MATH 480, Review for the Final Exam
Date and time: On Tuesday December 11, 2012, 8-10am in LO 1127.
Test coverage
The final exam will be cumulative; with questions similar to midterms.
New material (not included in the Midterms)
- Boundary value problems in cylindrical coordinates
- Laplace's operator in cylindrical coordinates (3.1)
- Laplace's equation in a circle (3.1)
- The vibrating drumhead problem and Bessel functions (3.3 and
parts of 3.2; lecture notes)
- Fourier Transforms
- Definition and basic properties (lecture notes posted; also 5.1)
- Solution of the heat equation on the real line (5.2)
For the Chapter 3 material you need to know the derivation of solutions for the drumhead,
the use of Bessel functions (Section 3.3) and the nodal lines diagrams for the
separable solutions. Problems 8 and 9 in Section 3.3 may be helpful.
For the Chapter 5 material you need to know how to prove the properties of the Fourier
transform formulated in Theorem 5.2 + the dilation property (proofs were done in class
and are in the posted lecture notes). You need to follow the computations of the Fourier
transform for the basic examples done in class. You also need to know how to solve the
heat equation with the Fourier transform (check the lecture notes for the corrected
version).
Exam questions
The exam will include two theoretical questions from the following list:
- Orthogonal sets: Bessel's inequalty; Parseval's equality and the Mean-square error
- Fourier convergence theorem: the following may be asked on the exam:
- Computation of the Fourier partial sum and definition of the Dirichlet kernel
- Lemma about the Dirichlet kernel (Lemma 2)
- Details of the proof from Section 1.2.3 and exercises
- Orthogonality of eigenfunctions of Sturm-Liouville problems
- Conditions for eigenvalues of Sturm-Liouville problems to be real and nonnegative (proofs)
- General solution of the wave equation and derivation of the D'Alembert solution
- Energy inequality and uniqueness of solutions for the heat equation
- Properties of the Fourier transform
Final exam problems
- Basic properties of PDEs; using changes of variables to transform PDEs
(example of uxx+uyy=0 being reduced
to uξη=0)
- Computing Fourier coefficients; using symmetry to simplify calculations;
manipulating Fourier series (differentiation, integration, identities);
computing values and plotting graphs for sums of the series (cf. problem
5 in Midterm 1).
- Series solutions for (some of) the following:
- 1D Heat equation: time-periodic or slab geometry
- 1D Wave equation (vibrating string)
- Laplace's equation in a rectangle with nonhomogeneous boundary conditions
- Example with multiple Fourier series (2D heat or wave)
- Sturm-Liouville problems: transcendental eigenvalues (cf. Problem 5 in Midterm 2)
- Solutions for the vibrating drumhead. Bessel functions. Nodal lines
- Properties of the Fourier transform and solution of the heat equation
Good luck on the exam!