Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 51

For the PDS

⎧
⎨

a 2 u xx u tt = + f ( x , t ) , ( 0 , l ) × ( 0 , + ∞ )

u x ( 0 , t )= u x ( l , t )= 0 ,

⎩

( , )= ( , )= ,

u x 0 u t x 0 0

we can also obtain its solution, similar to in Section 2.2.1,

t l

u ( x , t )= G ( x , ξ , t − τ ) f ( ξ , τ ) d ξ d τ .

0 0

is the Green function of the one-dimensional wave equation un-
der the boundary condition of the second kind and is defined by

Here G ( x , ξ , t − τ )

+ ∞
k = 1

t − τ

l

2

a cos k

πξ
l

cos k

π x

sin k

π a ( t − τ )

G ( x , ξ , t − τ )= + .

k π l l

2.3 Method of Separation of Variables
for One-Dimensional Mixed Problems

In the last section, we have applied the Fourier method to solve mixed problems
under the first kind or the second kind of boundary conditions. The method uses
the theory of Fourier series to obtain the solution of an odd or even continuation of
the solution based on the characteristics of the boundary conditions. It involves the

series expansion of the solution by orthogonal groups sin k

and cos k ,

π x π x

l l

respectively. However, it is not straightforward to knowwhat is the type of series that
should be used for the other boundary conditions, such as the third kind. For PDS
with homogeneous boundary conditions, we may appeal to the method of separation
of variables for the answer. In this section, we discuss this method by solving the
following PDS with the mixed type of boundary conditions
⎧
⎨

a 2 u xx ,

u tt = ( 0 , l ) × ( 0 , + ∞ )

u ( 0 , t )= 0 , u x ( l , t )= 0 ,

(2.15)

⎩

u ( x , 0 )= 0 , u t ( x , 0 )= ψ ( x ) .

2.3.1 Method of Separation of Variables

To understand the method of separation of variables, it is helpful to elucidate the
process of solving the PDS in Section 2.2.1 by using some physical results of a
vibrating string. In the field of vibration, the PDS in Section 2.2.1 describes the
free vibration of a string with two fixed ends being driven by an initial velocity. The
propagation of the vibration forms waves. When these waves arrive at the endpoints,