- #1
MisterX
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I've been teaching myself some thermodynamics, and I've been thinking about solving the heat equation.
\frac{\partial T}{\partial t} = K\frac{\partial ^2 T}{\partial x^2}
I haven't taken a course in PDEs.
I have noticed that if I assume an exponential solution, there are not non-decaying traveling wave solutions; the phase and group velocities are imaginary.
i\omega = k^2K
\omega = -ik^2K
non-traveling space oscillations decay in time, which isn't surprising since temperature tends to equalize:
e^{i(kx - \omega t) } = e^{ikx }e^{-k^2K t}
If we wanted to make omega real, we could have
k = \pm a(1+i)
e^{\mp ax} e^{\pm iax }e^{-i2a^2K t}
This indicates a phase speed of 2aK and a group speed of 4aK for the non-decaying factor, if I have done everything properly.
I have also seen there are solutions like
\frac{A}{\sqrt{t}}e^{-x^2/4Kt}
Also one can add a constant to any solution.
The problem I would like to solve is this: x = 0 is fixed at some temperature Ts. For x > 0 T(x, 0) is initially some other temperature Ti. If it made the solution simpler, the drop off needn't be so sharp. The point is that I would expect to see a solution so that every point with x>0 would become arbitrarily close to Ts if enough time was passed. Also, it should move along, so that it would take longer for a place with larger x to reach a given temperature than a place with smaller x.
The purpose of this thread is to solicit help in solving this problem, or any thoughts on my ideas about solutions to the heat equation.
\frac{\partial T}{\partial t} = K\frac{\partial ^2 T}{\partial x^2}
I haven't taken a course in PDEs.
I have noticed that if I assume an exponential solution, there are not non-decaying traveling wave solutions; the phase and group velocities are imaginary.
i\omega = k^2K
\omega = -ik^2K
non-traveling space oscillations decay in time, which isn't surprising since temperature tends to equalize:
e^{i(kx - \omega t) } = e^{ikx }e^{-k^2K t}
If we wanted to make omega real, we could have
k = \pm a(1+i)
e^{\mp ax} e^{\pm iax }e^{-i2a^2K t}
This indicates a phase speed of 2aK and a group speed of 4aK for the non-decaying factor, if I have done everything properly.
I have also seen there are solutions like
\frac{A}{\sqrt{t}}e^{-x^2/4Kt}
Also one can add a constant to any solution.
The problem I would like to solve is this: x = 0 is fixed at some temperature Ts. For x > 0 T(x, 0) is initially some other temperature Ti. If it made the solution simpler, the drop off needn't be so sharp. The point is that I would expect to see a solution so that every point with x>0 would become arbitrarily close to Ts if enough time was passed. Also, it should move along, so that it would take longer for a place with larger x to reach a given temperature than a place with smaller x.
The purpose of this thread is to solicit help in solving this problem, or any thoughts on my ideas about solutions to the heat equation.
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