Solve for the temperature underground u(x,t) assuming that it satisfies the heat equation (x measured downwards):
∂^2 u/ ∂ x^2 = ∂u/∂t
And the surface temperature can be written in the form
u(0,t)= u0+ u1 sinωt
Hints:
(i) Assume u(x,t) → u0 as x → ∞ .
(ii) Use the boundary condition in the form u(0,t) u u expiωt = 0 + 1 and look for a
solution in the form u(x,t) u F(x) expiωt
Solve Heat equation PDE?
2015-05-17 6:00 am
回答 (2)
2015-05-17 12:57 pm
✔ 最佳答案
Let U(x,t) be the solution. We write it as P(x) Q(t). Substitute this in the equation and divide it by U(x,t). You will get1/P(x) d^2 P/d x^2 = 1/Q(t) d Q/dt
Since LHS is independent of t and rhs is independent of x, we can equate this to - k^2
You will get
d Q/dt = - k^2 Q, Therefore Q = Q0 exp -k^2 t, Q0 is a constant.
The second equation is
d^2 P / d t^2 + k^2 P =0
The solution will be
P(x) = P1 sin k x + P2 cos kx,
P1 and P2 are constants.
The solution is
U(x,t) = Q0 e^(- k^2 t) ( P1 sin k x + P2 cos k x)
Now you put the conditions. The conditions given by you are not proper.
2015-05-17 6:02 am
I'll look through my ChemE notes and get back to you tomorrow.
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