Partial Differential Equations

Partial differential equations (PDEs) are the mathematical representation of many phenomena in various branches of applied physics. Therefore it is important to be able to solve this type of equations and interpret the solution in terms of the physics they describe. This course provides an introduction into several analytical solution and analysis techniques for PDEs. These analytical solution techniques are often especially suitable for analysing the general physical behaviour of solutions to PDEs and are therefore indispensable tools in addition to numerical techniques (taught in the course AP3001-FE). Hence, the way these analytical techniques may be used to gain insight into the physical interpretation of solutions to PDEs are an important part of this course.

The solution and analysis techniques that will be discsussed include:

• Method of eigenfunction expansion;

• Homogenisation for nonhomogeneous PDEs;

• Fourier transform method;

• Method of characteristics;

• Green's functions.
  

Our main examples of PDEs to apply these techniques are the linear wave and heat equations in 1, 2, and 3 dimensions. Additionally we will discuss linear and quasi-linear transport equations.