Math 322 Applied Mathematical Analysis II, FALL 2019

• 1.2: 3,5; 1.3: 2; 1.4: 1(aeg),2,11; 1.5:1,3,5,12,;
• 1.3: Explain the difference between eqns.(1.3.4) and (1.3.5);
• 2.2: 3; 2.3: 3b,4,7,8;

• 2.4: 1(ab), 2;
• u_t = κ u_xx, with u(0,t)=0, u_x=± u/c at x=L, where κ, c and L are positive constants. For both problems (±): Find eigensolutions; give and justify explicit bounds on decay rates. Consider sinh(αx) solutions also. Analyze and discuss physically.
• Derive an equation for ∫₀^L u² dx for the previous problem for both boundary conditions. Analyze and discuss.
• Solve u_tt = c² u_xx, with u(0,t)=0=u(L,t) and initial conditions u=f(x), u_t=g(x).
• 2.5: 1(bg), 2, 3a, 5d, 8b,

• Read Chapter 3: Fourier Series.
• Define piecewise smooth, even, odd
• Let S(θ) and C(θ) be the sine and cosine series of f(θ) in 0 < θ < π, respectively
3.W.:
1. Find S(θ) and C(θ) for f(θ) = θ. Make a nice sketch (preferably with computer software) of dS/dθ and dC/dθ truncated to 2, 4, and 6 terms in the series (and more if you wish, see example of figure 3.3.8). Do these derivatives yield the sine and cosine series of df/dθ? Discuss briefly.
2. Find S(θ) and C(θ) for f(θ) = 1 in 0 < a < θ < b < π and f(θ) = 0 otherwise. Sketch f(θ). How fast do the series converge?
3. Find S(θ) and C(θ) for f(θ)=θ². Sketch S(θ) and C(θ) in -2 π < θ < 2π,
4. Sketch f(θ)=θ(1-θ/π) and find its sine and cosine series. Sketch S(θ) and C(θ) in -2 π < θ < 2π. Which series converges faster and why? Can you explain/predict a priori the convergence rates? How?
5. Find C(θ) for f(θ)=e^θ in 2 ways: (1) directly, (2) using d²e^θ /dθ² = e^θ (see also problem 3.4.6).
6. Find the sine series of f(x)=x and f(x)=x(1-x/L) in 0 <x < L.
• 3.3.: 12, 15
• 3.4.: 9
• 4.4.: 3, 7, 9, 10
• Fourier series handout and partial homework solutions
• Is it Xmas?! All the book solutions, funded by lots and lots of commercials

Study Chapter 3: Fourier series convergence and differentiation. Do you understand (3.4.13)?

Read Chapter 4: Wave equation derivation

1. from Maxwell's equations,
2. vibrating string. Vibrating String Derivation

Study Chapter 5: Sturm-Liouville. What is that? Theorems? (P 157) Must know: proofs that SL operator is symmetric (`self-adjoint'), λ real, orthogonality, uniqueness of eigfcts, Rayleigh quotient.

• 5.3.: 2,3,4,8,9. 5.4.: 1. 5.5.: 8. 5.5.A:6. 7.3. 1(a), 4(c), 7(b). 7.7.: 2(a), 4, 6.

• Study Sec 7.7, 7.8. Where did λ go? isn't that what we are looking for?! (Sec 7.7.4)
• Bessel functions J_m(z), Y_m(z). Asymptotics as z → 0. Asymptotics as z → ∞.

• Bessel functions notes and exercises
• 7.8.: 2, 7, 8, 11. 7.9.: 1(d), 4(b). 7.10.: 2(bd), 9(a).

• Multipoles, Legendre, Helmholtz

• Legendre polynomials, Rodrigues formula, Associated Legendre functions Spherical Harmonics
• Prof. Thiffeault's nice notes on Legendre polynomials

• Review Problems: Active learning! Flip the classroom! You discuss in class, you go to the boards. Do not hand in.