WP10 Introduction to Partial Differential Equations (MA2)
The module first introduces the method of separation of variables and the Fourier method to solve initial value problems for the heat and wave equations. Then first order differential equations will be discussed. The module continues with the n-dimensional heat equation, especially with the representation of the solution, uniqueness and maximum principle. Next d'Alembert and Poisson formulas, Hadamard?s descent method, finite speed of propagation and Huygens? principle for n-dimensional wave equations will be introduced. Finally, the n-dimensional Poisson equation, the Green representation formula, the mean value property of the Poisson integral formula, the maximum principle, Perron?s method and variational methods will be discussed. A number of geometric problems and numerous phenomena that are modelled in the natural science and increasingly also in economic sciences, lead to partial differential equations. The main goal of the module is to explore the existence, uniqueness and fundamental properties of the classic solutions of three main types of nd order partial differential equations.