1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

5.10 Some Applications of Legendre Polynomials 347

Now, a series of constant multiples of product solutions is the most general
solution of our problem:

u(φ,t)=

∑∞

n= 0

bnPn

(

cos(φ)

)

e−n(n+^1 )kt/R^2. (10)

The initial condition, Eq. (9), now takes the form of a Legendre series,

∑∞

n= 0

bnPn

(

cos(φ)

)

=f(φ), 0 <φ<π. (11)

From the information in Section 5.9, we know that the coefficientsbnmust be
chosen to be

bn=^2 n 2 +^1

∫π

0

Pn

(

cos(φ)

)

f(φ)sin(φ)dφ.

Then iff(φ)is sectionally smooth for 0<φ<π, the series of Eq. (11) actually
equalsf(φ), and thus the functionu(φ,t)in Eq. (10) satisfies the problem
originally posed.
For instance, iff(φ)=T 0 in the northern hemisphere( 0 <φ<π/ 2 )and
f(φ)=−T 0 in the southern(π/ 2 <φ<π), then the coefficients are

bn=^2 n 2 +^1

∫π

0

f(φ)Pn

(

cos(φ)

)

sin(φ)dφ

=^2 n+^1

2

[∫ 1

− 1

f

(

cos−^1 (x)

)

Pn(x)dx

]

=T 02 n+^1

n+ 1

Pn− 1 ( 0 ),

as found in the previous section. Figure 15 shows graphs ofu(φ,t)as a func-
tion ofφin the interval 0<φ<πfor various times. The CD shows an ani-
mated version of the solution.

C. Spherical Waves

In Section 5.8, we solved the wave equation in spherical coordinates for the
case where the initial conditions depend only on the radial variableρ.Nowwe
consider the case where the variableφis also present. A full statement of the