This course will provide an introduction to some of the (numerous) techniques that are available to provide approximate solutions to differential (and other) equations when there are no exact solutions. Unsurprisingly many equations do not have exact solutions; although in many cases numerical solutions will be adequate, it is often valuable to obtain an approximate solution to gain insight into the behaviour of the system being described. Such approximate solutions can also be very valuable in validating numerical solvers.
There are many different techniques available and the precise choice will depend to a large extent on the exact nature of the equation being studied. The course will provide an overview of the more common techniques and examples. We will follow very closely the book ‘Introduction to Perturbation Methods’ by Holmes.