
"ANALYTIC
SOLUTIONS
OF PARTIAL DIFFERENTIAL EQUATIONS"
(MATH 3414)

Please, consult the Undergraduate
Module Catalogue to have a description of the course Math 3414.
This lecture aims to give a general feel of analytical solutions of
PDEs.
So, why should we study PDEs and, in particular, analytic methods of
PDEs? We study PDEs because most of mathematical physics is described
in terms of PDEs (fluid mechanics, and more generally continuous media
mechanics, electromagnetism, quantum
mechanics, etc.). It is the case that typically, a given PDE will only
be accessible to numerical solution. However, it is crucial that
we know the general theory in order to conduct a sensible numerical
approach. For example, certain types of equations need certain types
of boundary conditions; without a knowledge of the general theory it is
possible that a problem may be illposed or that the method of
solution is erroneous.
If you need more
information on the course, please contact Evy Kersalé.

I shall try to put
my lecture notes online; hopefully I shall manage to converge
toward a sensible document. The following chapters should be included:
Outlines & Course Summary
1Introduction
2First Order PDEs
2.1Linear & Semilinear
Equations
2.2Quasilinear Equations
2.3Wave Equation
2.4System of Linear
Equations
3Second Order Linear and Semilinear PDEs in Two
Variables
3.1Classification and Standard
Form
Reduction
3.2Extensions of the Theory
4Elliptic Equations
4.1Definitions
4.2Properties of
Laplace's and Poison's Equations
4.3Solving Poisson
Equation Using Green's Functions
4.4Extensions
of the Theory
5Parabolic Equations
5.1Definitions and
Properties
5.2Fundamental
Solution of the Heat Equation
5.3Similarity
Solution
5.4Maximum
Principles and Comparison Theorems
Please, feel free to report any
errors, typos or frenglish grammatical structures!
Many thanks to Dr. R. Sturman, Prof. D. W. Hughes
& Prof. J. H. Merkin for providing me with their lecture notes; it
really helps!

