MA3G1 Theory Of Partial Differential Equations Notes | Assignment Help | Syllabus
Country: United Kingdom
University: The University Of Warwick
MA250 Introduction to Partial Differential Equations describes the central and widespread importance of pdes in both pure and practical mathematics. In this module, I’ll show you how to solve (or at least prove that there is a solution!) different kinds of pdes. In contrast to odes, the domain on which a pde is to be solved is crucial. Most pdes were solved in the second year course MA250 on symmetric domains (e.g., round disks or squares) using specific techniques (e.g., variable separation) that are not applicable on generic domains. Much of the analysis you learned in the previous two years plays an important role in the general theory of pdes, as you will see in this module. You’ll also learn how sophisticated analytical subjects like MA3G7 Functional Analysis I arose from an abstract definition of pdes. This module covers the following topics:
For first order PDEs, the method of characteristics is used.
Green’s function is the fundamental solution to the Laplace equation.
Compactness and regularity are two characteristics of harmonic functions.
The concepts of comparison and maximization.
Diffusion equations with the Gaussian heat kernel
The fundamentals of wave equation (time permitting).
This course will expose students to broad issues about the existence, uniqueness, and characteristics of partial differential equation solutions.
Students who have completed this module should be familiar with a variety of pdes, as well as some of the methods used to discuss the existence and uniqueness of solutions to the Dirichlet problem for the Laplacian, as well as the properties of harmonic functions and a rudimentary understanding of solutions to parabolic and wave equations.
Partial Differential Equations, by Fritz John, Springer, 1982.
Partial Differential Equations, by L.C. Evans, American Mathematical Society, 1998.
Partial Differential Equations, E. Dibenedetto, Birkhauser, 2010.
Elliptic Partial Differential Equations of Second Order, D.Gilbarg and N.Trudinger, Springer, 2001 (more advanced, optional).
During the lectures, more comprehensive book recommendations will be provided.
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