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial t} \end{bmatrix}
\begin{bmatrix}
A & B/2 \\
B/2 & C
\end{bmatrix}
\begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial t} \end{bmatrix} \tag{2}
\end{align}
The determinant of the matrix in the middle \( AC -B^2/4 \), or equivalently the discriminant \( \Delta = B^2 -4AC \), then decides the PDEs types.
Parabolic/Heat Equation
If \( \Delta = B^2 -4AC = 0 \), then the PDE is known as a parabolic PDE. The standing example is the heat equation that models temperature conduction, which has the form of
\begin{equation}
\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2} \tag{3}
\end{equation}
where \( \kappa \) is the thermal diffusivity.
Elliptic/Laplace Equation
If \( \Delta = B^2 -4AC < 0 \), then the PDE is referred to as an elliptic PDE. The most representative example will be the Laplace equation, which has the simple form of
\begin{equation}
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \tag{4}
\end{equation}
here we use \( y \) instead of \( t \) as the second independent variable. In a coordinate-free way, we will simply write
\begin{align}
\nabla^2 u = 0 \tag{5}
\end{align}
where \( \nabla^2 \) is the Laplacian operator and is equal to \( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\) in a two-dimensional Cartesian coordinate system.
Hyperbolic/Wave Equation
The last case is \( \Delta = B^2 -4AC > 0 \), and the corresponding PDE is called a hyperbolic PDE. The most common physical example will be the wave equation that models the propagation of waves:
\begin{equation}
\frac{\partial^2 u}{\partial t^2} -c^2 \frac{\partial^2 u}{\partial x^2} = 0 \tag{6}
\end{equation}
where \( c \) is the wave travel speed.
Boundary Conditions
Since PDEs now involve more than one independent variable, we need some more general boundary conditions that (particularly for the Laplace equation, where the domain is a two-dimensional, maybe three-dimensional, space) dictate the behavior of inputs to the system along the boundary. The Dirichlet and Neumann boundary conditions are extended from the ODE counterpart: a Dirichlet B.C. imposes the value of \( u \) at the boundary, while a Neumann B.C. imposes the value of the normal derivative \( \partial u/\partial n \) across the boundary instead, where \( n \) is along the direction of normal to the boundary. Meanwhile, the radiation/Robin B.C. enforces a weighted sum of \( u \) and \( \partial u/\partial n \), e.g. \( \alpha u + \beta\partial u/\partial n = 0\). Finally, a mixed B.C. means that along different parts of the boundary, different B.C.s are used.
We note in passing that an initial condition at some time may be regarded as a sort of boundary condition if we treat time as another added dimension of space.
Exercise
Show that in general, the \(A,B,C\) part of a hyperbolic equation (e.g. (6)) can be written as
\begin{align}
(\frac{\partial}{\partial t} -\alpha\frac{\partial}{\partial x})(\frac{\partial}{\partial t} -\beta\frac{\partial}{\partial x})u \tag{7}
\end{align}
Answer
If \( \Delta = B^2 -4AC > 0 \), then the quadratic equation \( Av^2 + Bv + C = 0 \) has two real roots \(\alpha, \beta\) and can be factorized into \( (v-\alpha)(v-\beta) = 0 \). Informally substituting \( v = \frac{\partial}{\partial t}/\frac{\partial}{\partial x} \) then leads to the statement.
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