Page 169 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes We denote these eigenvalues by , ,..., ,..., and the eigenfunctions by (x), (x)..., (x),...
1 2 n 1 2 n
Next, we expand f(x) in terms of these eigenfunctions, as

f ( ) c n n ( ) ...(11)
x
x
n 1
By making use of the orthogonality of the eigenfunctions, after taking the inner product of (11)
with , we find that the expansion coefficients are
n
, f n
c n ...(12)
n , n

Next, we expand the solution of the boundary value problem in terms of the eigenfunctions, as

x
x
y ( ) d n n ( ), ...(13)
n 1
and substitute (12) and (13) into (4) to obtain

x
L d ( ) c ( ).
x
n n n n
n 1 n 1
From the linearity of L and the definition of this becomes
n
d n n n ( ) c n n ( ).
x
x
n 1 n 1
We have therefore constructed a solution of the boundary value problem with d = c / , if the
n n n
2
series (13) converges and defines a function in C (a, b). This process will work correctly and give
a unique solution provided that none of the eigenvalues is zero. When = 0, there is no
n m
solution if c 0 and an infinite number of solutions if c = 0.
m m
Example 1: Consider the boundary value problem
y” = f(x) subject to y(0) = y( ) = 0 ...(14)
In this case, the eigenfunctions are solutions of

y” + y = 0 subject to y(0) = y( ) = 0,
2
which we already know to be = n , (x) = sin nx. We therefore write
n n
f ( ) c n sin nx ,
x
n 1
and the solution of the inhomogeneous problem (14) is
c
y ( ) n sin nx ,
x
n 2
n 1
In the case f(x) = x,
x sin nx dx 2( 1) n 1
c n 0 ,
2
sin nx dx n
0

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