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(MATH 3414)

Monthy Python's Flying Circus: French Lecture on Sheep-Aircraft

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.

Examples Classes

Examples 1: Linear & Semilinear PDEs
Examples 2: Quasilinear PDEs & System of PDEs
Examples 3: Standard Form of Second Order Linear PDEs
Examples 4: Elliptic Equations

Examples 5: Parabolic Equations


Lecture Notes

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
 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
    4.2-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

Complete Document: Full size (88 pages) & Reduced size (44 pages)
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!