On The Positivity Of The Difference Operators With Nonlocal Boundary Conditions

Journal of Science and Technology

3 (1), 2009, 68 - 73

©BEYKENT UNIVERSITY

ON THE POSITIVITY OF THE DIFFERENCE

OPERATORS WITH NONLOCAL BOUNDARY

CONDITIONS

Afgan ASLANOV

Department of Mathematics and Computing, Beykent University, Istanbul, Turkey,

afganaslanov@beykent.edu.tr Received: 09.01.2009 Accepted: 02.02.2009

ABSTRACT

In this work we consider the positivity of difference operators. We established that the difference operators have some properties of the simplest self-adjoint operators since they actions can be considered as an action of operators in finite dimensional spaces. We proved theorems about the positivity and estimates of the resolvent.

Keywords: positive operato, resolvent set, difference operator.

POSITIVITY

Positivity of the operator is a fairly "pleasant" property. For positive operators the fractional powers can be defined, the square root of a positive operator generates the analytic semigroup exp(-tA1/2), etc. Hence the importance of establishing the positivity of specific operators.

Let X and Y be Banach spaces. A densely defined in X linear operator A: X ^ Y is said to be positive, if for all nonnegative numbers t the operator A+ tI invertible and the next estimate holds:

II (A

+

tl)-1

||< - + - (1)

1 +1

for some constant c and all t > 0. We consider a differential operator A:

Au = -a( x) + 5u,

dx

with domain

D(A) = {u e C

(2)

[0,1]: u(0) = u(1), u'(0) = u'(1), a ( x ) > 0, 5 > 0}.

We define the grid space

Afgan ASLANOV

[0,1], = {x

k

= kh, 0 < k < N, Nh = 1} = {0, h, 2h,...,1},

where N is a fixed positive integer. To the operator A we assign the difference operator Ah defined by the formula

C > N-1

A

h

u

h

=\-a(x

k

)

Uk 1

- h +

U k - 1 +

5u

k

\ ,

{ }

N

I

h

J1 u ={ uk }o

with u0 = uN, u1 - u0 = uN - uN-1.

Let Ch denotes the Banach space of grid functions u defined on [ 0'1 ] h with the norm

|| u

h ||C

=

max | u

k

|.

Ch 0<k < N k

m a x | uk

=

um ,

Let us show that the operators Ah are positive. If and m f N we have

u

m

„

-

2u

m

+

u

m

h

2

IJHCJ UM - ^ (UM+1 + UM-1 )|>|S + ^ UM | - | | ( +

>| S + ^ u J H ^ u

m

| > S | u

m

| = S | | u

h

||

C

.

I h2 m I I h2 m I I m I II IIC,

max | u

k

|= u

N

,

u

=

u

If m = N, that is if from the conditions 0 N' u1 - u0 = uN - uN-1 we conclude that u1 - uN-1 = 2 uN and so u1 = uN '

u

=

u

max | u

k

|= u

m

,

or N-1 N' That is again we can take m f N.

Thus,

K U " ^ > S | | u

h

| |

C

h

for 5 >0 regardless of h. Similarly, for any given X >0 we have

||(Ah + A)u

h

||

Ch

> ( S + A ) | | u

h

| |

C h

.

The action of Ah is equivalent to the action of the matrix

II A u

h

i k - a ( x

m

)

m+1

J "

m-1

+5u

n

On The Positivity Of The Difference Operators With Nonlocal Boundary Conditions

M

n

=

2a( X

0

)

+5

a( Xj)

0

a(XN )

h

2 a ( X0)

h

2

0

0

0

a ( X0) a ( XN - 1) 2 a ( XN - 1)

h

2

2a(

X N

)

h

2

+ 5

h

2 a ( XN )

h

2

+ 5

0...

h

2

2a( X

1

)

" h h

0

0

+ 5

A + 2

Thus the Kernel (Null space) of Ah , and h is zero set, if 1 >0 (even if 1

A + 2

>-5). Then there exists the inverse of h that we call the resolvent operator

1

* h

= ( +

2 )

a_and„d ₅' ^ h _{+ -}" ^

By the definition of the positive operators this means that A is positive operator. (If 0<5<1, then

1

_{< •}

1 / 5

5 + 2 1 + 2

and if 5 >1, then

1

• <•

1

(2)

5 + 2 1 + 2

for 2 > 0 )' It is easily seen that

I I ( A

-

+ 2 ) - , | I

= 5 7 I ' ( 2 > 0 ) .

u = u

Indeed, if k m for all k, m = 1,..., N, we obtain

(A

h

+2-u

k

= (5 + 2)u

k

and so (2) holds.

_{A +2 A}

Note that h and h are onto, that is in terms of algebra the system of linear equations

f- a ( Xk)u k+1 - 2,u2k + uk-1 +mk = fk 1 _{k I '}

k = 1,2,..., N -1,

(3)

Afgan ASLANOV

with u° U n , u u° U n U n - 1, has a solution for any given set of

{ f

f f

^

numbers V1'-'2»"-» JN-^ , where we denoted by ^=5+1 . Indeed the system (3) is equivalent to the matrix equation

MN ( uı, u 2 ,' ' ', uN ) T

=

( 0 , f1, f2 ^i".:' fN - 2 , fN - 1) T

and so the determinant of the matrix MN is not zero. (Otherwise, one of

eigenvalues of the matrix MN must be zero and then there exists a nonzero uh

such that

MN (u„ u

2

,.„, uN )

T

= 0

o r

(Ah + 2 ) u

h

= 0 • u

h

= 0,

which contradicts the inequality ||(Ah +2-h Ilch>(5 + 2 ) | | uh| | c h ) .

{ f I

Thus (3) has a unique solution for any given (N-1)-tuple

Now about some estimations of the resolvent operator Rh ( ( + 2 ) . Theorem 1. For an arbitrary positive function a(x) (continuous or

discontinuous) the next inequality holds

|| (Ah + 2 ) ^

1

| | < M ( ^ , 5 ) ( 1 + 1 2 |)

-1

,

where

2 e R

n/2

= {2 :| arg21< p , 0 < p < n/2},

and 1

if 5 > cos p

M (p,S) =

cosp

— if 8 < cos p.

8

That is M does not depend on h.

Proof. It is known that (see [1], Lemma 13.1) if 10 is a point from resolvent set, then all points from the circle

1

2 - 2

0 l < l l ( A + 2 )

- 1

II

-1

are the points of resolvent set and

II (A

h

+ 2)

- 1

1|< p =

1

II(Ah + 2 0 )

- 1

| |

- 1

- 1 2 - 2 0 1 5 + 2 0 - | 2 - 2 I

if just 10 > 0. It is clear that all points of right half plane are the points of resolvent set (of -Ah). Indeed, for large enough integer n there exists a circle

I Li- n I< n + 5 .•• - , ,

_{containing 1 and so we have}

On The Positivity Of The Difference Operators With Nonlocal Boundary Conditions

|| (A

h

+ A ) x ||=|| (A

h

+ n + A - n ) x ||>|| (A

h

+ n)x || - | ( A - n ) | • || x |

> ( 8 + n-1 n-A|)|| x ||,

which yields

|| (Ah+a)

- 1

||<-1

8

+

n-1 n

-

Ä |

As n tends to infinity it is clear that

| n - A | ^ n-1 A | c o s p ,

which gives

w ,, 1

||

(Ah + A ) ||< || ( A h + A )- 1| | <

1

8(1+1A |)

The inequality (5) implies

|| (Ah + A)

- 1

||< M ( ( A ) -

1

-1+ | A |

where

1

8 + n - n+1 A | c o s p 8+1 A | c o s p '

1

if S> cosp,

if 8

<

cos p.

cosp(1+1 A |)

1

M (p, A) =

c o s p if 8 > cosp

1 if 8 < cos p.

8

(4) (5) Theorem 1 is proved.

It is easily seen that the points of the line R e A = 0 are also the points of the resolvent set and

|| ( A h + a )

- 1

| | < 8

-We can easily extend the resolvent set:

Theorem 2. Let a(x) be any positive function defined on [0,1]. Then all points

of the half-plane R e A > - 8 are the points of resolvent set (for -Ah) and

Afgan ASLANOV

!!(A +ä)-

1

\\<

1

(5+1 21 cosp)

if

-5 < R e 2 < 0.

Proof is similar to the proof of Theorem 1. W e just need to repeat the proof of theorem 1 and consider that now c o s P < 0 .

REFERENCES

[1]. Krasnoselskiy M., Zabreyko P. P., Pustilnik E. I. and Sobolevskiy P. E., Intergral Operators in the Space of Summable Functions, (Russian), Nauka, Moscow, 1966

Journal of Science and Technology

3 (1), 2009, 74 - 81

©BEYKENT UNIVERSITY

RESIDUE METHOD FOR THE SOLUTION OF

WAVE EQUATION WITH NONLOCAL

BOUNDARY CONDITION

Bahaddin SİNSOYSAL

Beykent University, Faculty of Science and Letters, Department of Mathematics and Computing, Sisli-Ayazaga Campus, 34396, Istanbul, Turkey

E-mail: bsinsoysal@beykent.edu.tr Received: 05.12.2008 Accepted: 08.01.2009

ABSTRACT

In this paper, using the residue method the initial-boundary value problem with nonlocal condition for a string vibration equation is investigated. The residue representation for the solution is obtained. The obtained representation permits to prove the existence and uniqueness theorem of the studying problem.

Key Words: Residue representation, Nonlocal boundary condition, Expansion

formula.

LOKAL OLMAYAN SINIR KOŞULUNA SAHİP

DALGA DENKLEMİNİN REZİDÜ YÖNTEMİYLE

ÇÖZÜMÜ

ÖZET

Bu çalışmada rezidü metodu kullanılarak telin titreşim denklemi için yazılmış başlangıç ve lokal olmayan sınır değer problemi incelenmiştir. Çözüm için rezidü gösterimi elde edilmiştir. Elde edilen bu gösterim, incelenen problemin çözümünün varlık ve tekliğini ispatlamaya imkan vermektedir.

1. INTRODUCTION

Many problems of sciences and technology are reduced to the problem in terms of nonlocal boundary condition for partial equations. The differential problems for heat equations with integral boundary conditions are investigated in [2], [4], [5], [13]. It is known that differential operators generated by the problem with nonlocal boundary conditions are not self-adjoint. In this case

Bahaddin SINSOYSAL

the system of eigenfunctions is not complete and application the Fourier's method impossible. The use of the method suggested by Keldish [6] which constructs adjoint functions corresponding to eigenvalues obtained from spectral problem is defined in [4] and [5]. Using these adjoint functions, the Fourier representation of the investigated problem is written. Using this method in [1] and [9] the problem of existence of solution for a

one-1

dimensional wave equation with integral conditions as J u(x, t)dx = 0 is 0

investigated.

In [11], using the residue method proposed by M.L. Rasulov [10] for the solution of the one-dimensional heat equation with nonlocal condition, at the residue representation has been obtained. Later, this problem for the two-dimensional heat equation is studied in [12].

In this paper, the initial boundary problem with nonlocal condition for a string vibration equation is investigated.

Let DT = { ( x , y ) , 0 < x , y < / } x [0, T ) , here l and T are given constants.

In DT we consider the following problem

d

2

u d

2

u ,, s ,,,

a F

=

a ? +

f

' > •

(1)

u(x,0) = (p

1

( x), (2)

du(x,0) , ^

— ^

= ( ( x ) ,

(3)

dt

1

1

(u) = u(0, t) - u (/, t ) = 0 , 1

2

(u) ^

d u ( 1 , t)

= 0 .

(4)

dx

Here, the functions ( (x) and (2 (x) express the initial profile and the initial velocity of the string, respectively. Assume that, ( (x) e L2 [0, /], (i = 1,2)

are given functions and the source function f (x, t) is a known continuously differentiable function on [ 0 , l ] .

According to the general theory of the residue method developed in [10], we will split the problem (1)-(4) into two following auxiliary problems:

(i) The boundary value problem

Residue Method For The Solution Of Wave Equation With Nonlocal Boundary Condition

d

2

y( x)

dx

2

+ AA y( x) = h( x),

y (0) - y(/) = 0, y'(/) = 0

(ii) The Cauchy problem

d

2

z

+ AA z = f (x, t),

(Pl( x),

= (p2 ( x)

dt

2

z (0) = (pI( x),

dz (0)

dt

(5) (6) (7) (8) (9)

Here, the function h(x) is any function from L2 [0, l]. The problem (5),(6) is called spectral problem corresponding to the problem (1)-(4).

The solution of the Cauchy problem (7)-(9) can easily be obtained as (see [3])

1 1 '

z(x, t, A) = (p

1

(x) cos At +—(p

2

(x) sin At +— [ sin)(t - r)f (x, rfdr. (10)

A )

A

J

2. THE SOLUTION OF THE SPECTRAL PROBLEM

The exact solution of the spectral problem (5),(6) can be constructed as,

i

y(x, A, h) = J G(x, A ) h ( ) Ç ,

₍₁₁₎

(see [3, 10]). Here, G(x,£, A) is called the Green's function of the problem (5),(6), and has the following form

G (

^ = ^ •

(12)

where

A(x,£,A) =

g (x,£,A) cos Ax sin Ax

1

1

( g (x,£,A)) 1

1

(cosAx) 1

1

(sin Ax)

1

2

(g(x,£,A)) 1

2

(cos Ax) 1

2

(sin Ax)

(13)

Bahaddin SINSOYSAL

A(A) = 11(cos Ax) 11(sin Ax)

12 (cos Ax) 12 (sin Ax) 1 g ( x , ) , A) sin A ( x - ) ) , 0 <)< x < / 2 A — — s i n A ( x - ) ) , 0 < x <)< /, 2A (14) (15)

11 (cosAx) = 1 - cos Al, 11 (sin Ax) = - sin Al, (16)

12 (cos Ax) = - A sin Al, 12 (sin Ax) = A cos A , (17)

11 [ g ( x , # , A ) ] = g ( 0 , 0 , A) - g ( / A ) = - A [ s i n A ) - s i n A ( / - ) ) ] (18) 2 A

12 [g ( x , ) , A ) ] d g ( / ) A )=1c o s A ( / - ) ) . (19) dx 2

Here, cos Ax and sin Ax are the fundamental solutions of the problem (5),(6). Using the conditions (6) for A ( x , ) , A ) and A(A) we have

A ( x , ) , A) = g ( x , ) , A ) [ A c o s A - A]- 2 cos Ax cos A sin A)

+ 2 cos Ax cos Al sin A (/ - ) ) - 2 cos Ax sin Al cos A (/ - ))

- 2 s i n Ax sin Al sin A) + ~ s i n Ax sin Al sin A(/ - ))

- 2 sin Ax cosA(/ - ) ) + -2sin Ax cos A c o s A ( / - ) ) , (20)

and

A(A) = AcosAl-A. (21)

2

It is obvious that the numbers Av = —-— , (v = 0,±1,±2,...) are roots of

the second order of equation (21). The numbers Av are called eigenvalues of

the problem (5),(6) and the corresponding eigenfunctions to Av are

/ ^ 2nvx

yv (x) = c o s — / — . Consequently, the system of eigenfunctions are not

complete. Therefore, applying the Fourier's method to the problem (1)-(4)

Residue Method For The Solution Of Wave Equation With Nonlocal Boundary Condition

encounters some difficulties. In [8], the general scheme of Fourier's method for a second order hyperbolic type equation is studied.

In order to apply the residue method at first, it is necessary to prove the formula of expansion of any function.

Theorem 1. Let h(x) and h'(x) be continuous functions on [0,l] . Then the formula of expansion

- 1

i

^ ^ f X d X f G ( x , g , X ) h ( g ) d g =

2n-1

v

V

0

= - £ Re sX f G(x,Z,mOdt =

{ h(x)

'

i f s

=

1

(22)

U

X

0 I 0, if s = 0,

(s = 0,1) x e (0,1)

is valid. Here, cv is a closed contour enclosing only one pole of the function

A(X) , (see [10]).

Here, ResF (X) denotes the integral residue of F(X) with respect to poles

X , and the sum extend on all poles of F ( X ) .

In order to find the explicit representation for the function h(x) , we will calculate the residue in (22) using the formula given in [7]. Let g and y be

analytic at z

0

and let g(z

0

) ^ 0, y ( z

0

) = 0 , y'(z

0

) = 0 , and

g (z)

y ( z)

ff S» j

y " (z0) ^ 0 , then — — has a second order pole at z0 and the residue is

Res

f g ^— , z0 = 22 g'(Z0) 2 g ( z 0 ) y ' ( Z 0 ) "

y ) y"(z

0

) 3 [y"(z0)]

2

Since X ( v = 0,±1,±2,...) are roots of second order of the equation A ( X ) = 0 , it is clear that A ' (Xv) = 0 and A ' '(Xv) ^ 0 . By simple

calculation we have A ''(X) = -2nvl and A ' ''(X

v

) = -3l

2

for v ^ 0.

Bahaddin SINSOYSAL

But, the A = 0 is the pole of third order of the function G ( x , £ , A ) . By taking above residue formula into account, we have

h(x) = 4 t j cos-—--—- JChCC) cos + (l -x) s i n

2

^ Jh (£)sin I

I

¡,=0

[

l 0 l l 0 l

J

3. SOLUTION OF THE MIXED PROBLEM

According to [10], the solution of the mixed problem (1)-(4) can be represented by the formula

- 1

i

u(x, t) = -¡= V jAdAj G(x,£,A)z(t,£,A)d£, (23)

2 W - 1

v V 0

where the cv denotes as a closed contour enclosing only one pole AV of the

solution of problem (5),(6) and the summation with respect to V is constructed over all poles of that solution. If we substitute the expressions for G ( x , £ , A) and z (t,£, A) into (23), we have

u (x, t) =

- 1

V J Ad A J

A ( x ,

f

A )

p

1

(C) cos AtdC

2 W - 1

V c 0

A (A)

1=r

V j A d A

J -

A

A A A A

A

V— ® si

n

Adi

2 W - 1 V V 0 AA(A)

V J A A J

A ( x , C A )

J sin A(t - T ) f ( £ , ) ) ) £ .

(24)

2 W - 1

V

V

J

AA(A)

j

v ) j y h

'

* ( 2 4 ) By simple calculations, we obtain the following representation of the solution of the problem (1)-(4)

4 r

u ( x,T ) =-1 — L V = 1 0

2kvE 2nvx

n

, . 2nvC . 2nvx

ccos cos + (l - x)sin sin

l l l l

dC +

2nV

l . —nV l r . 2nv,

. _,„ . 7

V (C)cos - +-— p— ( C ) s i n — - + - — I sin—- (t - T)f (C, ))d)

l 2nv l —nvQ ' x x 7 9

Residue Method For The Solution Of Wave Equation With Nonlocal Boundary Condition 4 2kv% 2nvx J A ) 1 v = 1 0 sin- -cos-l . 2nvt 2nvt l2 P (Ç)sm— 1$2 (Ç)cos—— +-;-. l l 4n

V

. 2nvt (ç)sin h

4n

2

v

2

2 f *

J f ti.T) sin 2nV (t - z)dz l— J (t - T) f (Ç,z)cos 2nV (t - z)dz

l l

l

₀

l

P i ( a + t p

2

( a + J (t -T)f (Ç,r)dT

4. CONCLUSION

Using the residue method, the exact solution of the initial boundary value problem with nonlocal condition for one-dimensional linear wave equation was obtained. The obtained formula permits to prove the existence and uniqueness of the investigated solution. In addition to this, the formula gives us the possibility to compare the exact solution with the approximate solution obtained by using numerical methods.

l

+

REFERENCES

[1] S.A. Beilin, Existence of Solutions for one-Dimensional Wave Equations with Nonlocal Conditions, Electronic Journal of Differential Equations, Vol. 2001, No. 76, pp. 1-8, 2001.

[2] J.R. Cannon, The Solution of the Heat Equation Subject to Specification of Energy, Quart. Appl. Math., 21, No.2, pp. 155-160, 1963.

[3] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw- Hill Book Company, New York, Toronto, London, 1955. [4] N.I. Ionkin, Solutions of Boundary Value Problem in Heat Conductions Theory with Nonlocal Boundary Conditions, Differents. Uravn., Vol.13, No.2, pp 294-304, 1977.

[5] L.I. Kamynin, A Boundary Value Problem in the Theory of the Heat Conduction with Nonclassical Boundary Condition, Z. Vychisl. Mat. Fiz., 4, No.6, pp. 1006-1024, 1964.

[6] M.V. Keldysh, On Eigenvalues and Eigenfunctions of Certain Classes of Not Self-Adjoint Equations, Doklady Acad. Nauk SSSR, 87, pp. 11-14, 1951.

[7] J.E. Marsden and M.J. Hoffman, Basic Complex Analysis, W.H. Freeman, New York, 1999.

Bahaddin SINSOYSAL

[8] I.G. Petrovskii, Lecture of Partial Differential Equations, Moscow, 1961. [9] L.S. Pulkina, A Nonlocal Problem with Integral Conditions for Hyperbolic Equations, Electron. J. Diff. Eqns., Vol. 1999, No.45, pp. 1-6, 1999.

[10] M.L. Rasulov, Methods of Contour Integration North-Holland Publishing Company, Amsterdam, 1967.

[11] M. Rasulov and B. Sinsoysal, Residue Method for the Solution of Heat Equation with Nonlocal Boundary Condition, Beykent University J. Sci. and Tech., 2, No.1, pp. 146-158, 1998.

[12] B. Sinsoysal and M. Rasulov, Residue Method for the Solution of a 2D Linear Heat Equation with Nonlocal Boundary Condition, Int. J. Contemp. Math. Sciences, Vol.3, No.34, pp.1693-1700, 2008.

[13] N.I. Yurchuk, Mixed Problem with an Integral Condition for Certain Parabolic Equations, Differents. Uravn., Vol.22, No.12, pp. 2117-2126, 1986.