Temperature Distribution in a Cold Region With a Heated Surface

Dennis SilvermanMathematical Physics 212  BU . C .  Irvine

Temperature at surface is 1, length L is 1, diffusion constant κ = 1.

One Dimensional Plate Distribution

?Sum

Tsum := 
      1 - (4/Pi) 
          Sum[(1/n) Sin[n Pi x] Exp[-n^2 Pi^2 t], {n, 1, 11, 2}]

t =.

Tsum

tabplot  := Table[Tsum, {t, 0,  .5,  .05}]

Plot[Evaluate[tabplot], {x, 0, 1}, PlotRange -> {-0.2, 1}]

Two Dimensional Rectangular Box Distribution

?Sum

Tsum2 := 
      1 - (16/(π^2)) 
          Sum[1/(k m) Sin[k π x] Sin[m π y] 
              Exp[-((k^2) + (m^2)) (π^2) t], {k, 1, 9, 2}, {m, 
              1, 9, 2}]

Table[
      Plot3D[Evaluate[Tsum2], {x, 0, 1}, {y, 0, 1}, 
        PlotRange -> {-0.2, 1}], {t, 0, 0.1,  .02}]

ShowAnimation[%]