International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 5, August 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
Strum - Liouville Problems in Eigenvalues and Eigenfunctions
B. Kavitha1, Dr. C. Vimala2
1Assistant
Professor, 2Professor,
1Department of Mathematics, Sri Manakula Vinayagar Engineering College, Puducherry, India
2Department of Mathematics, Periyar Maniyammai University, Thanjavur, Tamil Nadu, India
ABSTRACT
This paper we discusses with Strum-Liouville problem of eigenvalues and
eigenfunctions, within the standard equation d  r d ( x)   (q   p) ( x)  0

dx 
How to cite this paper: B. Kavitha | Dr.
C. Vimala "Strum - Liouville Problems in
Eigenvalues
and
Eigenfunctions"
Published
in
International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456IJTSRD31721
6470, Volume-4 |
Issue-5,
August
2020,
pp.80-83,
URL:
www.ijtsrd.com/papers/ijtsrd31721.pdf

dx 
where p,q and r are given functions of the independent variable x is an
interval a  x  b. The  is a parameter and  ( x) is the dependent variable.
The method of separation of variable applied to second order liner partial
differential equations, the equation is known because the Strum-Liouville
differential equation. Which appear in the overall theory of eigenvalues and
eigenfunctions and eigenfunctions expansions is one of the deepest and
richest parts of recent mathematics. These problems are associate with
work of J.C.F strum and J.Liouville.
KEYWORDS: Strum-Liouville problem of eigenvalues and eigenfunctions,
eigenvalues and eigenfunctions, orthogonal, weight functions
INTRODUCTION
The method of separation of variables utilized within the
solutions of boundary value problems of mathematical
physics frequently gives rise so called Strum-Liouville
eigenvalue problems, our attention to small but significant
fragment of the theory of Strum-Liouville problems and
their solutions. The character of the any spectrum of
particular problem will be determined by actually finding
the eiganvalue. We shall establish the orthogonallity of
eigenfunctions corresponding to distinct eigenvalues,
orthogonal set with regard to weight functions that not
piecewise continuous.
1.1. Eigenvalue, Eigenfunction
The value of  , for that the strum-Liouville system (1.1)
feature a nontrivial solution are called the eigenvalues,
and thus corresponding solution are called the
eignfunctions.
1.2. Strum-Liouville system
The solution of partial differential equation by the method
of separation of variables.
Under separable conditions we’ll transform the second
order homogeneous partial differential equation into
ordinary differential equation.
a1 x ''  a2 x '  (a3   ) x  0
...(1.1)
b1 y  b2 y  (b3   ) y  0
...(1.2)
''
@ IJTSRD
'
|
Unique Paper ID – IJTSRD31721
Copyright © 2020 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/
by/4.0)
Which are of the form
a1 ( x)
d2y
dy
 a2 ( x)   a3 ( x)    y  0...(1.3)
2
dx
dx
i.e. two strum Liouville problems we introduce
 x a2 (t ) 
r ( x)  exp  
 dt
 a1 (t ) 
a ( x)
r ( x)
q( x)  3
r ( x), p ( x) 
a1 ( x)
a1 ( x)
...(1.4)
Equation (1.3) we obtain
d  dy 
 r   (q   p ) y  0
dx  dx 
...(1.5)
This is mentioned because the strum-Liouville equation.
In terms of the operator
L
d  d 
r  q
dx  dx 
Equation (1.5) are often written as
L( y )   p ( x) y  0
...(1.6)
Where  is a parameter independent of x, and p, q and r
are real-valued functions of x.
|
Volume – 4 | Issue – 5
|
July-August 2020
Page 80
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
To ensure the existence of solution, we let p and q be
continuous and r be continuously differential during a
closed finite interval (a, b).
1.3. Theorem
Let the coefficients p, q and r within the Strum-Liouville
system be continuous in (a, b).Let the eigenfunctions  j
and
k ,
corresponding to
differentiable, then
j
and
j
and
k ,
be continuously
k are orthogonal with regard
to the load function p(x) in (a, b).
Proof:
Since  j
corresponding to
j


j
satisfies the strum-
...(1.7)
and equation (1.8) by
a
 r

1.5. Theorem
If 1 ( x) and

 jk'  'jk
 r (a)  j (a)k' (a)   j' (a)k (a) 

...(1.9)
j
and k are
b1 j (b)  b2 (b)  0
b1k (b)  b2k' (b)  0
b2  0, we multiple the first condition by k (b) and the
second condition by  j (b), and subtract to obtain
If
...(1.10)
solutions
of
We have
d  d1 
r
  (q   p )1  0
dx  dx 
d  d2 
r
  (q   p )2  0
dx  dx 
...(1.14)
..(1.15)
Multiplying the equation (1.14) by 2 and the equation
(1.15) by 1 and subtracting, we obtain
1
d  d2 
d  d1 
r
  2  r
 0
dx  dx 
dx  dx 
Integrating this equation from a to x, we obtain
r ( x) 1 ( x)2' ( x)  1' ( x) 2 ( x) 
a2  0, we obtain
 j (a) (a)   (a)k (a)  0
two
Proof:
Since 1 and 2 are solution of
L( y )   p ( y )  0
'
j
'
j
any
a
The end condition for the eigenfunctions
 j (b)k' (b)   j' (b)k (b)  0
2 ( x) are
L( y )   p( y )  0 on (a, b), then r ( x) w ( x :1 , 2 )  cons tan t
Where w is the wronskian.
b
The right side of which is knows as the boundary term of
the Strum-Liouville system.
'
k
b
2i   p (u 2  v 2 ) dx  0
This implies that  must vanish for p>0 and hence the
eigenvalues are real.
dx
 r (b)  j (b)k' (b)   j' (b)k (b) 
In a similar manner, if
Then, because the coefficient of the equation are real.
a
and integrating yields
b
 j  u  iv
By using the relation (1.12), we have
k
 k  p jk   j d  rk'   k d  r j' 
j  k   p jk dx
Proof:
Suppose that there is a complicated eigenvalue
 j    i
Corresponding to the eigenvalue k    i
...(1.8)
d
 rk'  j  dxd  r j' k
dx

1.4. Theorem
All the eigenvalues are regular strum-Liouville system
with p(x) > 0 are real.
Thus,there exists an eigenfunction
k  u  iv
and subtracting we obtain
dx
...(1.13)
j k
a
The complex conjugate of the eienvalue is also eigenvalue.
For the same reason
d
 rk'    q  k p k  0
dx
j,
 p  dx  0
With eigenfunction
Liouville equation, we’ve
d
 r j'    q   j p  j  0
dx
Multiplying equation (1.7) by
b
 r (a) 1 (a)2' (a)  1' (a) 2 (a) 
...(1.11)
...(1.16)
= constant
We see by virtue of (1.10) and (1.11) that

This is called Abel’s formula.
b
j
 k   p jk dx  0
...(1.12)
a
if
 j and k
@ IJTSRD
|
1.6.
are district eigenvalues, then
Unique Paper ID – IJTSRD31721
|
Theorem
Volume – 4 | Issue – 5
|
July-August 2020
Page 81
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470


An eigenfunction of normal strum-Liouville system is
unique except for a constant factor.
d
r yy '  y y '  p y y (   )
dx
Proof:
Let 1 ( x) and 2 ( x) are eigenfunctions corresponding to an
eigenvalue .
Integrating both sides of (1.23) with respect to x from a to
b, we get
The according to Abel’s formula we have
r ( x) w ( x :1 , 2 )  cons tan t r ( x)  0
Where w is the wronskian.
b
' 2
 p y1 (   )dx  r
a
...(1.23)
 yy  y y 
'
'
'
x  ( a, b)
Now (1.21) x
Since 1 and 2 satisfy the end condition at x = a, we have
a1  y(a) y ' (a)  y(a) y '(a)   0
a11 (a)  a21' (a)  0
'
y ' (a ) - (1.18) x y (a ) gives
As a1  0, y(a) y ' (a)  y(a) y '(a)  0
a12 (a)  a22' (a)  0
Similarly y (b) y ' (b)  y (b) y '(b)  0
Since a1 and a2 are not both zero.
Hence (1.24) gives
Therefore
a
 r (b)  y(b) y (b)  y(b) y (b) 
 r (a)  y(a) y ' (a)  y(a) y '(a)  ...(1.24)
'
Thus, if w vanishes at a point in (a, b) it must vanish for all
b
1 (a) 1' (a)
 w(a : 1 , 2 )  0
2 (a) 2' (a)
We have
b
(   )  p( x) y ( x) dx  0
2
a
w( x : 1 , 2 )  0 for all x at (a, b), which is a
sufficient condition for the linear dependence of two
functions 1 and 2 . Hence 1 ( x ) differs from 2 ( x) only
Since p ( x)  0 and b  a, the integral in the L.H.S is
positive.
Hence (   )  0
i.e,    (or )  is real
by a constant factor.
1.7. Theorem
The eigenvalues of a strum-Liouville system are real.
Proof:
Let the strum-Liouville system be the second order
differential equation
d
(r y ' )  (q   p) y  0
...(1.17)
dx
In the interval a  x  b where p, q, r are real functions
and p ( x)  0, together with the boundary conditions.
a1 y (a)  a2 y (a)  0
...(1.18)
b1 y (b)  b2 y ' (b)  0
...(1.19)
'
 and y may be
Taking the complex conjugate (1.17), (1.18) and (1.19) we
have,
d
r y '  (q   p) y  0
...(1.20)
dx

a1 y(a)  a2 y ' (a)  0
...(1.21)
b1 y(b)  b2 y (b)  0
...(1.22)
'
 
|
Proof:
Let the strum-Liouville system by the second order
differential equation
d
(r y ' )  (q   p) y  0
...(1.25)
dx
In the interval a  x  b where p, q, r are real functions
together with the boundary conditions.
a1 y (a)  a2 y ' (a)  0
...(1.26)
b1 y (b)  b2 y ' (b)  0
...(1.27)
Unique Paper ID – IJTSRD31721
Let y1(x) and y2(x) be two eigenfunctions corresponding to
two different eigenvalues 1 and 2 of 
Then
d
(r y1' )  (q  1 p) y1  0
dx
d
(r y2 ' )  (q  2 p) y2  0
dx
...(1.28)
...(1.29)
Also
(1.17) x y - (1.20) x y gives
d
d
y (r y ' )  y
r y '  py y (   )  0
dx
dx
@ IJTSRD
a strum-Liouville system
different eigenvalues are
then a1, a2, b1 and b2 are real.
We assume that a1, a2, b1, b2 is real while
complex.

1.8. Theorem
The eigenfunctions of
corresponding to two
orthogonal.
|
a1 y1 (a)  a2 y1' (a)  0
...(1.30)
b1 y1 (b)  b2 y1' (b)  0
...(1.31)
Volume – 4 | Issue – 5
|
July-August 2020
Page 82
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
a1 y2 (a)  a2 y2' (a)  0
...(1.32)
b1 y2 (b)  b2 y2' (b)  0
...(1.33)
b
Hence p( x) y1 ( x) y2 ( x)dx  0

(1.29) x y1 - (1.28) x y2 gives
d
d
y1 (r y2' )  y2  ry1'   py1 y2 (2  1 )  0
dx
dx
d
'
i.e.,
(r y1 y2  ry2 y1' )  py1 y2 (1  2 ) ...(1.34)
dx
Integrating both sides of (1.34) with respect to x from a to
b
We have,

b
'
 py1 y2 (1  2 )dx  ry1 y2 ry2 y1
a
'

b
a
 r (b)  y1 (b) y2 (b)  y2 (b) y (b) 
 r (a)  y1 (a ) y2 ' (a)  y2 (a ) y1' (a )  ...(1.35)
'
'
'
1
'
As a1  0, y1 (a) y2 ' (a)  y2 (a) y1' (a)  0
References:
[1] Takashi Suzuki(1985) on the inverse strum –
Liouville problem of differential for spatilly
symmetric operators-1[Journal of differential
equations 56,165-194)
[4] Pergrnon (Oxford 1980) strum – liouville invers
iegenvalue problem[Mechanics today vol-5 281-295]
Similarly y1 (b) y2 ' (b)  y2 (b) y1' (b)  0
[5] James ward ruel and ruel c. Churchill brown(1993)
strum – liouville problem and Applications [Fourier
series and Boundary value problem]
Hence (1.35) gives
b
(1  2 )  p ( x) y1 ( x) y2 ( x)dx  0
[6] Dr. M. K. Venkataraman (1992) strum – liouville
system
eigenvalues,
eigenfunctin
[Higher
Mathematics for Engineering and Science]
a
But 1  2
Unique Paper ID – IJTSRD31721
Conclusion
In this article we’ve investigated Strum-Liouvillie
problems in eigenvalues and eigenfunctions, eigenvalue
and special function are discussed. Eigenfunctions
expantion and orthogonality of eigen function are
lustrated.
[3] Mohammed Al-Refai and Thabet Abdeljawad
Fundamental Results of Conformable Sturm-Liouville
Eigenvalue Problems [Hindwai Complicity]
a1  y1 (a ) y2 ' (a)  y2 (a) y1' (a)   0
|
Then (1.36) shows that y1(x) and y2(x) are orthogonal in
(a, b) with respect to the load function p(x).
[2] M H Annaby and Z S Mansour Basic strum-Liouville
problems Journal Of Physics A: Mathematical And
General
Now (1.30) x y2 ( a ) - (1.32) x y1 ( a ) gives
@ IJTSRD
...(1.36)
a
|
Volume – 4 | Issue – 5
|
July-August 2020
Page 83