324 Chapter 5 Higher Dimensions and Other Coordinates
u(r,θ, 0 )=f(r,θ), 0 <r<a, (15)
∂u
∂t(r,θ, 0 )=g(r,θ), 0 <r<a. (16)Following the procedure suggested in Section 5.4, we assume thatuhas the
product form
u=φ(r,θ)T(t)and we find that Eq. (11) separates into two linked equations:
T′′+λ^2 c^2 T= 0 , 0 <t,( 17 )
1
r∂
∂r(
r∂φ
∂r)
+^1
r^2∂^2 φ
∂θ^2=−λ^2 φ, 0 <r<a.( 18 )If we separate variables of the functionφby assumingφ(r,θ)=R(r)Q(θ ),
Eq. (18) takes the form
1
r(
rR′)′
Q+
1
r^2 RQ′′=−λ (^2) RQ.
The variables will separate if we multiply byr^2 and divide byRQ. Then the
preceding equation may be put in the form
r(rR′)′
R +λ
(^2) r (^2) =−Q′′
Q=μ
(^2).
Finally we obtain two problems forRandQ:
Q′′+μ^2 Q= 0 , −π<θ≤π, (19a)
Q(θ+ 2 π)=Q(θ ), (19b)
(
rR′
)′
−
μ^2
rR+λ(^2) rR= 0 , 0 <r<a, (20)
∣∣
R( 0 )
∣∣
bounded,
R(a)= 0.As we observed before, the problem (19) has the solutions
μ^20 = 0 , Q 0 = 1 ,
μ^2 m=m^2 , Qm=cos(mθ) and sin(mθ), m= 1 , 2 , 3 ,....Also, the differential equation (20) will be recognized as Bessel’s equation, the
general solution of which is (usingμ=m)
R(r)=CJm(λr)+DYm(λr).