Lecture 24 2nd Order Partial Differential Equations I

In this Lecture we discuss solutions of differential equations expressed as expansions in terms of the

appropriate functions we have learned about in the course. The specific choices of functions and

arguments correspond to the specific boundaries, and especially the boundaries with zero boundary

values. The coefficients in these expansions are then determined from fitting the non-zero boundary

conditions. While Mathematica has some facility for directly solving partial differential equations, the

most important application for us will be evaluating the integrals to find the coefficients and then making

plots of the resulting solutions - the desired distributions, typically in more than 1 dimension.

Consider first the 1-D case were Mathematica can easily do everything, including matching the bound-

ary conditions. For the example in the lecture we have

In[97]:= DSolve@ y''@xD � 0, y@xD, xDOut[97]= 88y@xD ® C@1D + x C@2D<<

and with boundary conditions

In[98]:= DSolve@8 y''@xD � 0, y@0D � c, y'@0D � 0<, y@xD, xDOut[98]= 88y@xD ® c<<

In[99]:= DSolve@ D@Y@x, yD, 8x, 2<D + D@Y@x, yD, 8y, 2<D � 0, Y@x, yD, 8x, y<DOut[99]= 88Y@x, yD ® C@1D@ä x + yD + C@2D@-ä x + yD<<

In this case Mathematica is telling us that we want solutions as functions of argument ±ä x +y, which is

true but not so useful. It is much easier to use the know expansions in Eq. (24.4) and use Mathematica

to find the coefficients. So consider the example starting on page 2 of the Lecture with a non-zero

boundary condition on the side y = Ly, which here we take to be the constant Y[x,Ly] = 100, with Lx = Ly

= 10.

In[100]:= Lx = 10; Ly = 10;

In[101]:= Y1@x_D := 100

In[102]:=

2

SinhA n Π Ly

LxE Lx

IntegrateBSinBn Π x

Lx

F Y1@xD, 8x, 0, Lx<F

Out[102]= -200 H-1 + Cos@n ΠDL Csch@n ΠD

n Π

In[103]:= a@n_D := -

200 H-1 + Cos@n ΠDL Csch@n ΠDn Π

Thus the (truncated) sum for the result distribution is

In[104]:= Y@x_, y_, m_D := SumBa@nD SinBn Π x

Lx

F SinhBn Π y

Ly

F, 8n, 1, m<F

In[105]:= Y@x, y, 10D

Out[105]=

400 Csch@ΠD SinA Π x

10E SinhA Π y

10E

Π

+

400 Csch@3 ΠD SinA 3 Π x

10E SinhB 3 Π y

10F

3 Π

+

80 Csch@5 ΠD SinA Π x

2E SinhA Π y

2E

Π

+

400 Csch@7 ΠD SinA 7 Π x

10E SinhA 7 Π y

10E

7 Π

+

400 Csch@9 ΠD SinA 9 Π x

10E SinhB 9 Π y

10F

9 Π

Now make a plot in 2-D - we'll use a contour plot

In[106]:= ContourPlot@Y@x, y, 10D, 8x, 0, Lx<, 8y, 0, Ly<, Contours ® Automatic,

ContourLabels ® Automatic, FrameLabel ® 8"x", "y"<, LabelStyle ® 8Large<D

Out[106]=

0 2 4 6 8 10

0

2

4

6

8

10

x

y

Clearly the function is near 100 at the top boundary and falls off into the interior, as expected. How-

ever, 10 terms is too few to get a smooth result. Try more terms and more contours

2 Lec24_228_09.nb

In[107]:= ContourPlot@Y@x, y, 100D, 8x, 0, Lx<, 8y, 0, Ly<,

Contours ® 8890<, 880<, 870<, 860<, 850<, 840<, 830<, 820<, 810<<,

ContourLabels ® Automatic, FrameLabel ® 8"x", "y"<, LabelStyle ® 8Large<D

Out[107]=

0 2 4 6 8 10

0

2

4

6

8

10

x

y

Note the expected symmetries.

Now consider a similar situation where we add a similar nonzero boundary condition on x = Lx, and

thus another sum of terms with x and y switched. Define

In[108]:=

2

SinhB n Π Ly

LyF Ly

IntegrateBSinBn Π y

Lx

F Y1@yD, 8y, 0, Ly<F

Out[108]= -200 H-1 + Cos@n ΠDL Csch@n ΠD

n Π

In[109]:= b@n_D := -

200 H-1 + Cos@n ΠDL Csch@n ΠDn Π

With a distribution

In[110]:= Y2@x_, y_, m_D :=

SumBa@nD SinBn Π x

Lx

F SinhBn Π y

Ly

F, 8n, 1, m<F + SumBb@nD SinhBn Π x

Lx

F SinBn Π y

Ly

F, 8n, 1, m<F

Lec24_228_09.nb 3

In[111]:= ContourPlot@Y2@x, y, 100D, 8x, 0, Lx<, 8y, 0, Ly<,

Contours ® 8890<, 880<, 870<, 860<, 850<, 840<, 830<, 820<, 810<<,

ContourLabels ® Automatic, FrameLabel ® 8"x", "y"<, LabelStyle ® 8Large<D

Out[111]=

0 2 4 6 8 10

0

2

4

6

8

10

x

y

A distribution that should be intuitively reasonable.

Next consider a similar situation but in cylindrical coordinates. The relevant functions are given in Eq.

(24.14). Here we will define a radius and temperature on the edge via

In[112]:= Ρ0 = 10; Y0@Φ_D := 100 Sin@ΦD;

The coefficients are

In[113]:=

1

Π

Integrate@Cos@n ΦD Y0@ΦD, 8Φ, 0, 2 Π<D

Out[113]= -200 Sin@n ΠD2

I-1 + n2M Π

In[114]:= a@n_D := -

200 Sin@n ΠD2

I-1 + n2M Π

i.e., really zero for all n and

In[115]:=

1

Π

Integrate@Sin@n ΦD Y0@ΦD, 8Φ, 0, 2 Π<D

Out[115]=

100 Sin@2 n ΠDI-1 + n2M Π

In[116]:= b@n_D :=

100 Sin@2 n ΠD

I-1 + n2M Π

4 Lec24_228_09.nb

In[117]:= b@1D

Power::infy : Infinite expression

1

0

encountered. �

Infinity::indet : Indeterminate expression 0 ComplexInfinity encountered. �

Out[117]= Indeterminate

So all zero except n = 1 but must be careful

In[118]:= Limit@b@nD, n ® 1DOut[118]= 100

In[119]:= Y3@Ρ_, Φ_D := 100Ρ

Ρ0

Sin@ΦD UnitStep@Ρ0 - ΡD

In[120]:= ContourPlotBY3B x2

+ y2

, ArcTan@x, yDF, 8x, -Ρ0, Ρ0<, 8y, -Ρ0, Ρ0<,

Contours ® 8880<, 860<, 840<, 820<, 80<, 8-20<, 8-40<, 8-60<, 8-80<<,

ContourLabels ® Automatic, FrameLabel ® 8"x", "y"<, LabelStyle ® 8Large<F

Out[120]=

-10 -5 0 5 10

-10

-5

0

5

10

x

y

Finally consider a 3-D rectangular situation as in the Lecture in side a cube of side 10.

In[121]:= Lx = 10; Ly = 10; Lz = 10;

Next assume all sides are at zero temperature except the side at z = 0 where Y2=100. The coefficients

are

Lec24_228_09.nb 5

In[122]:=

100

I Lx

2M I Ly

2M SinhBΠ Lz I m

LxM2

+ J n

LyN

2

F

IntegrateBSinBm Π x

Lx

F SinBn Π y

Ly

F, 8x, 0, Lx<, 8y, 0, Ly<F

Out[122]=

1600 CschB10m

2

100+

n2

100ΠF SinA m Π

2E2

SinA n Π

2E2

m n Π2

In[123]:= b@m_, n_D :=

1600 CschB10m2

100+

n2

100ΠF SinA m Π

2E2

SinA n Π

2E2

m n Π2

In[124]:= Clear@Y4D

In[125]:= Y4@x_, y_, z_, M_D := SumB

b@m, nD SinBm Π x

Lx

F SinBn Π y

Ly

F SinhBΠ HLz - zL Km

Lx

O2

+

n

Ly

2

F, 8m, 1, M<, 8n, 1, M<F

In[126]:= Y4@x_, y_, z_D := Y4@x, y, z, 10D

In[127]:= ContourPlot3D@Y4@x, y, zD == 50, 8x, 0, Lx<,

8y, 0, Ly<, 8z, 0, Lz<, LabelStyle ® 8Large<DOut[127]= $Aborted

ContourPlot3D@Y4@x, y, zD, 8x, 0, Lx<,

8y, 0, Ly<, 8z, 0, Lz<, Contours ® 2, LabelStyle ® 8Large<D$Aborted

6 Lec24_228_09.nb