Homework 4: Green’s functions and distributions Math 456/556 1. Show that Z b d[φ] = φ0 (x)dx a is a linear mapping from D → R. What is this distribution in terms of delta functions? 2. Recall that distributions can themselves have derivatives, simply by defining d0 by the linear functional Z ∞ 0 d(x)φ0 (x) dx = −d[φ0 (x)]. d [φ] = − −∞ What rule represents the derivative of the delta function δ(x − x0 )? 3. Find the distributional Laplacian (in R2 ) of f (r, θ) = ln(r) in terms of δ functions. You need to use the definition, regarding ∆f as a mapping of some smooth φ Z Z ∞ 2π f (r, θ)∆φ rdrdθ, ∆f [φ] = 0 0 and integrate by parts to see how this distribution acts on φ. 4. Consider the 1D problem d2 u = f (x), dx2 u(0) = 0, du (L) = 0. dx A. Find the Green’s function G(x; x0 ) which is continuous at x = x0 , satisfies the jump condition lim Gx (x; x0 ) − lim− Gx (x; x0 ) = 1. x→x+ 0 x→x0 and boundary conditions Gxx (x; x0 ) = 0 when x 6= x0 , G(0; x0 ) = 0, Gx (L; x0 ) = 0. B. Write the solution u in terms of the f and the Green’s function you found. Verify your formula for the source term f (x) = x2 . 5. Find the Green’s function for the following problem: d2 u 2 − 2 u = f (x), dx2 x u(0) = 0, lim u(x) = 0. x→∞ (Hint: the homogeneous equation is of Euler type, so solutions are powers of x) Write down the solution u(x) in terms of the Green’s function.

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