1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.5 Bessel’s Equation 311

5. Suppose the problems originally stated were to be solved in the half-disk
   0 <r<a,0<θ <π, with additional conditions:
   v(r, 0 ,t)= 0 , 0 <r<a, 0 <t,
   v(r,π,t)= 0 , 0 <r<a, 0 <t.
   What eigenvalue problem arises in place of Eqs. (9)–(11)? Solve it.


6. Suppose that the boundary condition

∂v

∂r

(a,θ,t)= 0 , −π<θ≤π, 0 <t

were given instead ofv(a,θ,t)=f(θ ). Carry out the steps involved in sepa-

ration of variables. Show that the only change is in Eqs. (6) and (13), which

becomeR′(a)=0.

7. One of the consequences of Green’s theorem is the integral relation
   ∫∫

R

(

f∇^2 g−g∇^2 f

)

dA=

∫

C

(

f

∂g

∂n−g

∂f

∂n

)

ds,

whereRis a region in the plane,Cis the closed curve that boundsR,and

∂f/∂nis the directional derivative in the direction normal to the curveC.

Use this relation to show that eigenfunctions of the problem

∇^2 φ=−λ^2 φ inR,

φ=0onC

are orthogonal if they correspond to different eigenvalues. (Hint: Use

f=φk,g=φm,m=k.)

8. Same problem as Exercise 7, except the boundary condition is

φ+λ∂φ

∂n

=0onC.

5.5 Bessel’s Equation

In order to solve the Bessel equation,

(

rR′

)′

−μ

2

r

R+λ^2 rR= 0 , (1)

we apply the method of Frobenius. Assume thatR(r)has the form of a power
series multiplied by an unknown power ofr:

R(r)=rα

(

c 0 +c 1 r+···+ckrk+···

)

. (2)