1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.6 Temperature in a Cylinder 319

Theorem.If f(r)is sectionally smooth on the interval 0 <r<a, then at every
point r on that interval,

∑∞

n= 1

anJ 0 (λnr)=

f(r+)+f(r−)

2 ,^0 <r<a,

where theλnare solutions of J 0 (λa)= 0 and

an=

∫a

(^0) ∫fa(r)J^0 (λnr)rdr
0 J^20 (λnr)rdr

. (12)

The CD shows an animation of a Bessel series converging.
Now we may proceed with the problem at hand. If the functionf(r)in the
initial condition (3) is sectionally smooth, the use of Eq. (12) to chose the
coefficientsanguarantees that Eq. (11) is satisfied (as nearly as possible), and
hence the function

v(r,t)=

∑∞

n= 1

anJ 0 (λnr)exp

(

−λ^2 nkt

)

(13)

satisfies the problem expressed by Eqs. (1), (2), (3), and (8).
By way of example, let us suppose that the functionf(r)=T 0 ,0<r<a.It
is necessary to determine the coefficientsanby formula (12). The numerator
is the integral
∫a

0

T 0 J 0 (λnr)rdr.

This integral is evaluated by means of the relation (see Exercise 6 of Sec-
tion 5.5)
d
dx

(

xJ 1 (x)

)

=xJ 0 (x). (14)

Hence, we find
∫a

0

J 0 (λnr)rdr=λ^1

n

rJ 1 (λnr)

∣∣

∣∣

a

0

= a

λn

J 1 (λna)=a

2

αn

J 1 (αn). (15)

The denominator of Eq. (12) is known to have the value (Exercise 5)
∫a

0

J 02 (λnr)rdr=a

2

2 J

12 (λna)

=a

2

2

J 12 (αn). (16)