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"ANALYTIC
SOLUTIONS
OF PARTIAL DIFFERENTIAL EQUATIONS"
(MATH 3414)
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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 ill-posed or that the method of
solution is erroneous.
If you need more
information on the course, please contact Evy Kersalé.
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I shall try to put
my lecture notes on-line; hopefully I shall manage to converge
toward a sensible document. The following chapters should be included:
Outlines & Course Summary
1-Introduction
2-First Order PDEs
2.1-Linear & Semilinear
Equations
2.2-Quasilinear Equations
2.3-Wave Equation
2.4-System of Linear
Equations
3-Second Order Linear and Semilinear PDEs in Two
Variables
3.1-Classification and Standard
Form
Reduction
3.2-Extensions of the Theory
4-Elliptic Equations
4.1-Definitions
4.2-Properties of
Laplace's and Poison's Equations
4.3-Solving Poisson
Equation Using Green's Functions
4.4-Extensions
of the Theory
5-Parabolic Equations
5.1-Definitions and
Properties
5.2-Fundamental
Solution of the Heat Equation
5.3-Similarity
Solution
5.4-Maximum
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!
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