A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. They are used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, or elasticity.

In this course, we will discuss four important PDEs in detail: the
transport equation, the Laplace equation, the heat equation and the wave equation. We will learn various techniques for finding explicit solutions of these PDEs,
and we'll study various properties of these solutions. We will describe the
method of characteristics and use it to solve a general first-order
nonlinear PDE. We will introduce distributions and discuss the notion of a 'fundamental solution'. We will also introduce Sobolev spaces, investigate their properties and use them to solve
the general second order elliptic equation.

Tuesday and Thursday

9:00 - 10:25 am

103 Downs Laboratory of Physics (DWN)

210-2 Math Building (Building 15)

Office hours: by appointment

Office hours: by appointment

The course will be offered Pass-Fail.

Class participation: 40%. Student presentations: 60%.

Partial Differential Equations, by Evans.

1. April 3, Tuesday: ODEs and PDEs: examples, important 2nd order linear PDEs, classification of ODEs, solution of general 1st order linear ODE, solution of general 2nd order linear homogenous ODE, linearly independent solutions, separation of variables method, Frobenius method (notes I).

2. April 5, Thursday: Solution of 2nd order linear non-homogenous ODE; Classification of PDEs, hyperbolic, parabolic and elliptic equations (notes II).

3. April 10, Tuesday: Transport equation, solving general 1st order PDEs: Method of characteristics I - derivation of characteristic equations, projected characteristic curves (notes III pp. 1-9).

4. April 12, Thursday: Method of characteristics II - explicit examples for linear, semi-linear and fully non-linear PDEs (notes III pp. 9-18).

5. April 17, Tuesday: Method of characteristics III - boundary conditions, compatibility conditions, noncharacteristic boundary conditions, local solution, local existence theorem (notes III pp. 19-33).

6. April 19, Thursday: Laplace equation I - harmonic functions, fundamental solution (notes IV pp. 1-8).

7. April 24, Tuesday: Laplace equation II - solution to Poisson equation in
R^{n}, Dirac distribution, mean-value formulas, strong maximum principle (notes
IV pp.
8-21).

8. April 26, Thursday: Laplace equation III - positivity, uniqueness, regularity, Liouville's theorem, Green's function and its properties, representation formula for Poisson's equation using Green's function, energy methods (notes IV pp. 22-32).

9. May 1, Tuesday:** **Dirichlet's principle (notes
IV pp. 33-36); Heat equation I - fundamental solution, initial value
problem (notes V pp. 1-11).

10. May 3, Thursday: Heat equation II - solution of nonhomogeneous heat equation (notes V pp. 12-19).

11. May 8, Tuesday: Heat equation III - mean-value formulas, strong maximum principle, uniqueness, regularity, Tychonov's solution, energy methods, backwards uniqueness (notes V pp. 20-36).

12. May 10, Thursday: Backwards uniqueness of heat equation (notes V pp. 37-39); Wave equation I - d'Alembert's formula, physical intuition, method of reflection (notes VI pp. 1-11).

13. May 15, Tuesday: Wave equation II - spherical means, Euler-Poisson-Darboux equation, Kirchhoff's formula (n=3), method of descent, Poisson's formula (n=2), solution of wave equation in odd/even dimensions, Huygen's principle (notes VI pp. 12-29).

14. May 17, Thursday: Wave equation III - nonhomogeneous wave equation, energy methods: uniqueness, domain of dependence (notes VI pp. 30-35); Sobolev spaces I - motivation, weak derivatives, Banach spaces, Hilbert spaces, Sobolev spaces (notes VII pp. 1-8).

15. May 22, Tuesday: Sobolev spaces II - norm, properties of weak derivatives, local and global approximation by smooth functions, extension theorem, Sobolev inequalities (notes VII pp. 8-22).

16. May 24, Thursday: Sobolev spaces III - Gagliardo-Nirenberg-Sobolev inequality, Poincare's inequality, Rellich-Kondrachov compactness theorem (notes VII pp. 22-32); 2nd order elliptic equations I - uniformly elliptic partial differential operators, weak solutions (notes VIII pp. 1-5).

17. May 29, Tuesday: 2nd order elliptic equations II - Lax-Milgram theorem, energy estimates (notes VIII pp. 5-17).

18. May 31, Thursday: 2nd order elliptic equations III - First Existence theorem for weak solutions, statement of the Second and Third Existence theorems, regularity, closing remarks (notes VIII pp. 18-28).

19. June 5, Tuesday: Student presentations: Jing (Similarity Solutions), Marc (Hamilton-Jacobi Formulation of Classical Mechanics), Zhi (Weak Solutions and Uniqueness for Hamilton-Jacobi Equations).

20. June 7, Thursday: Student presentations: Yeorgia (Fourier Transform
Methods for Schroedinger and Wave Equations), Eitan (Generalization of Maximum
Principle for Parabolic PDEs), Tynan (Intersection Numbers on M_{g,r} and the KdV Hierarchy).