MATH 220: FINAL EXAM – DECEMBER 11, 2009This is a closed book, closed notes, no calculators exam. There are 7 problems. Solve allof them. Write your solutions to problems 1, 2 and 3 in blue book(s) #1, and your solutionsto problems 4, 5, 6 and 7 in blue book(s) #2. Within each book, you may solve the problemsin any order. Total score: 200 points.Use blue book(s) #1 for Problems 1-3!Problem 1. (i) (20 points) For |y − 1| small, solvexux+ yuy= 1, u(x, 1) = x2.Sketch the characteristics, and discuss where in R2is the solution uniquely determinedby the initial data. Does the solution you found extend to this region? Does it extendto a larger region?(ii) (15 points) For |x| small, solveuux+ uuy= 1, u(0, y) = y2+ 1.Problem 2. Consider the wave equation utt= c2uxxon the half-line, i.e. on [0, ∞)x×[0, ∞)t,with homogeneous Dirichlet boundary condition u(0, t) = 0, and with initial conditionsu(x, 0) = φ(x) and ut(x, 0) = ψ(x) for x ≥ 0.(i) (10 points) Find u.(ii) (8 points) Suppose φ, ψ are both linear near 0 (i.e. φ(x) = cx for x small, and similarlyfor ψ), and are C∞away from a point x0> 0. Where can you say for sure that u isC∞?(iii) (7 points) Suppose that φ ≡ 0, and ψ(x) = x for x < 1, ψ(x) = 0 for x > 1. Findu(x, t) explicitly for t ≥ 0. (Hint: it is best to consider different cases depending onwhere (x, t) lies.) Does the location of the singularities (lack of being C∞) agree withwhat you found in (ii)?You may use in any part of the problem that if v solves vtt− c2vxx= 0 on R2thenv(x, t) =v(x − ct, 0) + v(x + ct, 0)2+12cZx+ctx−ctvt(x0, 0) dx0Problem 3. (25 points) Consider the (real-valued) damped wave equation on [0, `]x×[0, ∞)twith Robin boundary conditions:utt+ a(x)ut= (c(x)2ux)x, ux(0, t) = αu(0, t), ux(`, t) = −βu(`, t)where α, β ≥ 0 are constants, a ≥ 0 and c > 0 depend on x only, and there are constantsc1, c2> 0 such that c1≤ c(x) ≤ c2for all x. (Note that if α = 0 and β = 0 then this is justthe Neumann boundary condition! In general, this BC would hold for example for a string ifits ends were attached to springs.) Assume throughout that u is C2. LetE(t) =12Z`0(ut(x, t)2+ c(x)2ux(x, t)2) dx +12(c(0)2αu(0, t)2+ c(`)2βu(`, t)2).(i) Show that if a ≡ 0 then E is constant.(ii) Show that if a ≥ 0 then E is a decreasing (i.e. non-increasing) function of t, and thatthe solution of the damped wave e quation (under the conditions mentioned above )with given initial condition is unique.12 MATH 220: FINAL EXAM – DECEMBER 11, 2009Use blue book(s) #2 for Problems 4-7!Problem 4. (i) (8 points) Consider the following eigenvalue problem on [0, `]:−X00= λX, X(0) = 0, X0(`) = 0.Find all eigenvalues and eigenfunctions, and show that eigenfunctions correspondingdifferent eigenvalues are orthogonal to each other.(ii) (8 points) Using separation of variables, find the general ‘separated’ solution of theheat equation (with k > 0 fixed):ut= kuxx, u(0, t) = 0, ux(`, t) = 0.(iii) (6 points) Solve the heat equation with initial conditionu(x, 0) = φ(x),i.e. give a formula for the series coefficients in part (ii) in terms of φ.(iv) (8 points) Now suppose φ(x) = x(` − x)2. Give an estimate for the coefficients inthe series which implies the uniform convergence of the series on [0, `] × [0, ∞)t, andexplain how the estimate implies uniform convergence. You do not need to computethe coefficients, though that is one way of getting the desired estimate.Problem 5. (i) (15 points) For both of the following functions f on [0, `], state whetherthe Fourier sine series on [0, `] converges in each of the following senses: uniformly, inL2. State what the Fourier series converges to on all of R. Make sure that you givethe reasoning that led you to the conclusions.(a) f(x) = x2(` − x)4,(b) f(x) = 0, for 0 ≤ x ≤ `/2, and f(x) = x − `/2 for `/2 < x ≤ `.(ii) (10 points) For the function f in (b) above, we wish to approximate f by a functiong of the form a1sin(πx/`) + a3sin(3πx/`) on [0, `]. Find the constants a1and a3thatminimize the L2error,R`0|f − g|2dx, of the approximation.Problem 6. Recall that S(Rn) is the set of Schwartz functions on Rn.(i) (7 points) Show that if φ, ψ ∈ C0(Rn) with (1+|x|)Nφ(x), (1+|x|)Nψ(x) both boundedfor some N > n thenZRn(Fφ)(ξ) ψ(ξ) dξ =ZRnφ(x) (Fψ)(x) dx.(ii) (6 points) Define Fu if u is a tempered distribution, i.e. u ∈ S0(Rn), and show that thisis c onsistent with the standard definition if u = ιφ, φ ∈ C0(Rn) with (1 + |x|)Nφ(x)bounded for some N > n.(iii) (5 points) Recall that uj→ u in S0(Rn) means that for each φ ∈ S(Rn), uj(φ) → u(φ).Show that if uj→ u in S0(Rn) then Fuj→ Fu in S0(Rn).(iv) (5 points) Show that for φ ∈ C0(Rn) with (1 + |x|)Nφ(x) bounded for some N > n,Fφ(ξ) = (2π)n(F−1φ)(ξ).(v) (7 points) Show the Parseval/Plancherel formula, i.e. that for φ, ψ ∈ S(Rn),ZRnφ(x) ψ(x) dx = (2π)−nZRn(Fφ)(ξ) (Fψ)(ξ) dξ,and hence conclude that, up to a constant factor, the Fourier transform preservesL2-norms:kFφkL2(Rn)= (2π)n/2kφkL2(Rn).MATH 220: FINAL EXAM – DECEMBER 11, 2009 3Problem 7. In Rn+1= Rnx× Rt, we write points as (x1, . . . , xn, t), and also write x =(x1, . . . , xn). With ∆x=Pnj=1∂2∂x2j, consider the modified wave equation in Rn+1:(1) utt− c2∆xu − λu = f.(i) (12 points) Show that if f ∈ S(Rn+1), then (1) has a unique solution u in S(Rn+1)when Im λ 6= 0, and give an expression for u in terms of f. Your final formulamay involve the (inverse) Fourier transform. (Hint: use the Fourier transform in allvariables!)(ii) (12 points) Still assuming Im λ 6= 0, show that if φ, ψ ∈ S(Rn), f ≡ 0 then the PDE(1) together with the initial conditionsu(x, 0) = φ(x), ut(x, 0) = ψ(x),has a unique solution which is bounded as long as t varies in bounded intervals. Again,give an expression for u in terms of φ, ψ. Your final formula may involve the (inverse)partial Fourier transform.(iii) (6 points) Compare (i) and (ii): in (ii) we impose an arbitrary additional condition:why does this not violate the uniqueness of (i) (note that for different φ the solutionsare certainly