The 2nd National Conference on
Mathematics and Its Applications in Engineering Sciences 14 , 15 May 2015 Islamic Azad University Jouybar Branch
Representation the Numerical values of the Nodal Points and the Nodal Length Abdolali Neamaty1, a, Shahrbanoo Akbarpoor2,b and Abdolhadi Dabbaghian3,c 1
Department of Mathematics, University of Mazandaran, Babolsar, Iran 2
Islamic Azad University, Jouybar Branch, Jouybar, Iran 3
Islamic Azad University, Neka Branch, Neka, Iran
a
[email protected],
[email protected],
[email protected]
Keywords: Differential equation, Aftereffect, Nodal points.
Abstract. In this work, we consider a boundary value problem with aftereffect on a finite interval and study the asymptotic behavior of the solutions, eigenvalues, the nodal points and the associated nodal length and calculate the numerical values of the nodal points and the nodal length. Introduction In this paper, we consider the equation x
y( x ) q( x ) y ( x ) M ( x t ) y (t )dt y ( x ), 0
0 x ,
(1)
under the separated boundary conditions U ( y ) : y(0) hy (0) 0, V ( y ) : y( ) Hy ( ) 0,
(2)
where 2 and i is the spectral parameter and also q, M W 2,1 (0, ) are real functions. We denote the boundary value problem (1)-(2) by L( q, M , h, H ) . In fact, in this work, we consider the Sturm-Liouville operator disorganized by a Volterra integral operator. The form of the differential equation with aftereffect without the computation of the nodal points were perused in [4]. Computation of the nodal points was studied for Sturm-Liouville equations in [1, 2] and other works but it was not considered for the differential equations with aftereffect. Also in [3], we studied the equation (1) under the separated boundary conditions on a finite interval with discontinuity conditions in an interior point and proved the uniqueness theorem by using the nodal points but we did not obtain the numerical values of the nodal points. Whereas in this paper, we consider the differential equation (1) under the boundary conditions (1)-(2) but without discontinuity conditions in an interior point and obtain the numerical values of the nodal points and the nodal length. In section 2, we obtain the asymptotic form of the solution, the characteristic function, the eigenvalues and in section 3, we calculate the asymptotic form of the nodal points and the nodal length and the numerical values of them.
The 2nd National Conference on
Mathematics and Its Applications in Engineering Sciences 14 , 15 May 2015 Islamic Azad University Jouybar Branch
The asymptotic form of the solution and the eigenvalues Let C ( x, ) and S ( x, ) be solutions of (1) under the initial conditions S (0, ) C (0, ) 0 and S (0, ) C (0, ) 0 . In this case, the functions C ( x, ) and S ( x, ) satisfy the following integral equations respectively (see [4]):
C ( x, ) cos x
x
0
S ( x, )
x sin ( x t ) t sin ( x t ) q(t )C (t , )dt 0 M (t s)C ( s, )dsdt, 0
x sin ( x t ) x sin ( x t ) t sin x q(t )S (t , )dt 0 M (t s) S ( s, )dsdt. 0 0
(3)
(4)
Denote ( x, ) C ( x, ) hS ( x, ) . Thus, the function ( x, ) is solution of (1) under the initial conditions (0, ) 1 and (0, ) h and the solution of the integral equation
( x, ) cos x h
x sin ( x t ) t sin x q(t ) (t , ) M (t s ) ( s, )ds dt , 0 0
(5)
and hence x
t
( x, ) sin x h cos x cos ( x t ) q (t ) (t , ) M (t s ) ( s, ) ds dt. 0
0
(6)
Lemma 1. For , the formula 1 ( x, ) cos x O e x O e x
holds, uniformly with to x 0, as Im . Proof. See [4]. Substituting (7) into (5) and (6), we get
(7)
The 2nd National Conference on
Mathematics and Its Applications in Engineering Sciences 14 , 15 May 2015 Islamic Azad University Jouybar Branch
( x, ) cos x q1 ( x ) 1
sin x 1 2
x
0
q( t )sin ( x 2t )dt
1 x 0 sin ( x t ) 0 M (t s) cos sdsdt O 2 e , x
(8)
t
and also
1 x q(t ) cos ( x 2t )dt 2 0 1 x x t cos ( x t ) M (t s ) cos sdsdt O e , 0 0
( x, ) sin x q1 ( x ) cos x
where q1 ( x ) h
(9)
1 x q( t ) dt. 2 0
Integration by parts results
x
0
q(t )sin ( x 2t )dt
1 1 1 q( x ) cos x q(0) cos x 2 2 2
x
0
q(t ) cos ( x 2t )dt ,
(10)
and
x
0
t
sin ( x t ) M (t s ) cos sdsdt
1
0
x
0
1
x
0
M ( x s ) cos sds
t M cos ( x t ) M (0) cos t ( t s ) cos sds dt. 0 t
(11)
We obtain from (8)-(11) that
( x, ) cos x q1 ( x )
1 x sin x O 2 e ,
(12)
and 1 ( x, ) sin x q1 ( x ) cos x O e x .
Now, substituting (12) into (5) and applying integration by parts, we get
(13)
The 2nd National Conference on
Mathematics and Its Applications in Engineering Sciences 14 , 15 May 2015 Islamic Azad University Jouybar Branch
( x, ) cos x q1 ( x )
sin x 1 1 2 q( x ) cos x 2 q(0) cos x 4 4
1 cos x x M (0) q(t )q1 (t )dt x cos x O 3 e x , 2 0 2 2 2
(14)
similarly, we get
( x, ) sin x q1 ( x ) cos x
3 1 q( x )sin x q(0)sin x 4 4
1 x sin x x M (0) q ( t ) q ( t ) dt x sin x O 2 e . 1 2 0 2
(15)
Let ( x, ) and ( x, ) be solutions of (1) under the initial conditions (0, ) 1 , (0, ) h , ( , ) 1 and ( , ) H . Denote ( ) : ( x, ), ( x, ) ,
(16)
where
( x, ), ( x, ) ( x, ) ( x, ) ( x, ) ( x, ), is the Wronskian of ( x, ) and ( x, ) . Since the function ( ) called the characteristic function for the boundary value problem L( q, M , h, H ) does not depend on x , hence, substituting
x into (16), we get
( ) V ( ) ( , ) H ( , ).
(17)
Lemma 2. For , the representation ( ) sin cos sin 2
0
3 1 q( )sin q(0)sin 4 4
1 M (0) sin q(t )q1 (t )dt sin Hq1 ( ) O 2 e , 2
holds where H q1 ( ) .
(18)
The 2nd National Conference on
Mathematics and Its Applications in Engineering Sciences 14 , 15 May 2015 Islamic Azad University Jouybar Branch
Since the eigenvalues
n n 1 of
the boundary value problem coincide with the zeros of the
function ( ) , using some straightforward calculations ([4]), we obtain
n n
1 O 2 , n n
n .
(19)
The form of the nodal points and the nodal length Denote n ( x ) ( x, n ), where n is the eigenfunction corresponding to the eigenvalue n . Let
0 1 ... be the eigenvalues of the aftereffect problem (1)-(2) and also let the nodal points of the nth eigenfunction n be shown with j 1, 2,..., n 1 .
0 x1n xn2 ... xnj ,
Theorem 3. The nodal points of the aftereffect problem (1)-(2) are 1 1 1 xnj xnj ( j ) 2 q1 ( xnj ) 2 q( t ) cos 2ntdt 2 n n 2n 0 j 1 xn t 1 2 M (t s ) cos nt cos nsdsdt O 3 , 0 0 n n
(20)
and the nodal length is lnj
1 xnj 1 1 2 j q1 (t )dt 2 x n n 2n 2n
1 2 n
xnj 1 xnj
xnj 1 xnj
q(t )cos 2 ntdt
(21)
1 0 M (t s) cos nt cos nsdsdt O n3 . t
Proof. Since the nodal points are the zeroes of the eigenfunctions, then we get from (5) that
n ( x ) cos n x h
x sin ( x t ) t sin n x n q (t ) n (t ) M ( t s ) n ( s )ds dt 0 0 n n
cos n x h
t sin n x sin n x x cos( n t ) q(t ) n (t ) M (t s ) n ( s ) ds dt 0 0 n n
t cos n x x sin( n t ) q( t ) n ( t ) M (t s ) n ( s ) ds dt. 0 0 n
Now, we set n ( x ) 0 . Thus, we obtain
(22)
The 2nd National Conference on
Mathematics and Its Applications in Engineering Sciences 14 , 15 May 2015 Islamic Azad University Jouybar Branch
cot n x
h 1 n n
x
0
t
cos( nt ) q (t ) n (t ) M ( t s ) n ( s )ds dt 0
t cot n x x sin( nt ) q (t ) n (t ) M ( t s ) n ( s )ds dt 0, 0 0 n
(23)
then, for n , we get 1 h 1 xnj ( j ) 2 2 2 n n n 1 2 n
xnj
0
xnj
0
q(t ) cos( n t ) n ( t )dt
(24)
t
cos nt M (t s ) n ( s ) dsdt. 0
From (19) and (24), we obtain 1 1 1 xnj xnj ( j ) 2 q1 ( xnj ) 2 q( t ) cos 2ntdt 2 n n 2n 0 1 xnj t 1 2 M (t s ) cos nt cos nsdsdt O 3 . 0 0 n n
(25)
Also, the nodal length is lnj xnj 1 xnj ,
(26)
Therefore lnj
1 2 n 2n
1 2 n
xnj 1 xnj
xnj 1 xnj
q1 (t )dt
1 2n 2
xnj 1 xnj
q(t ) cos 2 ntdt
1 0 M (t s) cos nt cos nsdsdt O n3 , t
and consequently, the theorem is proved. Example 4. Let h 1 and q( x ) M ( x ) x . Using (20) and (21), we obtain the numerical values of the nodal points and the nodal length (see the table 1 and the table 2).
(27)
The 2nd National Conference on
Mathematics and Its Applications in Engineering Sciences 14 , 15 May 2015 Islamic Azad University Jouybar Branch
Table 1: The numerical values of the nodal points
n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10
j=1
j=2
j=3
j=4
j=5
j=6
j=7
j=8
j=9
1.0581 0.6391 0.4566 0.3547 0.2899 0.2450 0.2121 0.1869 0.1671
0.7473 1.2613 0.9909 0.8166 0.6955 0.6058 0.5366 0.4817
2.0852 1.6351 1.3483 1.1486 1.0011 0.8873 0.7969
2.2870 1.8821 1.6030 1.3970 1.2384 1.1124
2.4220 2.0606 1.7949 1.5907 1.4287
2.5188 2.1931 1.9432 1.7451
2.5935 2.2971 2.0624
2.6510 2.3798
2.6982
Table 2: The numerical values of the nodal length
n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10
j=1
j=2
j=3
j=4
j=5
j=6
j=7
j=8
1.0895 0.8000 0.6345 0.5267 0.4505 0.3937 0.3497 0.3146
0.8268 0.6452 0.5317 0.4532 0.3953 0.3507 0.3152
0.6493 0.5338 0.4544 0.3960 0.3511 0.3155
0.5398 0.4576 0.3978 0.3523 0.3162
0.4582 0.3983 0.3525 0.3164
0.4004 0.3539 0.3173
0.3539 0.3173
0.3183
References [1] P. J. Browne and B. D. Sleeman, Inverse nodal problems for sturmliouvile equations with eigenparameter dependent boundary conditions, Inverse problems, 12 (1996), 377-381. [2] Y. H. Cheng, C. K. Law and J. Tsay, Remarks on a new Inverse nodal problem, J. Math. Anal., 248 (2000), 145-155. [3] A. Dabbaghian, Sh. Akbarpour and A. Neamaty, The uniqueness theorem for discontinuous boundary value problems with aftereffect using the nodal points, Iranian Journal of Science and Technology, 7 (2012), 391-394. [4] G. Freiling and V. A. Yurko, Inverse Sturm-Liouville problems and their applications, NOVA science publishers, New York (2001).
