# Math 400 - Notes

Thanks to several faculty members and other sources for these notes.

Here are some of the slides used in class: ODErevise, PDEintroLaplace, LinearFirstOrderQuasiLinear, ShocksTrafficFlow, Classify2ndOrder, GreensFn, EigenFunctions

Notes will be listed by topic, roughly in the order covered.

1. Introduction: Some Basics, Intro1, Derivation of Basic 2nd order Equns. Intro2, Initial & boundary conditions Intro3, Some basics, derive heat equn.Intro4, Good summary, Intro5
2. Separation of Variables: Good summary of basics, Basics,
3. First order: Method of characteristics, general solution FirstOrder1
4. Quasi-linear: First order, shocks, expansion fan, traffic flow Quasi1, characteristics, traffic flow Quasi2, shocks, Burger's equn, equal area rule Quasi3, kinematic waves, shock fitting Quasi4, traffic flow from first principles TrafficFlow
5. Classify 2nd Order PDE: classify and canonical form Classify1, good summary Classify2
6. Nondimensional equations: theoretical background Nondimen1
7. Eigenfunction expansions: Fourier series and Laplace equn. Eigen1, Legendre polynomials Eigen3, Sturm-Liouville, sphere and cylinder Eigen4, Good basics, Eigen5
8. Green's functions: introduction Greens1, variation of parameters, heat equn. Greens2, delta function, Green's identities, Laplace equn. Greens3, good summary Greens4
9. Potential equation: good summary Potential1, rectangle and circle Potential3, conformal mappings Conformal1
10. Fourier Transform: heat and Laplace Fourier1, heat equn. Fourier2, basics for circuits Fourier3
11. Laplace Transform: basics, heat equation, similarity solutions Laplace1, small time solutions Laplace2
12. Heat equation: derivation, basic separation of variables, Fourier series Heat1, LT and Green's function Heat2, basics plus Sturm-Liouville theory Heat3, good summary Heat4
13. Wave equation: Physical derivations Wave1, derivation and D'Alembert Wave2, basic linear results Wave3, basic linear results Wave4, good summary Wave5.