Classification
Partial differential equations have multiple inputs
In the previous section of the course we studied
Partial differential equations (PDEs) have multiple inputs (independent variables). For example, think about a sheet of metal that has been heated unevenly across the surface. Over time, heat will diffuse through the 2-dimensional sheet. The temperature depends on both time and position - there are two inputs.
Because PDEs have multiple inputs they are generally much more difficult to solve analytically than ODEs. However, there are a range of numerical methods that can be used to find approximate solutions.
PDEs have application across a wide variety of topics
The same type of PDE often appears in different contexts. For example, the diffusion equation takes the form:
\[\begin{equation} \nabla^2T = \alpha \frac{\partial T}{\partial t} \end{equation}\]
When used to describe heat diffusion, this PDE is known as the heat equation. This same PDE however can be used to model other seemingly unrelated processes such as brownian motion, or used in financial modelling via the Black-Sholes equation.
Another type of PDE is known as Poisson’s equation:
\[\begin{equation} \nabla^2\phi = f(x,y,z) \end{equation}\]
Poisson’s equation can be used to describe electrostatic forces, where \(\phi\) is the electric potential. It can also be applied to mechanics (where \(\phi\) is the gravitational potential) or thermodynamics (where \(\phi\) is the temperature). When \(f(x,y,z)=0\) this equation is known as Laplace’s equation.
The third common type of PDE is the wave equation:
\[\begin{equation} \nabla^2r = \alpha \frac{\partial^2 r}{\partial t^2} \end{equation}\]
This describes mechanical processes such as the vibration of a string or the motion of a pendulum. It can also be used in electrodynamics to describe the exchange of energy between the electric and magnetic fields.
In this course we will look at techniques for solving the diffusion equation and Poisson’s equation, but many of the topics we will discuss - such as boundary conditions, and finite difference methods - can be transferred to PDEs more generally.
The Laplacian operator corresponds to an average rate of change
But what is the operator \(\nabla^2\)?. This is the Laplacian operator. When applied to \(\phi\) and written in full for a three dimensional cartesian coordinate system with dependent variables \(x\), \(y\) and \(z\) it takes the following form:
\[\begin{equation} \nabla^2\phi = \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}. \end{equation}\]
We can think of the laplacian as encoding an average rate of change. To develop an intuition for how the laplacian can be interpreted physically, we need to understand two related operators - div and curl. We will not explore these operators further in this lesson, but a video is included under external resources.