1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

310 Chapter 5 Higher Dimensions and Other Coordinates

R(r)Q(θ ). After some algebra, we find that

(rR′)′

rR +

Q′′

r^2 Q=−λ

(^2) , (5)
R(a)= 0 , (6)
Q(θ )=Q(θ+ 2 π), (7)
R(r)bounded asr→ 0. (8)
The ratioQ′′/Qmust be constant; otherwise,λ^2 could not be constant. Choos-
ingQ′′/Q=−μ^2 , we get a familiar, singular eigenvalue problem:
Q′′+μ^2 Q= 0 , (9)
Q(θ+ 2 π)=Q(θ ). (10)
We found (in Chapter 4) that the solutions of this problem are
μ^20 = 0 , Q 0 (θ )= 1 ,
μ^2 m=m^2 , Qm(θ )=cos(mθ) and sin(mθ),

(11)

wherem= 1 , 2 , 3 ,....
There remains a problem inR:
(
rR′

)′

−μ

2

r

R+λ^2 rR= 0 , 0 <r<a, (12)

R(a)= 0 , (13)

R(r)bounded asr→ 0. (14)

Equation (12) is calledBessel’s equation, and we shall solve it in the next sec-
tion.

EXERCISES

1.State the full initial value–boundary value problems that result from the

problems as originally given when the steady-state or time-independent so-

lution is subtracted fromv.

2.Verify the separation of variables that leads to Eqs. (1) and (2).

3.Substitution ofv(r,θ,t)in the form of a product led to the problem of

Eqs. (1)–(4) for the factorφ(r,θ). What differential equation is to be satis-

fied by the factorT(t)?

4.Solve Eqs. (9)–(11) and check the solutions given.