PDE I

AMCS 231. Applied Partial Differential Equations I, Fall 2010, KAUST
Instructor: Aslan Kasimov
Office: 4-3217
Office hours: TBA
Lectures: Sa/M 10:30-noon, Room 9-3224

Prereq: Advanced and multivariable calculus, basic complex variables
Homework: 7-8 psets, due about every 10 days
Exams: mid-term exam on Oct. 16 and the final exam (time/place TBA)
Grading: homework – 30%, mid-term exam – 30%, final exam – 40%
Textbook: No single book is followed strictly in the lectures, but the following texts contain most of the material, with Salsa's book getting closest in content and level:

  1. S. Salsa, Partial Differential Equations in Action: From Modelling to Theory
  2. R. Guenther & J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations
  3. W. Strauss, Partial Differential Equations: an Introduction
  4. T. Myint-U & L. Debnath, Linear PDE for Scientists and Engineers

Catalog description: First part of a sequence of courses on partial differential equations (PDE) emphasizing theory and solution techniques for linear equations. Origin of PDE in science and engineering. Equations of diffusion, heat conduction, and wave propagation. The method of characteristics. Classification of PDE. Separation of variables, theory of the Fourier series and Fourier transform. The method of Green’s functions. Sturm-Liouville problem, special functions, eigenfunction expansions. Higher dimensional PDE and their solution by separation of variables, transform methods, and Green’s functions. Fractional PDE. Introduction to quasi-linear PDE and shock waves.

To give a better idea of specific topics introduced in the class, the following lists of topics are handed out to students before the midterm and the final exams. These can vary slightly from year to year.

Midterm Exam

No books, notes, or calculators are allowed. A list of topics/questions covered by the exam follows. Wherever the word “derivation” or “proof” appears, I expect you to be able to derive or prove the given fact.

1. Basic definitions; linear, quasi-linear, nonlinear PDE; superposition principle, initial and boundary value problems, well-posedness.
2. First-order quasi-linear PDE, its geometric interpretation, characteristic directions, the method of characteristics, the Cauchy problem.
3. Derivation of the general solution of the first-order quasi-linear PDE in two independent variables; first integrals.
4. Classification of the second order linear PDE, the canonical forms, general properties of hyperbolic, parabolic, and elliptic PDE.
5. Derivation of the wave equation for the vibrating string.
6. Derivation of the heat and diffusion equations; the heat and diffusion flux; Fourier’s law of heat conduction; Fick’s law of diffusion; Laplace’s equation.
7. Wave equation on an infinite and semi-infinite lines; derivation of D’Alember’s solution; signaling problem; Duhamel’s principle; general solution of the non-homogeneous wave equation with non-homogeneous initial data on an infinite line.
8. Separation of variables for the heat equation and the Fourier series; orthogonal functions; convergence of the Fourier series: point-wise, uniform, and mean-square convergence.
9. Approximation by trigonometric polynomials. Derivation of Bessel’s inequality; Parseval’s equality; completeness.
10. Gibbs phenomenon.
11. Fourier sine and cosine series; complex Fourier series;
12. Proof of the Riemann-Lebesgue Lemma; point-wise convergence theorem; Dirichlet kernel.
Differentiation and integration of the Fourier series.

Final Exam

The final exam is comprehensive and will be structured similarly to the midterm exam with about eight problems in total. No books, notes, or calculators of any kind are allowed. In addition to the topics handed out for the mid-term exam, the following topics/questions should be expected in the final exam. Again, understanding derivations and proofs is important.

1. Separation of variables for the one-dimensional heat/wave equations on bounded domains.
2. Non-homogeneous problems by separation of variables/e-function expansions.
3. Sturm-Liouville problem; self-adjoint operators; orthogonality and completeness of e-functions; reality and positivity of e-values of S-L problems; regular vs singular SL problems.
4. Green’s function for boundary value problems for ODE; conditions for the existence of the Green’s function. Fredholm alternative
5. Boundary value problems in several dimensions: equations of Laplace, Poisson, Helmholtz; Dirichlet and Neumann problems in bounded domains.
6. Maximum/minimum principle for the Laplace’s equation.
7. Uniqueness of solutions of the Laplace’s equation.
8. Dirichlet problem for the Laplace’s equation in a circle; mean-value theorem.
9. Solution of Dirichlet problem for Laplace’s/Poisson’s equation in a rectangle by separation of variables.
10. Separation of variables for the wave/heat equations in several dimensions.
11. E-function expansion for the Poisson’s equation with non-homogeneous boundary conditions.
12. Distribution theory: test functions; definition and properties of distributions; definition of a delta distribution and its sifting property; derivative of a distribution; a dipole distribution.
13. Solution of the wave equation in the sense of distributions.
14. Green’s second identity; representation of the solution of non-homogeneous Dirichlet problem for the Poisson’s equation in terms of the Green’s function.
15. Fundamental solutions of Laplace’s equation in two and three dimensions.
16. Symmetry of the Green’s function.
17. Green’s functions for Laplace’s equation and representation of the solution in terms of: a volume potential, single-layer, and double-layer potentials.
18. Method of images; solution of Poisson’s equation in the first quadrant subject to Dirichlet/Neumann boundary conditions.
19. Fundamental solution of the Helmholtz equation.
20. Fundamental solution of the heat equation.
21. Non-homogeneous problems for the heat equation; Duhamel’s principle.
22. Wave equation in 3D; Huygen’s principle; wave equation in 2D.
23. Hyperbolic systems of linear PDE; diagonalization.
24. Fourier transforms; basic properties; transform of distributions: Heaviside step function and delta distribution; convolution theorem; sine and cosine transforms.
25. Solution of heat/wave equations by Fourier transforms. Laplace’s equation in a half-plane.
26. Shocks and rarefaction solutions of Burgers’ equation. Rankine-Hugoniot conditions. Lax entropy conditions.