# Partial Differential Equations (PDEs)

In contrast to ODEs, PDEs are the governing equations for mathematical models in which the system has spatial dependence as well as time dependence (think of a vibrating guitar string, whose displacement depends on position, compared to an idealized point mass suspended by a spring and undergoing oscillations).

You may be wondering why we use a different symbol for derivatives (∂ instead of d) when working with PDEs. The symbol d indicates an ordinary derivative and is used for the derivative of a function of one variable, y = y(t). The symbol ∂ indicates a partial derivative, and is used when differentiating a function of two or more variables, u = u(x,t). For example  means differentiate u(x,t) with respect to t, treating x as a constant. Partial derivatives are as easy as ordinary derivatives!

There are three famous PDEs that you will encounter in our foundational course on PDEs. We have already see the wave equation. The second of the "big three" is the diffusion equations: where k is a constant. Here u = u(x,t) is an unknown function of position and time.

If a drop of dye falls into a container of clear water it will gradually diffuse throughout the container. This process is described by the diffusion equation, with u = u(x,t) representing the concentration of dye. This PDE also describes other processes of diffusion, for example the diffusion of heat. Think of holding a metal toasting fork in a camp fire. Eventually your hand will feel the heat. In this situation  u = u(x,t) represents the temperature of the fork.

The applications of the diffusion equation are not confined to science, however. Surprisingly, in recent years the diffusion equation has also played an important role in the area of mathematical finance.

The third member of the "big three" is Laplace's equation: In addition to describing systems that evolve in time, PDEs also describe systems in a state of equilibrium, and here Laplace's equation comes into play.

As a simple example, think of a square sheet of metal, with three edges in contact with ice (0° Celsius) and the fourth edge in contact with steam (100° C). Heat will diffuse through the sheet, but eventually the temperature of the sheet will reach a steady state u(x,y), that depends on position (x,y) but not on time. This function will satisfy Laplace's equation. What would you expect the temperature to be at the center of the plate in this example?

The preceding examples merely illustrate the "tip of the iceberg" as regards the subject of PDEs. Mathematical models based on partial differential equations (PDEs) are ubiquitous these days, arising in all areas of science and engineering, and also in finance and economics.