Differential equations occupy a central role in mathematics because they allow us to describe a wide variety of real-world systems. The module will aim to stress the importance of both theory and applications of differential equations.
The module begins by revisiting some of the material from the first year module on differential equations focussing attention on boundary value problems and also on equations with a source term. We then look at how one can express a general periodic function in terms of Fourier series of sine and cosine functions.
The second section of the module introduces some of the basic concepts of partial differential equations (PDEs). It is shown how PDEs may be used to model situations in a wide variety of situations including biology, finance and applied mathematics. The three important classes of second order PDE appropriate for modelling different sorts of phenomena are introduced and the appropriate boundary conditions for each of these are considered. The technique of separation of variables will be used to reduce the problem to that of solving the sort of ordinary differential equations seen at the start of the module and writing the general solution using Fourier series. Throughout the module there will be a strong emphasis on problem solving and examples.
Prerequisites (MATH1056 or MATH1059 and MATH1060) and MATH2039 or (MATH1006 and MATH1007)