dominic.tsy
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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