MATH 480, Review for Midterm Test 2

Test coverage

2. Fourier series
1. Complex form of Fourier series (1.5)
2. Sturm-Liouville eigenvalue problems (1.6; lecture notes)
• Eigenvalues, eigenfunctions, orthogonality (1.6.1-2)
• Examples; transcendental eigenvalues (1.6.3)
• Positivity of eigenvalues (1.6.4)
3. Boundary-value problems in rectangular coordinates
   1. Derivation of the heat equation in multiple dimensions (2.1.1-2)
   2. Boundary conditions (2.1.3)
   3. Time-periodic solutions (2.1.5-6)
   4. Homogeneous boundary conditions: solution with Fourier method (2.2.1)
   5. Asymptotic behavior and relaxation time (2.2.3)
   6. Uniqueness of the solutions (2.2.4)
   7. Nonhomogeneous boundary conditions (2.3)
   8. Temporally nonhomogeneous problems (2.3.3)
   9. The vibrating string: derivation and linearized model (2.4.1-2)
   10. Solution for the vibrating string with the Fourier method (2.4.3)
   11. D'Alembert solution (2.4.5, lecture notes)
   12. Multidimensional problems: Laplace's and heat equations in 2 and 3D (2.5.2)
   13. Multiple Fourier series (2.5)
   14. Vibrations of a rectanguler membrane. Nodal lines (2.5.4)

Important concepts

• Sturm-Liouville eigenvalue problem. General form; eigenvalues, eigenfunctions
• Orthogonality of eigenfunctions
• Conditions of nonnegativity of eigenvalues
• Fourier series method for solving boundary-value ptoblems
• Relaxation rate; relaxation time
• Multiple Fourier series
• Normal modes
• Nodal lines

Theoretical material

• Orthogonality of eigenfunctions
• Eigenvalues of a (symmetric) Sturm-Liouville problem are real and nonnegative
• Uniqueness of solutions for the heat equation
• Derivation of the wave equation based on a model of vibrating string
• Derivation of D'Alembert solution based on a general solution of the 1D wave equation

Important types of problems

• Fourier series in complex form (Homework 6)
• Bring a second order ODE to Sturm-Liouville form (Homework 6)
• Find eigenvalues and eigenfunctions of a Sturm-Liouville problem
• Transcendental eigenvalues in Sturm-Liouville problems
• Heat equation in 1D: time-periodic solutions
• Heat equation in 1D: solutions in slab geometry; relaxation time
• Heat equation in 1D: nonhomogeneous boundary conditions
• Laplace's equation in a rectangle
• 1D wave equation: Fourier method
• 1D wave equation: D'Alembert's solution. Graphing solution snapshots and (x,t) diagrams
• D'Alembert's solution for boundary-value problems
• Laplace's eqution with nonhomogeneous boundary conditions
• Multidimensional heat and wave equations. Nodal lines