simultaneous quadruple integral equations involving g(xn)

Scientific Research Journal (SCIRJ), Volume I, Issue IV, November 2013 31 ISSN 2201-2796

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© 2013, Scientific Research Journal

Simultaneous Quadruple Integral Equations

Involving G(xn)

KULDEEP NARAIN

School of Quantitative Sciences, UUM College of Arts and Sciences

Universiti Utara Malaysia

06010 UUM, Sintok, Malaysia

Abstract- Integral and Series equations are very useful in the theory of elasticity, elastostatics, diffraction theory and acoustics.

Particularly these equations are very much useful in finding the solution of crack problems of fracture mechanics .In the present

section fractional integral operators are used to obtain a formal solution of simultaneous quadruple Integral equations involving

Meijer’s G-function of n-variables i.e. ( )nG x as kernel by reducing them to one having a common kernel.

Index Terms - Integral Equations, Fractional Integration, G-Functions of n variables.

I. INTRODUCTION

Khadia and Goyal [4] have introduced the Meijers G-function of n-variables i.e. ( )nG x . Rewriting ( )nG x in a slightly

different form as

( )

( );( )

, ; ( ), ( ) ( )(( ))1, 2,......., , 1 ,.......,, ; ( ), ( )

(( ))1,........, , 1 ,.........,

1'( ) ( ) ( )

(2 ) 1n

n n

n n n n

kkk n k k

L

a bp m q

n n n n nm o M N x C C C C CG n M M p Mp m q p M q N n n n nn n n nd d d d

N N q Nn n n n

n SS S x dS

kni k

(1.1)

where, a repeated suffix represents sum from 1 to n , i.e. .1

kkk

n

k

SS

( )1

'( )

(1 ) ( )1 1

na S

j kkj

Skk p q

a b Sj m j kk

j j

(1.2)


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( ) ( )1 1

( )1

( ) ( )1 1

M Nk kk kC S d S

j k j kn j jS

n k p qk k kk kC S d S

j M k j N kj jk k

(1.3)

and

( ) . ........1 2

1

ndS dS dS dS

k nk

(1.4)

Also ( )q

b represents the sequence 1

b , 2

b ,……, q

b ; ( )n

L are n suitable contours and the positive integers p : 1

p ,

2p ,…….,

np ; q :

1q ,

2q ,……..,

nq ; m :

1M ,

2M , …….,

nM ;

1N ,

2N ,…………,

nN satisfy the following inequalities

0, 0;( ) 1;k

p q q 0 ( ) ( );k k

M p ( ) ( ) ; 1 , 2 , . . . . . . . , .k kp p q q k n The values ( ) 0kx ,

( 1,2,...., )k n are excluded.

An empty product is interpreted as unity. The contour kL is in the

kS – plane and runs from i to i with loops, if

necessary to ensure that the poles of kk

k

j NjSd ,........2,1);( lie to the right and the poles of

kk

k

j MjSC ,........2,1);( and ( ); 1,2,........j kka S j m lie to the left of the contour kL where 1,2,.......,k n

,hereafter.

The function ( )nG x is an analytic function of (xn) under the following set of conditions :

1 1 1 1arg ( )

2 2 2 2k k k k kX m M N q q p p , 2( )k k k km M N q q p p .

II. NOTATIONS AND KNOWN RESULTS

( )1

[ ]

(1 )1

ma SII j kk

jS

kk pa SII j m kk

j

(1.5)


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( ) ( )1 1

( )1

( ) ( )1 1

M Nk kk kl S f S

j k j kn j jS

n k p qk k kk kl S f S

j M k j N kj jk k

% (1.6)

( ) ( )1 1

( )1

( ) ( )1 1

M Nk kk kl S d S

j k j kn j jS

n k p qk k kk kc S f S

j M k j N kj jk k

%% (1.7)

when kS is replaced by

kS in )( kn S , )(~kn S and )(

~~kn S , they will be replaced by )( kn S , )(~

kn S and

)(~~

kn S respectively. n denotes ……………n integrals.

III. RESULTS USED IN THE PROOF OF THE SEQUEL

Mellin transform of n variables under similar suitable conditions as due to Reed [7] for two variables, we restate parseval

theorem for n variables identical to the one by Fox[3] for one variable. If

0

1[ ( )] ( )] ( ) ( )

1

SnkM f x F S n g x x dS

n n n k kk

(1.8)

then 1[ ( )] ( )M F S g x

n n

( ) ( )(2 ) 1

i Snn kF S x dSn k kni ki

(1.9)

Also, if [ ( )] ( )]M h u H Sn n

and [ ( )] ( ) ,1

SnkM f x u F S x

n n n kk

where, ( ) ( ),M f u F Sn n

Then,


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0

( ) ( ) ( )1

( ) (1 ) ( )(2 ) 1

nn h x u f u du

n n n kk

i Snn kH S F S x dSn n k kni ki

(1.10)

Extending Erdelyi’s [1] fractional integral operators of one variable in to corresponding operators in n variables, we have

0

'( ), ( ) : ( ) : ( )

1' ' ' ' '1[ ] [ ] ( ) ( )

( )1

xk

J S w xn n n n

kS S S Snk k k k k k k kx x v v dv w v

k k k k k nk k

(1.11)

'( ), ( ) : ( ) : ( )

1' ' ' ' ' 1[ ] [ ] ( ) ( )

( )1 xk

R S w xn n n n

kS S S S Snk k k k k k k kx v x v dv w v

k k k k k nk k

(1.12)

where ( ), ( ) ( , ), ( , ),........................( , ).1 1 2 2n n n n

Equations (3.1.11) and (3.1.12) will reduce to Kober [ 5] operators for ( )nx x , ( ) 1nS S , )( n = and )( n = . In

the contracted form, we write

( ), ( 1) : (1) : ( ) ( )k k kJ c l l x J xi j j j n j n

(1.13)

* *( ) ( ), ( 1) : (1) ( ) ( )k k kJ f d d x J xj j N j N j N n j n

k k k

(1.14)

( ), ( ) : (1) : ( ) ( ) ,k k kR f d d x R xj j j j n j n

(1.15)


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(1.16)

using (3.1.1), we have

( );

,0;( ), ( )(( , ,........., ))( ) 1 2 , 1,...........,0;( );(

)(( ))

1,........., , 1,...............

[ ] ( )

am p

m M N n n n n nn n c c c C CM G x M M p Mm p p M q N n n n n nn n n nn n n nd d d d

N N q Nn n n n

S Skk k

(1.17)

IV. RESULTS TO BE PROVED

In this section we shall establish that the formal solution of the simultaneous quadruple integral equations

' '

0

' '

( );

,0;( ), ( )(( ,........, ))( )

1 , 1 ,........,,0;( ), ( )

(( ))1,......., , 1 ,.........,

' ( ) ( )' 111

k k

k k

am p

m M N n n n nn n c c c cn G x uM M p Mm p p N q N n n

n n n nn n n nn mn nd d d d

N N q Nn n n n

n na f u du

k n khk kh

( ), 0 ( ) , ' 1, 2,....., ;'

x x a k nn nk

(1.18)

0

( );

,0;( ), ( )(( ,........., , ))( ) 1 1 ,...........,0;( ), ( )

(( ))1,........., , 1 ,...............

' ( ) ( )' '211

n

am p

m M N n n n nn n l l c cn G x u M M p Mm p p M q N n n n n nn n n nn n n nd d f f

N N q Nn n n n

n nb f u du

h n khk kkh

( ), ( ) , ' 1, 2,....., ;x a x b k nn n

(1.19)

* *( ), ( ) : (1) : ( ) ( )k k kR c l l x R xj j M j M j M n j n

k k k


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0

( );

,0;( ), ( )(( ,........., , ))( ) 1 1 ,.........,,0;( ), ( )

(( ))1,........., , 1 ,..........,

' ( ) ( ) ( ),' '311

n

am p

m M N n n n nn n l l c cn G x u M M p Mm p p M q N n n n n nn n n nn n n nd d f f

N N q Nn n n n

n nb f u du x

h n k nhk kkh

( ) , ' 1, 2,....., ;b x c k nn

(1.20)

and

' '

0

' '

( );

,0;( ), ( )( ) (( ,......., , )),0;( ), ( ) 1 1 ,........,

(( ))1,........, , 1 ,.........,

( ) ( )' 411

k k

k k

am p

m M N n nn n n nn G x u l l l lm p p M q N n n M M p Mn n n n n n n nn n n nf f f f

N N q Nn n n n

n nc f u du

h n khk kkh

( ), ( ), ' 1, 2,....., ;'

x c x k nn n

(1.21)

is

0

'

(1 ), (1 );1,

,0;( ), ( )(1 )( 1 );( ) ( ) 1 ,,0;( ), ( )

( 1 ), (1 )1 ,

' ( ) ( ) , 1,2,....., ;'

11

n a n am p m

p p q n nn n c lf x n G x u M p Mh n m p p M q N n n n n nn n n nn nf d

N q Nn n n

n nt k u du h n

n khk kk

(1.22)

where, '

'hkt is the element of the matrix

' 1

'[b ]hk

and


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' * * *......... ....... ( ) ,0 ( ) ,' 1 2 1 2 1 '

1

( ) ( ) , ( ) ,2 '

( ) , ( ) ,3 '

' * * *......... ....... ( ) , ( )' 1 2 1 2 4 '

1

nd J J J J J J x x a

q M k n nhkh k k

K u x a x bn k n n

x b x ck n n

nl R R R R R R x c x

p N k n nhkh k k

(1.23)

We shall assume that ( )nG x of (1.18), (1.19), (1.20) and (1.21) satisfies all the conditions given earlier in section (1.1).

V. PROOF

To obtain the solution of (1.18) to (1.21), we observe that1 '( )k nx ,

2 '( )k nx , 3 '( )k nx ,

4 '( )k nx are given and we

have to determine ( )n nf x . Using (1.10), 1.17), (1.18), (1.19) (1.20) and (1.21), we respectively obtain

1 '( ) ( ) (1 )'

(2 ) 1 1

( ), 0 ( ) , ' 1,2,3,....... .1 '

i nSnkS S x dS a F S

kk n k k k h nn hki k hi

x x a k nk n n

(1.24)

1 '( ) ( ) (1 )'

(2 ) 1 1

( ), ( ) , ' 1,2,3,....... .2 '

i nSnkS S x dS b F S


x a x b k nk n n

%% (1.25)

1 '( ) ( ) (1 )'

(2 ) 1 1

( ) , ( ) , ' 1,2,3,....... .3 '

i nSnkS S x dS b F S


x b x c k nk n n

%% (1.26)

1 '( ) ( ) (1 )'

(2 ) 1 1

( ), ( ), ' 1,2,3,....... .4 '

i nSnkS S x dS c F S


x c x k nk n n

% (1.27)


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To reduce the equations (1.24), (1.25), (1.26) and (1.27) into three others with a common kernel, we shall transform

( ) ( )1 1

int1 1

( ) ( )1 1

N Nk kk kd S f S

j k j kn nj jo

p pk kk kk kc S l S

j N k j M kj jk k

to make the first transformation in (1.24) replace 'x s by 'v s and then multiply both sides by

' 11

.[ ] ( ) ,1

0

kk kc l kx M M lk k k Mnkx v v dv

k k k kk

evaluate the inner integral with the help of Erdelyi [ 2 ] to obtain

1'( ) ( ) ( )

1 1[ ] . ( ) ' (1 )

'(2 ) 1 1'( ) ( )

1 1

'1

[ ][ ]

'( )10

M Nk kkk k kc S d S l Si j k J k M k nSnn j j k kS x dS a F S

kk k hk h nn p qi k hk kk kki c S d Sj M k j N k

j jk k

kkckM x ck kx Mn

k kx vk kkk kc lk

M Mk k

' 1 1

[ ] ( ) ( ) ,1 '

k k kl lM M

k kv dv J xk k M k n

k

similarly applying the operator jJ successively for 1,.............,1kj M ; we have

( ) ( )1 1

int ,1 1

( ) ( )1 1

M Mk kk kc S l S

j k j kn nj jo

q qk kk kk kd S f S

j N k j N kj jk k


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( ) ( )1 1 '[ ] ( ) . (1 )

'(2 ) 1 1'( ) ( )

1 1

' . ...... ( ) , ' 1,2,.......; , 0 (' 1 2 1 '

1

M Nk kk kl S d S Sj k j ki nn kn j j

S x dS a F Skk k k hk h nn p qi k hk ki k kkc S d S

j M k j N kj jk k

nd J J J x k n

hk M k nkh

)x an

(1.28)

where the matrix ['

'hkd ]= ['

'hkb ] ['

'hka ]-1

.

also applying *

jJ operator successively for ,. .,1 ;kj q ; we obtain

'1 2

( ) ( )1 1 '[ ] ( ) (1 )

'(2 ) 1 1

( ) ( )1 1

' * * *....... ............ ( )' 1 2 1 '

1

k

k

s

M

M Nk kk kl S d S

j k j ki nnn j jS x dS a F S

kk k hk h nn p qi k hk ki k kc S f Sj M k j N k

j jk k

nd J J J J J J x

hk q k nkh

, 0 ( ) , ( ' 1,2,......, ),nx a k n

(1.29)

Now applying the operators jR and *

jR to (4.1.27) for ,. .,1 ;kj N and for 1, . .,1 ;k kj p p respectively , we have

'1 2

( ) ( )1 1 '[ ] .( ) (1 )

'(2 ) 1 1

( ) ( )1 1

' * * *....... ............ ( )' 1 2 3 '

1kN

M Nk kk kl S d S

j k j ki nSnn j j kS x dS a F Skk k hk h nn p qi k hk ki k kc S f S

j M k j N kj jk k

nl R R R R R R xhk p k n

kh

, ( ) , ( ' 1,2,......., )nx c k n

(1.30)

Where 'hkl are the element of the matrix ' '

1' '

hk hkb c

, on setting


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' * * *......... ....... ( ) ,0 ( ) ,' 1 2 1 2 1 '

1

( ) ( ) , ( ) ,2 '

( ) , ( ) ,3 '

' * * *......... ....... ( ) , ( )' 1 2 1 2 4 '

1

nd J J J J J J x x a

q M k n nhkh k k

K x x a x bn k n n

x b x ck n n

nl R R R R R R x c x

p N k n nhkh k k

The equations (1.25), (1.26), (1.29) and (1.30) transformed into one with a common kernel can be written as

n

( ) ( )1 1

[ ] ( )(2 ) 1

( ) ( )1 1

'. (1 ) K x'

1n

M Nk kk kl S d S

j k j ki Snn j j kS x dSkk kn p qi k k ki k kc S f S

j M k j N kj jk k

na F Shk h

h

(1.31)

on treating the kernel of (1.31) as an unsymnetric Fourier kernel and following a procedure similar to the one adopted by Fox[3] for

one variable,(1.31) becomes

(1 )1

( )'

(2 )' 1 ( )1

(1 ) ( 1 )1 1

( ) (1 )1

( 1 ) (1 )1 1

pn a S

j m kkin n jf x th n hk n mik i n a S

j kkj

p qk kk kc S f S

j M k j N k Sn j jk k kx dS K Sk k nM N

k k kk kl S d Sj k j k

j j

(1.32)

where 1,2,3,.......,h n .

'hkt are the element of the matrix '

1'

hkb

and ( ) ( )n nM K x K S . Equation (1.32) is the formal solution of (1.24), (1.25),

(1.26) and (1.27). Applying (1.10) to (1.32), we get


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0

1

,0;( ), ( )( )

' ,0;( ), ( )1

(1 ), (1 );1,

(1 )( 1 );( ) ( ) {( )}1 ,

( 1 ), (1 )1 ,

n

n kk

n p p qn nf x t n G

n hk m p p M q Nk n n n n

n a n am p m

n nc lx u K u duM p Mn n n n nn nf d

N q Nn n n

(1.33)

written out in full form (1.33) becomes

,0;( ), ( )( ) ( )

' ,0;( ), ( )' 01

' * * *....... ......... ( ) ( )' 1 2 1 '1 11

,0;( ), ( )( ) (

,0;( ), ( ) 2 '

an p p qn nf x t n G x u

h n hk m p p M q N n nn n n nk

n nd J J J J J x du

q M n khk k kh k k

b p p qn nn G x u x

m p M p N q n n kn n n na

2

) ( )1

,0;( ), ( )( ) ( ) ( )

,0;( ), ( ) 3 '1

,0;( ), ( )( )

,0;( ), ( )

' * * *....... ......... ( )' 1 2 1 '4 11

ndu

n kk

c p p q nn nn G x u x du

m p M p N q n n k n kkn n n nb

p p qn nn G x u

m p M p N q n nn n n nc

n nl R R R R R R x

p N nhk k kh k k

( ) , 1,2,3,......,du h nk

VI. PARTICULAR CASE

Let c in equations (1.18), (1.19),(1.20) and (1.21), then they are reduced to the triple integral equations considered by Narain

et. al.[6] and the above solution agrees with that solution.

REFERENCES

[1] Erdelyi, A , On some functional transform , Re Semin Mat . Univ, Torino , 10 , 217-34 (1950-51) .

[2] Erdelyi, A . ,Tables of Integral Transform, Vol. II, Mc Graw Hill Book co , Inc, New York , 185 (1954) .

[3] Fox , C ., A formal solution of certain dual integral equations , Trans Amer. Math. Soc ; 119 , 389 -95 (1965) .

[4] Khadia , S. S. and Goyal , A. N. , On generalized function of n-variables , Anusandhan Patrika , 13 , 119-201 (1970) .


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[5] Kober , H. , On fractional Integrals and derivatives , Q. J . Math (Oxford Ser) , 11 , 193-211 (1940) .

[6] Narain Kuldeep ; Singh , V.B. and Lal , M , , On a class of simultaneous triple integral equations involving ( )nG x , Journal of

Jiwaji University , 10 , 56 -65 (1986) .

[7] Reed , I. S. , The Mellin type of double integrals , Duke Math . J , 11 , 565 (1944) .

simultaneous quadruple integral equations involving g(xn)

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