Determine an equilibrium temperature distribution (if one exists) for ди Әt д? и дх2 +x - В for 0 < x < L subject to the boundary conditions ди - (0,t) = 0, дх ди (L, t) = 0, дх and initial condition и(x, 0) = 1. For what values of B are there solutions?
Solve the heat flow problem: ди ди - (x, t) = 2 — (x, t), 0<x<1, t> 0, д дх2 и(0, t) = (1,1) = 0, t>0, и(x, 0) = 1 +3 cos(x) – 2 cos(3лх), 0<x<1.
— дt ! [points=4] Q4. Solve the heat equation subject to the given conditions: д?u ди 0<х «п, t> о дх2 ди ди - (0,t) = 0, - (п,t) = 0, t>0 дх дх и(x,0) = п - 3x
Show that the following PDE for u(x,y) is linear in u and homogeneous. ди ду ди = 3- дх Ә2 и + sin(у) дх2
[points=4] Q4. Solve the heat equation subject to the given conditions: д?u ди О<x<п, to дх? at' ди ди - (0,t) = 0, - (п,t) = 0, t>0 дх дх u(x,0) = п-х Paragraph В І := =
Solve the heat flow problem: au t> 0, ди (x, t) = 2 (x, t), 0<x< 1, ot дх2 uz(0, t) = uz(1,t) = 0, t>0, u(x,0) = 1- x, 0 < x < 1.
5. Solve the heat equation x< T, t > 0 5ихх — бих, и(п,t) — 0 sin (x) 0 и(0, t) и (х,0) t 0 0 x T 5. Solve the heat equation x 0 5ихх — бих, и(п,t) — 0 sin (x) 0 и(0, t) и (х,0) t 0 0 x T
3. Finish the following problem we discussed in class today: Utt - и(х, 0) — 0, и (х, 0) — е-1e1 5 and then plot u(r, 5) for (a) Choose t do it 10 < x < 10. Use a program to (b) Try to figure out what happens as t -» o0, that is find lim u(r, t) t->oo 3. Finish the following problem we discussed in class today: Utt - и(х, 0) — 0, и (х, 0) —...
Find the constant a and all the similarity solutions of the form х u (x, t) = F ta Ut = de 0 дх и
2. Let и(x, y, 2) ='y+y23, t = rse", y = rse- = r’s sint Compute ди ди at the point r = 2, 8= 1, t= 0 дѕ' де ət ди