Reference no: EM133028468

Questions -

Q1. Consider the solution u(x, t) = 1-x²-2kt of the diffusion equation.

Q2. Show O < u(x, t) < 1 for all t > 0 and 0 < x < 1.

Q3. Show that u(x, t) = u(1-x,t) for all t ≥ 0 and 0 ≤ x ≤ 1.

Q4. Use the energy method to show that ₀∫¹u²dx is a strictly decreasing function of t.

Q5. Prove the Comparison Principle. If u and v are 2 solutions and if u ≤ v for t = 0, x = 0, and x = L, then u ≤ v for t ≥ 0, 0 ≤ x ≤ t.

Q6. Solve

1. u(x, t) = 1/√(4πkt)_-∞∫^∞e^{-((x-y)^2/4kt)φ(y)dy}

2. _-∞∫^∞|u(x, t)|dx

3. If S(x, t) = (1/4πkt)e^{-x^2/4kt}, then for fixed δ > 0

lim max_{(t→0|x|≥δ)}S(x, t) = 0.

4. Given u_t = Ku_xx

u(x, 0) = x²

is u (x, t) = x² + kt

5. v(x, t) = e^bt u(x, t)