� � � � 18.303 Problem Set 5 Due Friday, 15 October 2010. Problem 1: Min–max theorem (a) Consider the operator − d2 on the space of functions u(x) for x ∈ [0, 1] and u(0) = u(1) = 0,dx2 � 1with the usual inner product �u, v� = ¯uv. Integrating by parts, the Rayleigh quotient can 0 be written R(u) = �u�, u��/�u, u� as in class. Consider the function: x/a x ≤ a ua(x) = 1−x , x > a 1−a where a is some number in (0, 1). For what a is R(ua) minimized? How does the minimum of R(ua) compare with the smallest eigenvalue of −d2/dx2? (b) Consider −�2u = λu for functions u(x) in the 2d triangular domain Ω given by x ≥ 0, y ≥ 0, |x| + |y| ≤ 1 (a square cut in half diagonally) with Dirichlet boundary conditions u|dΩ = 0. Sketch contour plots of the eigenfunctions for the smallest 3 eigenvalues, making reasonable guesses based on the fact that these minimize R(u) = |�u|2/ |u|2 (constrained by the fact that they must be orthogonal). (In your plots, label peaks with a “+” and dips with a “-”.) Problem 2: Green’s functions In this problem, you will solve for the 1d Green’s function of the operator ˆd2 + κ2 for some A = − dx2 real κ, on the space of functions u(x) for x ∈ [0, L] and Dirichlet boundaries u(0) = u(L) = 0. Note that this operator is real-symmetric positive-definite. Helpful information: the ODE −y�� +κ2y = α, where α is a constant, is solved by functions y(x) of the form y(x) = c1e−κx + c2e+κx + κα 2 for arbitrary constants c1 and c2. (a) First, as in class, solve the “finite-delta” problem ˆs(xΔ−xx�) where1 x ∈ [0, Δx] ,Ag(x) = s(x) = 0 otherwise and x� ∈ [0, L − Δx] for g(x) by breaking it up into three regions (x < x�, x ∈ [x�, x� + Δx], x > x�) and enforcing continuity of g and g�. [Trick: you will get four equations in four unknowns, but by dividing two of the equations by the other two you can eliminate two of the unknowns immediately.] Plot your solution in Matlab for x� = 0.25, 0.5, and 0.75 with Δx = 0.1 and 0.01, with the parameters L = 1, κ = 5. (b) Take the limit Δx 0 to obtain the Green’s function G(x, x�). Alternatively, you may →compute G(x, x�) directly by solving ˆAG(x, x�) = δ(x − x�) as in class and the notes (breaking G into two regions where ˆAG = 0, and then matching the two regions by requiring G to be continuous and its slope to have a discontinuity = 1 at x�). Plot it for L = 1, κ = 5. (c) Verify that G(x, x�) = G(x�, x). (d) Verify that ˆAG(x, x�) = δ(x − x�), using the rules for differentiating discontinuous functions in the distribution sense. x(e) Optional: Using u(x) = � G(x, x�)f(x�)dx�, solve ˆ.Au = f for f(x) = eProblem 3: Distributions Define sε(x) by: � sΔx(x) = 0 1 |x| > Δx. x2Δx | | ≤ Δx As a distribution, Δx 0. What is the distributional derivative s�Show that s�sΔx{φ} → δ{φ} = φ(0) as →→ 0. Δx{φ}? Δx{φ} → δ�{φ} = δ{−φ�} = −φ�(0) as Δx 1MIT 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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