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18.303 Final Exam, Fall 2010 There are five problems with equal weight. You have 3 hours, and may bring one page of notes. Problem 1: Derivatives and differences (30 points) Consider functions u(x) on x ∈ [0, L], and the operator Aˆ= d4 dx4 . (a) Give one example of boundary conditions that make Aˆself-adjoint. (b) If we make a finite difference approximation u(mΔx) ≈ um, give a second-order accurate finite-difference ap-proximation of Aˆ. (Hint: use a second-order accurate difference approximation four times.) Problem 2: No more scalars (30 points) Let Ω ⊆ R3 be some 3d region, and consider 3-component vector-valued functions u(x) with Dirichlet boundary conditions ´u|dΩ = 0. In class, we showed that the curl operator �× is then self-adjoint for the inner product �u, v� = Ω u¯v. Consider the operator Aˆfor some real-valued function c(x), where: · Aˆu = � × (c� × u) ∂u(a) Under what conditions on c(x) does the equation Aˆu = ∂t have only solutions that decay exponentially to some limiting values (possibly nonzero)? (Hint: you should not need to do any messy integrals; the fact that �× is self-adjoint should simplify things.) ∂u(b) What quantities are conserved over time by solutions of Aˆu = ? (Hint: the nullspace of �× is �φ for any φ.)∂t Problem 3: Guided waves (30 points) Consider the scalar wave equation �2u = ∂2 u in two dimensions for x ∈ [0, L] and y ∈ (−∞, ∞), with the Neumann �∂t2 boundary conditions ∂u � x=0,L = 0. That is, Ω is a width-L strip extending infinitely in the y direction, with Neumann ∂x boundaries. (a) If we look for separable eigenfunctions u(x, y, t) = uk(x)ei(ky−ωt), what equation and what boundary conditions does uk satisfy? (b) Solve your equation from the previous part to obtain the eigenfunctions and the dispersion relation ω(k). (c) In this geometry, it possible to propagate a wavepacket (e.g. a Gaussian-envelope pulse) in the y direction without it spreading out (becoming broader in time and/or y)? Why or why not? CONTINUED ON NEXT PAGE... 1� � Problem 4: Timestepping and stability (30 points) Consider the equation ∂u = ∂u , where c is a constant, on an infinite domain x ∈ (−∞, ∞). Suppose that we ∂t −c ∂x ndiscretize this as u(mΔx, nΔt) ≈ u by m n+1 nn+1 n+1 n n um − um um+1 − um−1 um+1 − um−1 Δt = −c α 2Δx + (1 − α) 2Δx, where α is some real constant with 0 ≤ α ≤ 1. (a) Show that this discretization is unconditionally stable when α = 0.5. (Recall von Neumann analysis. e.g. look nfor solutions um = λneikm and show that λ satisfies....) (b) For what other values of α is it unconditionally stable? (c) For the remaining values of α, is it conditionally stable (and if so, what are the conditions on Δx and Δt?) or always unstable? Problem 5: Green’s functions (30 points) Suppose that we have an operator Aˆon a domain Ω with Dirichlet boundaries u0|dΩ = 0, and we know the corresponding Green’s function G0(x, x�) [i.e. ˆAG0(x, x�) = δ(x − x�) and G0(x, x�) = 0 for x ∈ dΩ]. Suppose that we now want to solve the problem with some nonzero Dirichlet boundary condition: u|dΩ = b(x) for some given function b(x). (a) Write the solution u of ˆAu = f (satisfying the b boundary condition on u) as some integral expression involving G0, f, and b. (b) Suppose ˆ2. In your expression from the previous part, integrate by parts on the term involving b to show A = �´that your total solution u is exactly the zero boundary-condition solution G0(x, x�)f(x�) plus a bunch of extra “source” terms from dΩ involving b. (Careful with integration by parts: b is not zero on dΩ.) Useful “integration by parts” formula from class: ´ ‚uv dA − ´(�u) v.Ω u� · v = dΩ · Ω· 2MIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit:


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