Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

−L L

f(x)

0 x

−HL/k

Figure 21.4 The appropriate continuation for a Fourier series containing

only sine terms.

in order to satisfy∂v(L, t)/∂x=0werequirecosλL=0,andsoλis restricted to the
values

λ=

nπ

2 L

,

wherenis an odd non-negative integer, i.e.n=1, 3 , 5 ,....
Thus, to satisfy the boundary conditionv(x,0) =−Hx/k, we must have

∑

nodd

Bnsin

(nπx

2 L

)

=−

Hx

k

,

in the rangex=0tox=L. In this case we must be more careful about the continuation
of the function−Hx/k, for which the Fourier sine series is required. We want a series that
is odd inx(sine terms only) and continuous asx=0andx=L(no discontinuities, since
the series must converge at the end-points). This leads to a continuation of the function
as shown in figure 21.4, with a period ofL′=4L. Following the discussion of section 12.3,
since this continuation is odd aboutx= 0 and even aboutx=L′/4=Lit can indeed be
expressed as a Fourier sine series containing only odd-numbered terms.
The corresponding Fourier series coefficients are found to be

Bn=

− 8 HL

kπ^2

(−1)(n−1)/^2

n^2

fornodd,

and thus the final formula foru(x, t)is

u(x, t)=

Hx

k

−

8 HL

kπ^2

∑

nodd

(−1)(n−1)/^2

n^2

sin

(nπx

2 L

)

exp

(

−

kn^2 π^2 t

4 L^2 sρ

)

,

giving the temperature for all positions 0≤x≤Land for all timest≥0.

We note that in all the above examples the boundary conditions restricted the

separation constant(s) to an infinite number ofdiscretevalues, usually integers.

If, however, the boundary conditions allow the separation constant(s)λto take

acontinuumof values then the summation in (21.16) is replaced by an integral

overλ. This is discussed further in connection with integral transform methods

in section 21.4.