I'm trying to solve a very simple heat conduction differential equation with NDSolve in Mathematica, but the solution I get is quite strange…
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13-12-2019 - |
Question
This is a very simple one-dimensional solid-phase heat conduction differential equation, here is my code:
a = NDSolve[{D[721.7013888888889` 0.009129691127380562` tes[t, x],
t] == 2.04988920646734`*^-6 D[tes[t, x], x, x],
tes[t, 0] == 298 + 200 t, tes[t, 0.01] == 298,
tes[0, x] == 298}, {tes[t, x]}, {t, 0, 0.005}, {x, 0, 0.01}]
Plot3D[tes[t, x] /. a, {t, 0, 0.005}, {x, 0, 0.01}, PlotRange -> All]
(Plot[(tes[t, x] /. a) /. t -> 0.0005, {x, 0, 0.01},
PlotRange -> All])
After you run it, you will see: the temperature (in the equation it's named as tes) is lower than 298! It's ridiculous, it's against the second law of thermodynamics…how does this error come out? How can I correct it?
Solution 2
This problem has been solved here,
I should admit that I haven't catch the nature yet at the time I posted this question…
OTHER TIPS
I'll deal only with the numerical aspects of this. First, scale time and space so that your equation becomes $\partial_t f=\partial_{x,x}f$ in the dimensionless units. then, for instance,
a = NDSolve[{D[ tes[t, x], t] == D[tes[t, x], x, x],
tes[t, 0] \[Equal] 1,
tes[t, 1] \[Equal] 1,
tes[0, x] \[Equal] Cos[2 \[Pi]*x/2]^2},
tes[t, x],
{t, 0, 1},
{x, 0, 1}
]
Plot3D[tes[t, x] /. a, {t, 0, .2}, {x, 0, 1}, PlotRange -> All,
AxesLabel \[Rule] {"t", "x"}]
so heat just diffuses inwards (note I changed the boundary and initial conditions).