Finding Solution of Inhomogeneous Heat Equation

[dominic.tsy]

Homework Statement

Show that if u(x,t) and v(x,t) are solutions to the Dirichlet problems for the Heat equation

u_t (x,t) - ku_xx (x,t) = f(x,t), u(x,0) = Φ₁(x), u(0,t) = u(1,t) = g₁(t)

v_t (x,t) - kv_xx (x,t) = f(x,t), v(x,0) = Φ₂(x), v(0,t) = v(1,t) = g₂(t)

and if Φ₂(x) ≤ Φ₁(x) for 0 ≤ x ≤ 1, g₂(t) ≤ g₁(t), t > 0, then for all 0 < x < 1, t >0, we have u(x,t) ≥ v(x,t)

Homework Equations

The Attempt at a Solution

Following steps of example 2 and 3 of the following link, but I don't really understand what they are doing

http://www.math.mcgill.ca/jakobson/courses/ma264/pde-heat.pdf

 

[dominic.tsy]

Please help me... bump...