Page 212 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 13: Orthogonality of Solutions

There are a few theorems about the eigenvalues and eigenfunctions as follows: Notes
Theorem 1: Let the functions P, Q, R in the Sturm Liouville equation be real and continuous on
the interval a x b. Let y (x) and y (x) be given functions of the Sturm Liouville problem
m n
corresponding to different eigenvalues and respectively, and let the derivatives y (x),
m n m
y (x) be also continuous on the interval. Then y and y are orthogonal on that interval with
n m n
respect to the weight function P i.e.,
b
P ( )y m ( )y n ( )dx = 0 for m n
x
x
x
a
Theorem 2: The eigenvalues of the Sturm Liouville problem are all real.
Theorem 3: If R(a) > 0 or R(b) > 0, the Sturm Liouville problem cannot have two linearly
independent eigen functions corresponding to the same eigenvalue.

Example: The simpler example of a Sturm Liouville equation is the Fourier’s equation
l
y ( , ) y ( , ) = 0 subject to (0)y y ( ) 0
x
x
which has solutions cos (x ) and sin (x ). Using the boundary conditions, we have for y (0)
= 0, only sin (x ) term is present. From the second consideration we have
l = n , n 0,1,2,...
So the eigenfunctions are given by

n x
y (x) = A sin , for n = 1, 2, 3,...
n n
l
The eigenvalues are given by
n 2 2
= , n 0,1,2,3,....
n l 2
Self Assessment

2. Find the eigenvalues ad eigenfunctions of the equation
2
y (x) + k y(x) = 0
with the boundary conditions
y(0) = 0 and y (1) = 0

13.3 Review of Bessel’s Inequality and Completeness Relation

Let { (x), [n = 1, 2, 3, ...]} be an orthonormal set of functions on an interval (a, b) and let an
n
arbitrary function on the same interval be a linear combination of these functions, in the form

x
f(x) = C n n ( ) a x b
n 1
If the series converges and represents f(x), it is called a generalized Fourier series of f(x). The
coefficient C , = 1, 2, .... given by
b
x
x
x
C = , f ( ) f ( ) ( )dx ...(i)
a
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