Reference no: EM131077369
Math 121A: Homework 5-
Part 1: Multiple integration
1. (a) Calculate the volume of the three-dimensional object given by the points (x, y, z) that satisfy x2 + y2 ≤ 1 and x2 + z2 ≤ 1.
(b) Optional for the enthusiasts. Calculate the volume of the three-dimensional object given by x2 + y2 ≤ 1, x2 + z2 ≤ 1, y2 + z2 ≤ 1.
Part 2: Sturm-Liouville theory
2. Find the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville problem
d/dx[-e-2xdy/dx] - e-2x y = λe-2xy
on the interval 0 ≤ x ≤ π, subject to the boundary conditions
y(0) = 0, y'(π) - y(π) = 0.
Verify that the eigenfunctions satisfy the orthogonality relation, so that (ym, yn) = 0 for any m ≠ n.
3. The concentration of gas in a channel over the range 0 < x < a follows the diffusion equation
∂f/∂t = b ∂2f/∂x2.
At x = 0 any gas is removed, and at x = a there are is a wall that is impermeable to gas, so suitable boundary conditions are
f(0, t) = 0, ∂f/∂x (a, t) = 0.
(a) Solve the system by searching for separable solutions of the form f(x, t) = X(x)T(t). (This will result in a Sturm-Liouville problem for X.)
(b) Write down the solution f(x, t) to the equation for the initial condition
f(x, 0) = sin πx/2a + 1/5 sin 17πx/2a
For the case of a = π and b = 1, plot f(x, t) for the values of t of 0, 0.2, 0.5, 1.0, 2.0, and 5.0. Interpret the graphs physically.
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