This post describes a relatively general notation for discussing differential equation problems of all shapes and sizes. It’s very similar to the notation of Dissanayake and Phan-Tien (1994), Lagaris et al. (1997) or Berg and Nyström (2017).
Let’s start with an example. Perhaps the simplest partial differential equation (PDE) of interest is the two-dimensional Laplace equation, $$u_{xx}(x,y) + u_{yy}(x,y) = 0.$$ Here, $x$ and $y$ are coordinates of some vector $\vec{x}$ in a domain $\Omega$. The problem is to find the function $u(\vec{x})$ that satisfies Laplace’s equation everywhere in $\Omega$. We can also write this PDE more succinctly using $\nabla$, as $$\nabla^2 u(\vec{x}) = 0.$$
Most PDEs are more complicated than Laplace’s equation. For instance, a simple version of the heat equation is $$\frac{\partial u(\vec{x},t)}{\partial t} = \nabla^2 u(\vec{x},t),$$ where now $u$ is a function of a vector $\vec{x}$ in some spatial domain $\Omega$, but also depends on time $t$. Many PDEs contain this mixture of spatial derivatives and derivatives with respect to time. For present purposes, however, we can ignore the distinction between space and time. It is more convenient to simply think of $u$ as a function of a single combined space-and-time vector $\vec{x}$. Then we can write down a general $n$th-order PDE as $$G(\vec{x},\nabla u(\vec{x}),\nabla^2 u(\vec{x}),\ldots,\nabla^n u(\vec{x})) = 0,$$ where $n$ is the highest order of derivative taken with respect to any of the independent variables in $\vec{x}$. More conveniently still, we can think of $G$ as a differential operator, and write the same relation as $$G[u](\vec{x}) = 0$$ to describe the PDE at a point $\vec{x}$.
I’ll note in passing how this relates to non-linear PDEs. These are generally a particularly challenging class of equations. In the current notation, non-linearity of the PDE amounts to non-linearity of the function $G$.
Ordinary differential equations
Ordinary differential equations (ODEs) are differential equations that only have one independent variable. For the present discussion, it is most convenient to think of ODEs as special cases of PDEs where the vector of variables $\vec{x}$ is just a single scalar $x$. In other words, the rest of this discussion applies directly to ODEs as well as PDEs.
Vector-valued solutions
So far, we have talked about $u$ assuming that it is a scalar function. More generally, we might consider PDEs that describe vector or tensor functions $u$. For instance, the Navier-Stokes equations describe the evolution of a three-dimensional fluid flow along with its density, pressure, stress tensor, etc. These PDEs can still be written in the general form given above, but the derivatives will generally be higher-order tensors.
Boundary conditions
On their own, most PDEs actually have infinitely many solutions. Generally, PDE problems will also come with boundary conditions (BCs). These specify that the solution $u$ must behave in some way on the boundary $\partial \Omega$ of the domain. For instance, Dirichlet BCs specify the required values of $u$ on $\partial \Omega$. On the other hand, Neumann BCs state the values of the normal derivative of $u$ on $\partial \Omega$.
Many other types of boundary conditions exist. In some applications, boundary conditions can get pretty complicated. We can denote BCs in general as $$B[u](\vec{x}) = 0\quad\mathrm{for}\quad\vec{x}\in\partial\Omega,$$ where $B$ represents some differential operator. For homogeneous Dirichlet BCs, the operator $B$ would be the identity operator, so that $$B[u] = u.$$ For homogeneous Neumann BCs, the operator $B$ would be $$B[u] = u_{\hat{n}},$$ namely the derivative of $u$ in the direction normal to the boundary.
Recap
With this notation, we can denote a general $n$th-order PDE problem as $$G[u](\vec{x}) = 0\quad\mathrm{for}\quad\vec{x}\in\Omega,$$ $$B[u](\vec{x}) = 0\quad\mathrm{for}\quad\vec{x}\in\partial\Omega.$$ Most (if not all) PDEs can be written in this form for suitable choices of $G, \Omega$, and $B$. In this notation,
- Time is treated on equal footing with spatial variables.
- Non-linear PDEs amount to non-linear functions $G$.
- ODEs are just special cases of PDEs.
- Vector-valued solutions $u$ are no problem.
- Complicated BCs are captured by the operator $B$.