Powers D.L. Boundary Value Problems and Partial Differential Equations

Подождите немного. Документ загружается.

348 Chapter 5 Higher Dimensions and Other Coordinates

Figure 15 Graphs of the solution of the example problem, with u(φ, 0) positive

in the north and negative in the south. The function u(φ, t) is shown as a function

of φ in the range 0 to π for times chosen so that the dimensionless time kt/R

takes the values 0, 0.01, 0.1, and 1; for convenience, T

=100.

problem is

∂

∂ρ



∂u

∂ρ



sin(φ)

∂

∂φ



sin(φ)

∂u

∂φ



∂

∂t

0 <ρ<a, 0 <φ<π, 0 < t,

u(a,φ,t) = 0, 0 <φ<π, 0 < t,

u(ρ, φ, 0) =f (ρ, φ), 0 <ρ<a, 0 <φ<π,

∂u

∂t

(ρ, φ, 0) =g(ρ, φ), 0 <ρ<a, 0 <φ<π.

(12)

As usual, we require in addition that u be bounded as ρ → 0andasφ → 0

and φ →π .

First, we seek solutions in the product form u(ρ, φ, t) = R(ρ)(φ)T(t).

Inserting u in this form into the partial differential equation (12) and manip-

ulating, we ﬁnd



(ρ



)



(sin(φ)



)



sin(φ)





. (13)

Both sides of this equation must have the same and constant value, say, −λ

Thus, we must have

(ρ



)



(sin(φ)



)



sin(φ)

=−λ

5.10 Some Applications of Legendre Polynomials 349

Again we see that the ratio containing  must be constant, say, −µ

.Hence,

we have two separate problems for the functions  and R :



sin(φ)







+µ

sin(φ) = 0, 0 <φ<π,

(φ) bounded at φ =0,π,







−µ

R +λ

R =0, 0 <ρ<a,

R(a) =0,

R(ρ) bounded at 0.

The ﬁrst of these problems is now quite familiar, and we know its solution to

=n(n +1), 

(φ) = P



cos(φ)



, n =0, 1, 2,....

The second problem is less familiar. In standard form, the differential equa-

tion is





−

R +λ

R =0.

Comparison with the four-parameter form of Bessel’s equation (Eq. (1) of Sec-

tion 5.8) shows α =−1/2, γ = 1, and p

= µ

+ α

.Sinceµ = n(n + 1),

= n

+n +

,andthenp = n +

. Thus, the general solution of the differ-

ential equation is

(ρ) = ρ

−1/2



n+1/2

(λρ) + BY

n+1/2

(λρ)



The fact that the Bessel functions of the second kind, Y

(λρ),areun-

bounded at ρ =0 allows us to discard them from the solution, leaving

(ρ) = ρ

−1/2

n+1/2

(λρ)

as the bounded solution. These functions occur frequently in problems in

spherical coordinates. Sometimes the functions

(z) =



n+1/2

(z),

called spherical Bessel functions of the ﬁrst kind of order n,areintroduced.As

noted in Section 5.8, there is a relation to sines and cosines:

(z) = sin(z)/z,

(z) =



sin(z) − z cos(z)



(z) =



(3 −z

) sin(z) −3z cos(z)



We have yet to satisfy the boundary condition R

(a) = 0. This cannot be

done by formula, except for n = 0. In this case, R

(a) = 0comesdownto

350 Chapter 5 Higher Dimensions and Other Coordinates

sin(λa)/λa = 0, so λ

= mπ/a for m = 1, 2,....Forothern’s, solutions of

n+1/2

(λa) =0 must be found numerically. For instance, for n = 1theequation

sin(λa) −λa cos(λa) = 0,

with solutions λa = 4.493, 7.725, 10.904,....(SeeHandbook of Mathematical

Functions by Abramowitz and Stegun, listed in the Bibliography.)

Finally we can put together some product solutions. Clearly the factor func-

tion T(t) will be a sine or cosine of λct. Thus our product solutions have the

form

−1/2

n+1/2

(λ

ρ)P



cos(φ)



sin(λ

ct),

−1/2

n+1/2

(λ

ρ)P



cos(φ)



cos(λ

ct).

The solution u(ρ, φ, t) will be an inﬁnite series of constant multiples of these

functions. We will not write it out.

Let us summarize some of the information we have obtained. First, the fre-

quencies of vibration of a sphere are λ

c (radians per unit time), where λ

is the mth positive solution of

n+1/2

(λa) =0.

Second, the nodal surfaces (loci of points where a product solution is 0 for all

time) are the values of ρ and φ for which

n+1/2

(λ

ρ)P



cos(φ)



=0.

One or the other factor must be 0, so these surfaces are either concentric

spheres, ρ = const., determined by J

n+1/2

(λ

ρ) =0, or else cones φ = const.,

determined by P

(cos(φ)) =0.

Finally, let us observe that, because P

(cos(φ)) ≡ 1, the product solutions

with n = 0 are precisely what we found as product solutions of the problem in

Section 5.8, Part B.

EXERCISES

1. Solve the potential equation in the sphere 0 <ρ<1, 0 <φ<π with the

boundary condition

u(1,φ)=



1, 0 <φ<π/2,

0,π/2 <φ<π,

together with appropriate boundedness conditions.

5.10 Some Applications of Legendre Polynomials 351

2. Solve the potential equation in a hemisphere, 0 <ρ<1, 0 <φ<π/2,

subject to boundedness conditions at ρ = 0andφ = 0, and the boundary

conditions

u(1,φ)= 1, 0 <φ<π/2,

u(ρ, π/2) = 0, 0 <ρ<1.

Hint: Use odd-order Legendre polynomials.

3. Solve this heat problem with convection on a spherical shell of radius R :

sin(φ)

∂

∂φ



sin(φ)

∂u

∂φ



−γ

(u −T) =

∂u

∂t

0 <φ<π, 0 < t,

u(φ, 0) = 0, 0 <φ<π.

Think carefully about the physical situation before attempting a solution.

4. Solve this heat problem on a hemispherical shell of radius R:

sin(φ)

∂

∂φ



sin(φ)

∂u

∂φ



∂u

∂t

, 0 <φ<π/2, 0 < t,

∂u

∂φ

(π/2, t) = 0, 0 < t,

u(φ, 0) = cos(φ), 0 <φ<π/2.

5. Solve the eigenvalue problem



∂

∂ρ



∂u

∂ρ



sin(φ)

∂

∂φ



sin(φ)

∂u

∂φ



=−λ

0 <ρ<a, 0 <φ<π/2,

u(a,φ)=0, 0 <φ<π/2,

u(r,π/2) =0, 0 < r < a,

subject to boundedness conditions at ρ =0andatφ = 0.

6. In Part C of this section we mention nodal surfaces (i.e., surfaces where

the function is 0). Find the nodal surfaces of the function

−1/2

3/2

(λρ)P



cos(φ)



if λ is the second positive solution of J

3/2

(λ) =0.

7. Describe in words the nodal surfaces for

−1/2

5/2

(λρ)P



cos(φ)



352 Chapter 5 Higher Dimensions and Other Coordinates

if λ is the second positive solution of J

5/2

(λ) = 0.

8. Solve the potential problem in the exterior of a sphere.



∂

∂ρ



∂u

∂ρ



sin(φ)

∂

∂φ



sin(φ)

∂u

∂φ



=0,

R <ρ, 0 <φ<π,

u(R,φ)= f (φ), 0 <φ<π.

9. L.M. Chiappetta and D.R. Sobel [Temperature distribution within a hemi-

sphere exposed to a hot gas stream, SIAM Re view, 26 (1984): 575–577]

analyze the steady-state temperature in the rounded tip of a combustion-

gas sampling probe. The tip is approximately hemispherical in shape. Its

outer surface is exposed to hot gases at temperature T

, and its base

is cooled by water at temperature T

circulating inside the probe. If

T(ρ, φ) is the temperature inside the tip, it should satisfy the condi-

tions



∂

∂ρ



∂T

∂ρ



sin(φ)

∂

∂φ



sin(φ)

∂T

∂φ



=0,

0 <ρ<R, 0 <φ<

T(ρ, π/2) = T

, 0 <ρ<R,

∂T

∂ρ

(R,φ)= h



−T(R,φ)



, 0 <φ<

together with boundedness conditions at ρ =0andatφ = 0.

The authors then change the variables to simplify the problem. Let

r = ρ/R, u(r,φ)= T(ρ, φ) − T

, and show that the problem for u be-

comes

∂

∂ρ



∂u

∂ρ



sin(φ)

∂

∂φ



sin(φ)

∂u

∂φ



=0, 0 < r < 1, 0 <φ<π/2,

u(r,π/2) =0, 0 < r < 1,

∂u

∂r

(1,φ)+u(1,φ)= D, 0 <φ<π/2,

where K = k /hR and D = T

−T

10. Solve the problem in Exercise 9. Hint: Use odd-indexed Legendre polyno-

mials to satisfy the boundary condition at φ =π/2.

5.11 Comments and References 353

5.11 Comments and References

We have seen just a few problems in two or three dimensions, but they are

sufﬁcient to illustrate the complications that may arise. A serious drawback

to the solution by separation of variables is that double and triple series tend

to converge slowly, if at all. Thus, if a numerical solution to a two- or three-

dimensional problem is needed, it may be advisable to sidestep the analytical

solution by using an approximate numerical technique from the beginning.

One advantage of using special coordinate systems is that some problems

that are two-dimensional in Cartesian coordinates may be one-dimensional

in another system. This is the case, for instance, when distance from a point

(r in polar or ρ in spherical coordinates) is the only signiﬁcant space variable.

Of course, nonrectangular systems may arise naturally from the geometry of a

problem.

As Sections 5.3 and 5.4 point out, solving the two-dimensional heat or wave

equation in a region

R of the plane depends on being able to solve the eigen-

value problem ∇

φ =−λ

φ in R with φ = 0 on the boundary. The solution

of this problem in a region bounded by coordinate curves (that is, in a gen-

eralized rectangle) is known for many coordinate systems. We have discussed

the most common cases; others can be found in Methods of Theoretical Physics

by Morse and Feshbach. Information about the special functions involved is

available from the Handbook of Mathematical Functions by Abramowitz and

Stegun and also from Special Functions of Mathematics for Engineers by L.C.

Andrews. Eigenfunctions and eigenvalues are known for a few regions that are

not generalized rectangles. (See Miscellaneous Exercises 20 and 21 in the text

that follows.)

Eigenvalues of the Laplacian in a region can be estimated by a Rayleigh quo-

tient, much as in Section 3.5. Furthermore, we have theorems of the follow-

ing type. Let λ

be the lowest eigenvalue of ∇φ =−λ

φ in R with φ = 0

on the boundary. Let

have the same meaning for another region,

R.If

ﬁts inside R,then

≥ λ

. [The smaller the region, the larger the ﬁrst eigen-

value. For further information, see Methods of Mathematical Physics, Vol. 1,

by Courant and Hilbert. In the famous article “Can one hear the shape of a

drum?,” American Mathematical Monthly, 73 (1966): 1–23], Mark Kac shows

that one can ﬁnd the area, perimeter, and connectivity of a region from the

eigenvalues of the Laplacian for that region. However, Kac’s title question has

been answered negatively. In the Bulletin of the American Mathematical Soci-

ety, 27 (1992): 134–138, authors C. Gordon, D.L. Webb, and S. Wolpert display

two plane regions, or “drums,” of different shapes, on which the Laplacian has

exactly the same eigenvalues.

The nodal curves of a membrane shown in Fig. 9 can be realized physically.

Photographs of such curves, along with an explanation of the physics of the

354 Chapter 5 Higher Dimensions and Other Coordinates

vibrating membrane, will be found in The Physics of Musical Instruments,by

Fletcher and Rossing.

Chapter Review

See the CD for review questions and special exercises.

Miscellaneous Exercises

1. Solve the heat problem

∇

u =

∂u

∂t

, 0 < x < a, 0 < y < b, 0 < t,

∂u

∂x

(0, y, t) = d0,

∂u

∂x

(a, y, t) = 0, 0 < y < b, 0 < t,

u(x, 0, t) = 0, u(x, b, t) = 0, 0 < x < a, 0 < t,

u(x, y, 0) =

, 0 < x < a, 0 < y < b.

2. Same as Exercise 1, but the initial condition is

u(x, y, 0) =

, 0 < x < a, 0 < y < b.

3. Let u(x, y, t) be the solution of the heat equation in a rectangle as stated

here. Find an expression for u(a/2, b/2, t).Writeouttheﬁrstthree

nonzero terms for the case a = b.

∇

u =

∂u

∂t

, 0 < x < a, 0 < y < b, 0 < t,

u =0 on all boundaries,

u(x, y, 0) = T, 0 < x < a, 0 < y < b.

4. Find the nodal lines of the square membrane. These are loci of points

satisfying φ

(x, y) = 0, where φ

satisﬁes ∇

φ =−λ

φ in the square

and φ =0 on the boundary.

5. Find the solution of the boundary value problem





=−1, 0 < r < a,

u(0) bounded, u(a) = 0,

Miscellaneous Exercises 355

both directly and by assuming that both u(r) and the constant function 1

haveBesselseriesontheinterval0< r < a:

u(r) =

∞



n=1

(λ

r), 0 < r < a,

1 =

∞



n=1

(λ

r), 0 < r < a.



Hint:



(λr)



=−λ

(λr).



6. Suppose that w(x, t) and v(y, t) are solutions of the partial differential

equations

∂

∂x

∂w

∂t

∂

∂y

∂v

∂t

Show that u(x, y, t) = w(x, t)v(y, t) satisﬁes the two-dimensional heat

equation

∂

∂x

∂

∂y

∂u

∂t

7. Use the idea of Exercise 6 to solve the problem stated in Exercise 1.

8. Let w(x, y) and v(z, t) satisfy the equations

∂

∂x

∂

∂y

=0,

∂

∂z

∂

∂t

Show that the product u(x, y, z, t) = w(x, y)v(z, t) satisﬁes the three-

dimensional wave equation

∂

∂x

∂

∂y

∂

∂z

∂

∂t

9. Find the product solutions of the equation

∂

∂r



∂u

∂r



∂u

∂t

, 0 < r, 0 < t,

that are bounded as r →0+ and as r →∞.

10. Show that the boundary value problem



(1 −x

)φ







=−f (x ), −1 < x < 1,

φ(x) bounded at x =±1,

356 Chapter 5 Higher Dimensions and Other Coordinates

hasasitssolution

φ(x) =



1 −y



f (z) dz dy,

provided that the function f satisﬁes



−1

f (z) dz =0.

11. Suppose that the functions f (x) and φ(x) in the preceding exercise have

expansions in terms of Legendre polynomials

f (x ) =

∞



k=0

(x), −1 < x < 1,

φ(x) =

∞



k=0

(x), −1 < x < 1.

What is the relation between B

and b

12. By applying separation of variables to the problem

∇

u =0, 0 <ρ<a, 0 ≤φ<π,

with u bounded at φ =0, π and u periodic (2π) in θ,derivethefollow-

ing equation for the factor function (φ):

sin(φ)



sin(φ)







−m

 +µ

sin

(φ) = 0,

where m = 0, 1, 2,... comes from the factor (θ).

13. Using the change of variables x = cos(φ), (φ) = y(x) on the equation

of Exercise 12, derive a differential equation for y(x).

14. Solve the heat conduction problem

∂

∂r



∂u

∂r



∂u

∂t

, 0 < r < a, 0 < t,

∂u

∂r

(a, t) = 0, 0 < t,

u(r, 0) = T

−





, 0 < r < a.

Miscellaneous Exercises 357

15. Solve the following potential problem in a cylinder:

∂

∂r



∂u

∂r



∂

∂z

=0, 0 < r < a, 0 < z < b,

u(a, z) = 0, 0 < z < b,

u(r, 0) = 0, u(r, b) = U

, 0 < r < a.

16. Find the solution of the heat conduction problem

∂

∂r



∂u

∂r



∂u

∂t

, 0 < r < a, 0 < t,

u(a, z) = T

, 0 < t,

u(r, 0) = T

, 0 < r < a.

17. Find some frequencies of vibration of a cylinder by ﬁnding product so-

lutions of the problem

∂

∂r



∂u

∂r



∂

∂z

∂

∂t

, 0 < r < a, 0 < z < b, 0 < t,

u(r, 0, t) = 0, u(r, b, t) = 0, 0 < r < a,α<t,

u(a, z, t) =0, 0 < z < b, 0 < t.

18. Derive the given formula for the solution of the following potential equa-

tion in a spherical shell:

∇

u =0, a <ρ<b, 0 <φ<π,

u(a,φ)=f



cos(φ)



, u(b,φ)= 0, 0 <φ<π,

u(ρ, φ) =

∞



n=0

2n+1

−ρ

2n+1

−a

2n+1





n+1



cos(φ)



2n +1



−1

f (x)P

(x) dx.

19. Show that the function φ(x, y) = sin(πx ) sin(2πy) − sin(2πx) sin(πy)

is an eigenfunction for the triangle

T bounded by the lines y = 0, y = x,

x = 1. That is,

∇

φ =−λ

φ in T ,

φ = 0 on the boundary of

T .

What is the eigenvalue λ

associated with φ?