Page 211 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes The set {f (x)} is said to be orthonormal set if
n
b
x
f m ( ) f n ( )dx = mn ,
x
a
Where the Kronecker delta,
0 if m n
=
mn 1 if m n

Orthonormal Set of Functions with Respect to a Weight Function
Let { (x)} (n = 1, 2, 3, ...) be a set of real functions defined on the interval a x b and p(x) 0.
n
Then the set { (x)} is said to be orthonormal set of functions on the interval a x b if
n
b
0 when m n
x
p ( ) m ( ) n ( )dx =
x
x
1 when m n
a
b
x
x
i.e., p ( ) ( ) ( )dx = .
x
m n mn
a
Self Assessment
1. Show that the function f (x) = 1, f (x) = x are orthogonal on the interval ( 1, 1) and determine
1 2
2
the constants A and B so that the function f (x) = 1 + Ax + Bx is orthogonal to both f (x) and
3 1
f (x) on the interval ( 1, 1).
2
13.2 Review of Sturm-Liouville Problem - Eigenvalues and
Eigenfunctions
Various important orthogonal sets of functions arise in the solution of second-order differential
equation

x
x
R ( )y [ ( ) P ( )]y = 0 ...(i)
Q
x
on some interval 0 x b satisfying boundary conditions of the form
(a) a y a y = 0 at x = a
2
1
...(ii)
(b) b y b y = 0 at x = b
1
2
The boundary value problem given by (i), (ii) is called a Sturm Liouville problem. Here is a
parameter and a , a , b , b are given real constants at least one in each of conditions (ii) being
1 2 1 2
different from zero. The equation (i) is known as the Sturm Liouville equation.
You may recall that Bessel’s differential equation, Legendre’s equation, Hermite equation and
other important equations can be written in the form (i).
The solution y = 0 is the trivial solution. The solution y 0 are called the characteristic functions
or eigenfunctions and are called characteristic values or eigenvalues of the problem.

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