Reference no: EM133029933

Questions -

Q1. Consider the diffusion equation u_t = u_xx in {0 < x < 1, 0 < t < ∞} with u(0, t) = u(1, t) = 0 and  u(x, 0) = 4x(1 - x).

(a) Show that 0 < u(x, t) < 1 for all t > 0 and 0 < x < 1.

(b) Show that u(x, t) = u(1 - x, t) for all t ≥ 0 and 0 ≤ x ≤ 1.

(c) Use the energy method to show that ₀∫¹u² dx is a strictly decreasing function of t.

Q2. The purpose of this exercise is to show that the maximum principle is not true for the equation u_t = xu_xx, which has a variable coefficient.

(a) Verify that u = -2xt - x² is a solution. Find the location of its maximum in the closed rectangle {-2 ≤ x ≤ 2, 0 ≤ t ≤ 1}.

(b) Where precisely does our proof of the maximum principle break down for this equation?

Q3. Prove the comparison principle for the diffusion equation: If u and v are two solutions, and if u ≤ v for t = 0, for x = 0, and for x = l, then u ≤ v for 0 ≤ t ≤ ∞, 0 ≤ x ≤ l.