Mathematica appendix to lectures on PDEs:
part 1 (Cartesian variables)
Example problem Sec 13.2 #2
Checking Fourier series representation of temperature along lower edge.
In[853]:=
In[854]:=
Clear@LLD; LL = 20;
Tlower@nmax_, x_D :=
Sum@Sin@ Pi n x ê LLD 200 HCos@n Pi ê 2D - Cos@n PiDL ê HPi nL, 8n, 1, nmax<D
Showing the convergence of the Fourier series with nmax=3, 10 and 100:
In[855]:=
Plot@8Tlower@3, xD, Tlower@10, xD, Tlower@100, xD<, 8x, 0, LL<,
AxesLabel Ø 8x, T<, PlotStyle Ø 88Thick, Black<, 8Thick, Blue<, 8Thick, Red<<D
T
120
100
80
Out[855]=
60
40
20
x
5
10
15
20
Here’s the full result, with the sum truncated at nmax terms:
In[856]:=
ttfull@nmax_, x_, y_D := Sum@
Exp@- n Pi y ê LLD Sin@ Pi n x ê LLD 200 HCos@n Pi ê 2D - Cos@n PiDL ê HPi nL, 8n, 1, nmax<D
Here’s a “3D plot” showing the temperature with nmax=100.
Note that the temperature distribution smooths out quickly at y increases,
since the higher harmonics damp most quickly.
2
PDE1.nb
In[857]:=
Plot3D@ttfull@100, x, yD, 8x, 0, LL<, 8y, 0, 2 LL<,
AxesLabel Ø 8x, y<, PlotRange Ø 8- 20, 120<, PlotLabel Ø "THx,yL"D
Out[857]=
This plot shows cross sections as y increases, showing the smoothness and the decay of the
temperature:
In[858]:=
Plot@8ttfull@200, x, 1D, ttfull@200, x, 5D, ttfull@200, x, 10D, ttfull@200, x, 20D<,
8x, 0, LL<, AxesLabel Ø 8x, T<, PlotLabel Ø "Temp profile at y=1, 5, 10 and 20"D
Temp profile at y=1, 5, 10 and 20
T
80
60
Out[858]=
40
20
x
5
10
15
20
This plot shows how, for y=5, only nmax =10 is needed
PDE1.nb
In[859]:=
Plot@8ttfull@2, x, 5D, ttfull@10, x, 5D, ttfull@50, x, 5D, ttfull@100, x, 5D<,
8x, 0, LL<, AxesLabel Ø 8x, T<,
PlotStyle Ø 88Thick, Black<, 8Thick, Blue<, 8Thick, Red<, 8Thick, Green<<,
PlotLabel Ø "Temp profile at y=5 for nmax=2, 10, 50, 100"D
Temp profile at y=5 for nmax=2, 10, 50, 100
T
35
30
25
Out[859]= 20
15
10
5
x
5
10
15
20
Sec 13.3 #5
Here’s the plot of time dependence, with alpha=1, and using nmax=100
In[862]:=
In[866]:=
uu5@nmax_, x_, t_D :=
100 + Sum@Exp@- HH2 n - 1L Pi ê 2L ^ 2 tD HSin@H2 n - 1L Pi ê 2D - H2 n - 1L Pi ê 2L
Sin@H2 n - 1L Pi x ê 2D 800 ê HPi H2 n - 1LL ^ 2, 8n, 1, nmax<D
Plot3D@uu5@100, x, tD, 8x, 0, 2<, 8t, 0, 1<, AxesLabel Ø 8x, t<, PlotRange Ø 80, 100<D
Out[866]=
Sec. 13.4 #4
Here is the result (with v=h=l=1)
3
4
PDE1.nb
In[867]:=
yy4@nmax_, x_, t_D :=
Sum@16 H2 Sin@n Pi ê 8D - Sin@n Pi ê 4DL Sin@n Pi xD Cos@n Pi tD ê Hn PiL ^ 2, 8n, 1, nmax<D
Here is the string shape at t=0---checking that the Fourier series coefficients are correct:
In[868]:=
Plot@yy4@100, x, 0D, 8x, 0, 1<, AxesLabel Ø 8x, y<,
PlotStyle Ø 8Thick, Red<, PlotLabel Ø "t=0", PlotRange Ø 8- 1, 1<D
t=0
y
1.0
0.5
Out[868]=
x
0.2
0.4
0.6
0.8
1.0
-0.5
-1.0
This animation shows the motion of the string for one full cycle
We see that this is nothing other than a travelling wave bouncing off the ends.
In[869]:=
Animate@Plot@yy4@100, x, tD, 8x, 0, 1<, AxesLabel Ø 8x, y<, PlotRange Ø 8- 1, 1<,
PlotStyle Ø 8Thick, Red<, PlotLabel Ø tD, 8t, 0, 2<, AnimationRate Ø 0.05D
t
1.54808
y
1.0
0.5
Out[869]=
x
0.2
-0.5
-1.0
0.4
0.6
0.8
1.0