Some Practice Problems November 4, 2014 To help prepare for the upcoming midterm, you may wish to try out some of the following exercises. You may find some of these are quite challenging. There’s nothing to hand in, these are obviously optional, and an opportunity for those of you wishing to work on more problems. I suggest also reviewing previous recommended & required problems from throughout the course. 1. Let c, ρ, K0 ∈ C 2 ([0, L]) and let c, ρ, K0 > 0 on [0, L]. Consider the general heat equation c(x)ρ(x) ∂ ∂ ∂u u(x, t) = (K0 (x) ), ∂t ∂x ∂x x ∈ (0, L), t>0 with initial condition u(x, 0) = f (x) and boundary conditions u(0, t) = u(L, t) = 0. Using what we’ve learned about Sturm-Liouville problems, solve this PDE IBVP by separating variables. 2. True or false: the spatial ODE one obtains when applying separation of variables to the PDE ut = kuxx − V0 ux , V0 > 0, x ∈ (a, b) is a Sturm-Liouville type. In the above, V0 is constant. 3. Regardless of your answer to the preceding question, solve ut = kuxx − V0 ux , V0 > 0, x ∈ (a, b) with vanishing Dirichlet conditions u(0, t) = u(L, t) = 0 and u(x, 0) = f (x). 4. Consider φ00 = −λφ for x ∈ (0, 1) subject to φ(0) − φ0 (0) = 0 φ(1) + φ0 (1) = 0 √ Show that λ ≥ 0. Can λ = 0? Show that tan λ = √ 2 λ λ−1 . 5. Solve ut = kuxx for k > 0 constant, and u(0, t) − ux (0, t) = 0, u(L, t) + ux (L, t) = 0 and u(x, 0) = f (x) 6. Is the Neumann problem for Laplace’s equation well-posed? Why or why not? As a sanity check you can think about this when n = 1. 7. Make sure you follow the derivation and application of d’Alembert’s formula. In particular, I went through it fast, it’s worth going through again. 1 p 8. Solve Laplace’s equation inside of an annulus 0 < a < x2 + y 2 < b subject to u(a, θ) = f (θ) and u(b, θ) = g(θ). How does your solution behave as a & 0? 9. Show that the solution to Poisson’s equation ∆u = f (x) for x ∈ Ω ⊂ Rn with u |∂Ω = g(x) is unique. Hint: You can use the maximum principle for solutions of Laplace’s equation for this. 10. Let c > 0 constant and solve utt (x, t) = c2 uxx (x, t) with u(x, 0) = 0 and 1 |x| ≤ 1 ut (x, 0) = 0 |x| > 1 Draw a few snapshots of the solutions at a few different values of t. Also draw a space-time diagram of the solution. Include the domain of dependence and range of influence for an arbitrary point in space-time. 11. (Hard, but fun). The drum equation. Consider the wave equation in a two-dimensional disk utt (r, θ, t) = c2 ∆u, r < a, θ ∈ (0, 2π], t>0 where a, c > 0 are constants, subject to the Dirichlet boundary data u(a, θ, t) = 0, ∀t and initial conditions u(r, θ, 0) = f (r, θ), ut (r, θ, 0) = g(r, θ) Separate variables. You should obtain the r equation r2 R00 + rR0 + (λr2 − m2 )R = 0 Show this can be rewritten as a so-called Bessel equation of order m z 2 R00 + zR0 + (z 2 − m2 )R = 0 Next, take my word for the fact that the solution to Bessel’s equations can be written as c1 Jm (z) + c2 Ym (z) where Ym (0) = ∞ and Jm (z) = 0 has infinitely many solutions. Unfortunately1 , the functions Ym and Jm cannot be written in terms of elementary functions like sin or cos or polynomials or similar things. They are their own functions. If you’re interested (I definitely wouldn’t ask for something like this on an exam), you can continue working out the solution to this wave equation in terms of Bessel functions. If you don’t like working with Bessel functions you can still work in the abstract formalism of high-dimensional Sturm-Liouville theory and write the solution to the PDE IBVP in terms of eigenfunctions of the Laplacian. 12. (Hard, but fun). Consider a harmonic function u ∈ C ∞ (Rn ) which is homogeneous of degree r, i.e. u(λx) = λr u(x), for all λ > 0. Use the form of the Laplacian we saw back in problem set #4 ∆= n−1 ∂ 1 ∂2 + + 2L 2 ∂r r ∂r r where L is the spherical Laplacian. Show that such a function, when restricted to S n−1 is an eigenfunction of L. What are the associated eigenvalues? Notice I’m asking to show that L(u ||x|=1 ) = λr u ||x|=1 1 Actually this is very common. 2