Solutions of Laplace's Equation – Assignment | MATH 454, Assignments of Mathematics

Download Solutions of Laplace's Equation – Assignment | MATH 454 and more Assignments Mathematics in PDF only on Docsity! MATH 454: HOMEWORK 4 WINTER 2005 NOTE: For each homework assignment observe the following guidelines: • Include a cover page. • Always clearly label all plots (title, x-label, y-label, and legend). • Use the subplot command from MATLAB when comparing 2 or more plots to make com- parisons easier and to save paper. Solution of Laplace’s Equation 1. Using separation of variables solve the following BVP: PDE: uxx +uyy = 0 BCs: u(0,y) = 0 u(L,y) = 0 u(x,0) = 0 u(x,H) = f (x). 2. Using separation of variables, solve Laplace’s equation inside a 60◦ wedge of radius a, sub- ject to the BCs: u(r,0) = 0 u(r,π/3) = 0 u(a,θ) = f (θ). Properties of Laplace’s Equation 1. Using the maximum principle for Laplace’s equation, prove that the solution of Poisson’s equation, ∇2u = g(~x), subject to u = f (~x) on the boundary, is unique. 2. Show that the “backward” heat equation ∂u ∂t = −κ ∂2u ∂x2 subject to u(0, t) = u(L, t) = 0 and u(x,0) = f (x), is not well-posed. (HINT: Solve the problem with separation of variables; actually the solution is the same as heat equation, but with κ replaced by −κ. Then show that if the initial conditions are changed by an arbitrarily small amount, for example, f (x) =⇒ f (x)+ 1 m sin (mπx L ) where m is an arbitrarily large integer, then the solution u(x, t) changes by a large amount.) 1 Separation of Variables: 1. Consider the following homogeneous PDE and BCs: ut = uxx in 0 < x < 1 u(x,0) = f (x) u(0, t) = 0, u(1, t)+ux(1, t) = 0 (a) Make the substitution u(x, t) = φ(x)G(t), separate variables, and find the equations for φ(x) and G(t). Be sure to include the boundary conditions appropriately. (b) Show that only λ > 0 produces non-trivial solutions. If λ > 0, find the equation satisfied by the eigenvalues. Unlike the previous examples that we have seen, you will not be able to solve for the eigenvalues explicitly. (c) Write the equation that the eigenvalues satisfy as F(λ) = 0. Find approximate values for the first four eigenvalues by using the MATLAB function fzero to compute the roots of F(λ). You will need to provide fzero with initial guesses; create a MATLAB plot of F(λ) to get an idea where the roots are (Make sure you turn this plot in with the rest of the assignment). Fourier Series: 1. Find the Fourier sine series for f (x) = 1− x defined on the interval 0 ≤ x ≤ 1. (a) In MATLAB, plot the first 20 terms and the first 200 terms of the sine series in the interval −3 ≤ x ≤ 3. (b) To what value does the series converge at x = 0? (c) Find the Fourier cosine series for f (x) = 1− x defined on the interval 0 ≤ x ≤ 1. (d) In MATLAB, plot the first 20 terms and the first 200 terms of the cosine series in the interval −3 ≤ x ≤ 3. (e) To what value does the series converge at x = 0? (f) Find the Fourier series for f (x) = { 0 if −1 ≤ x < 0 1− x2 if 0 < x ≤ 1 defined on the interval −1 ≤ x ≤ 1. (g) In MATLAB, plot the first 20 terms and the first 200 terms of the Fourier series in the interval −3 ≤ x ≤ 3. (h) To what value does the series converge at x = 0? 2