Principles and Techniques of Applied Mathematics
MATH 373, Exam #2
Wednesday, April 18, 6:00 - 7:30 pm, Swords 359

The second exam covers all of Chapters 3 and 4 and Sections 34 - 35 in Chapter 5, of the course text. It is highly recommended that you go over homework problems and your class notes. Some sample problems from the text as well as some good review problems are included below.

Note: You will be allowed one "cheat sheet," a single 8.5 x 11 piece of paper, front and back, full of whatever formulas, graphs, etc. you wish.

Exam Review: We will review for the exam during class on Monday, April 16th. Please come prepared with specific questions.

The following terms, concepts, definitions, equations, formulae, and theorems are important material for the exam:

1. General PDE terminology: operators, linear operators, homogeneous and inhomogeneous equations, classifying second-order linear PDE's (hyperbolic, elliptic, parabolic), steady-state or static solutions (no time dependence), harmonic functions, Principle of Superposition

2. Important PDE's: Heat equation, Wave equation, Laplace's equation, (know each in different spatial dimensions), the Laplacian (in standard coordinates, cylindrical, polar, spherical), physical interpretation of a given PDE, connections to music via the wave equation

3. Initial and Boundary Conditions: physical interpretations of initial conditions (position, velocity, etc.), describing given conditions mathematically, boundary conditions (Dirichlet, Neumann, Robin), physical interpretations of boundary conditions, flux, normal derivative, Newton's law of cooling, homogeneous vs. inhomogeneous conditions

4. Solving PDE's: Finding steady-state or static solutions, general solution to the wave equation, d'Alembert's formula for the wave equation on the entire real line (no boundary conditions), separation of variables technique for solving boundary value problems, solving PDE's with different boundary conditions (Dirichlet, Neumann, mixed) as infinite series, solving Sturm-Liouville problems (finding eigenvalues and eigenfunctions)

5. Fourier Series: definitions of, finding coefficients of (for cosine, sine and full Fourier series on the standard intervals (0,Pi) or (-Pi,Pi) or on the non-standard intervals (0,c), (-c,c)), linearity of Fourier coefficients, how to go from a Fourier series on a standard interval to one on a non-standard interval

Some Practice Problems:

Chapter 3
pp. 63 - 64 :   2
pp. 71 - 73 :   1, 5, 7
pp. 76 - 77 :   2
pp. 79 - 80 :   6, 7

Chapter 4
pp. 93 - 94 :   4, 6
pp. 102 :   3

1. Classify the PDE y u_xx + x u_yy - 2x u_xy + 3u_x = xy^2 as either hyperbolic, elliptic or parabolic in different regions of the xy-plane. Label and sketch the different regions.

2. Solve the 1d wave equation over the entire real line satisfying the conditions u(x,0) = cos x, u_t(x,0) = 2x . Simplify your answer.

3. Using the separation of variables technique, write down the two ODE's in x and t, with appropriate boundary conditions, for the following PDE. Do NOT solve them! If the PDE given modeled a vibrating string, describe physically what the boundary and initial conditions mean.
u_xx + (sin t)u_tt + 2x^2 u_x - 4t u_t = 0
u(0,t) = 0, u_x(2,t) = 0       u(x,0) = sin(Pi x), u_t(x,0) = 0.

4. Using the separation of variables technique, write down the complete series solution with the correct Fourier coefficients for the wave equation u_tt = 4 u_xx, 0 < x < Pi with u(0,t) = 0, u(Pi,t) = 0 and u(x,0) = 4 sin(3x), u_t(x,0) = 1.

5. Consider the Sturm-Liouville problem X''(x) + L X(x) = 0, X'(0) = 0, X'(pi) = -X(pi) which arises from the 1d heat equation with certain boundary conditions.
(a) Describe physically what the boundary conditions mean.
(b) Is L = 0 an eigenvalue? If so, what is the corresponding eigenfunction? If not, explain why.
(c) Are there any negative eigenvalues to this problem? Explain.