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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.
G( xn ) 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.
G( xn ) . Rewriting G( xn ) in a slightly
different form as
(a );(b ) pm q m, o;( M n ), ( N n ) Cn Cn Cn )) ( x )((C nC n G 1, 2,......., M , 1 M ,......., p M p m, q;( pn M n ), (qn N n ) n n n n n n n n n ((d d d d )) 1,........, N , 1 N ,........., q N n n n n n 1 S '( Skk ) n ( Sk ) x k (dSk ) k n (2 i ) ( Ln ) k 1
n
where, a repeated suffix represents sum from 1 to
n , i.e.
k 1
S k S kk .
n (a j Skk ) j 1 '( S ) kk p q (1 a j m ) (b j Skk ) j 1 j 1
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(1.2)
(1.1)
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M N k k k k (C j Sk ) (d j Sk ) n j 1 j 1 (S ) n k q k 1 pk k k k (C j M Sk ) (d j N Sk ) j 1 j 1 k k
(1.3)
and
n (dSk ) dS1.dS2 ........dSn k 1 Also
p2 ,……., pn
(bq ) ;
(1.4)
represents the sequence
b1 , b2 ,……, bq
q : q1 , q2 ,…….., qn ; m : M 1 , M 2 ,
p 0, q 0;(qk ) 1; 0 (M k ) ( pk );
;
( Ln )
…….,
are
n
suitable contours and the positive integers
M n ; N1 , N 2 ,…………, N n satisfy
p ( pk ) q q(k
p : p1 ,
the following inequalities
1 , 2n, . The . . .values . . . ,( xk. ) 0 ,
)k ;
(k 1,2,...., n) are excluded. An empty product is interpreted as unity. The contour necessary
to
ensure
that
the
poles
of
Lk
is in the
Sk
– plane and runs from i to i with loops, if
(d kj S k ); j 1,2,........N k
lie
to
the
right
and
the
poles
of
(C kj S k ); j 1,2,........M k and (a j Skk ); j 1, 2,........m lie to the left of the contour Lk where k 1,2,......., n ,hereafter. The function
G( xn )
is an analytic function of (xn) under the following set of conditions :
1 1 1 1 arg X k (m M k N k q qk p pk ) , 2(m M k Nk ) q qk p pk . 2 2 2 2
II. NOTATIONS AND KNOWN RESULTS
[S
kk
]
m II (a j Skk ) j 1
(1.5)
p II (1 a j m Skk ) j 1
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M N k k k k (l j Sk ) ( f j Sk ) n j 1 j 1 % ( S ) n k q k 1 pk k k k (l j M Sk ) ( f j N Sk ) j 1 j 1 k k
(1.6)
M N k k k k (l j Sk ) (d j Sk ) n j 1 j 1 % (S ) % n k q k 1 pk k k k (c j M S k ) ( f j N S k ) j 1 j 1 k k
(1.7)
when
~
~n (S k )
Sk
is replaced by
respectively. n
Sk
denotes
~ ~
in n ( S k ) , ~n ( S k ) and n ( S k ) , they will be replaced by n ( S k ) , ~n ( S k ) and ……………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 n S 1 M [ f ( x )] F (S )] n g ( x ) x k (dS ) n n n k 0 k k 1
then
(1.8)
M 1[ F ( S )] g ( x ) n n
Also, if
i n S F ( S ) x k (dS ) n k k (2 i )n i k 1 n
n S M [h(u )] H (S )] and M [ f ( x u )] F ( S ) x k n n n n n k k 1
(1.9)
,
where, M f (u ) F ( S ),
n
n
Then,
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n n h( x u ) f (u ) (du ) n n n k 0 k 1 n n i S H ( S ) F (1 S ) x k (dS ) n n k k (2 i)n i k 1
(1.10)
Extending Erdelyi’s [1] fractional integral operators of one variable in to corresponding operators in n variables, we have
J ( ),( ) : ( S ' ) : w( x ) n n n n
1 ' S ' ' 1 xk S ' S' k n Sk x k v k [x ] k k k k [v ] k (dv ) w(v ) k k k k n 0 k k 1 ( k )
(1.11)
R ( ), ( ) : ( S ' ) : w( x ) n n n n
1 ' S' S' k S ' S ' 1 n Sk k k k (dv ) w(v ) [ x ] k v k x k [v ] k k k k k n xk k k 1 ( k )
(1.12)
where ( ), ( ) ( , ), ( , ),........................( , ).
n
n
1 1
2 2
n n
Equations (3.1.11) and (3.1.12) will reduce to Kober [ 5] operators for
( xn ) x , (Sn ) S 1 , ( n ) =
the contracted form, we write
J (c k l k ), (l k 1) : (1) : ( x ) J ( x ) i j j j n j n
(1.13)
J * ( f k ) (d k ), (d k 1) : (1) ( x ) J * ( x ) j j N jN jN n j n k k k
R ( f k d k ), (d k ) : (1) : ( x ) R ( x ) , j j j j n j n www.scirj.org © 2013, Scientific Research Journal
(1.14)
(1.15)
and
( n ) = . In
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R* (c k lk ), (l k ) : (1) : ( x ) R* ( x ) j j M jM jM n j n k k k
(1.16)
using (3.1.1), we have
(a ); m p m,0;( M ),( N ) n , c n ,........., c n C n n n n (( c C )) M G (x ) 1 2 M , M 1,........... p M m p,0;( p M );(q N n n n n n n n n n) n n n n ((d d d d )) 1,........., N , N 1,............... q N n n n n [ S ] ( S ) kk k
(1.17)
IV. RESULTS TO BE PROVED In this section we shall establish that the formal solution of the simultaneous quadruple integral equations
(a ); m p m,0;( M ),( N ) n' n' n n n cn )) n G ( x u ) ((c k ,........, c k c 1 M , 1 M ,........, p M m p,0;( p N ),(q N ) n n 0 n n n n n n n n (1.18) n m n n d k' k' (( d d d )) 1,......., N , 1 N ,........., q N n n n n n n f (u ) (du ) ( x ), 0 ( x ) a, k ' 1, 2,....., n ; a' ' k n k n 1k ' n k 1 h 1 hk
(a ); m p m,0;( M ),( N ) n ,........., l n , c n n n n (( l c )) n G (x u ) 1 M n 1 M ,........... p M m p,0;( p M ),(q N ) n n 0 n n n n n n n n n n n ((d d f f )) 1,........., N , 1 N ,............... q N n n n n n ' n ( x ), a ( x ) b, k ' 1, 2,....., n ; b f (u ) (du ) ' h n k n hk 2k ' n k 1 h 1
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(1.19)
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(a ); m p m,0;( M ),( N ) n ,........., l n , c n n n n (( l c )) n G (x u ) M n 1 M ,........., p M m p,0;( p M ),(q N ) n n 1 0 n n n n n n n (1.20) n n n n ((d d f f )) 1,........., N , 1 N ,.........., q N n n n n n n ' ( x ), b ( x ) c, k ' 1, 2,....., n; b f (u ) (du ) ' h n k n hk 3k ' n k 1 h 1
and
(a ); m p m,0;( M ),( N ) n' n' n n n n k k n G (x u ) m p,0;( p M ),(q N ) n n ((l1 ,......., lM , l1 M ,........,l p M )) 0 n n n n n n n n n' n' n (( f k f k f fn )) 1,........, N , 1 N ,........., q N n n n n n n ( x ), c ( x ), k ' 1, 2,....., n; c f (u ) (du ) ' h n k n 4k ' n k 1 h 1 hk
(1.21)
is
(1 n a ),(1 n a ); m 1, p m p,0;( p ),(q ) n n ); n n (1 c )( 1 l f (x ) n G (x u ) 1 M , p M h n m p,0;( p M ),(q N ) n n 0 n n n n n n n (1 f n ),(1 d n ) 1 N , q N n n n n n ' t k (u ) (du ) , h 1, 2,....., n ; ' n k k 1 k ' 1 hk
where,
t'
hk '
is the element of the matrix
[b' ' ]1 and hk
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(1.22)
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n d ' J * J *.........J * J J .......J ( x ) , 0 ( x ) a, q 1 2 M 1k ' n n h 1 hk ' 1 2 k k K (u ) ( x ) , a ( x ) b, n 2k ' n n ( x ) , b ( x ) c, 3k ' n n n ' R* R*.........R* R R .......R l ( x ) , c (x ) ' 1 2 p 1 2 N 4k ' n n hk h 1 k k
We shall assume that
G( xn )
(1.23)
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 that 1k ' ( xn ) , have to determine
2k ' ( xn ) , 3k ' ( xn ) , 4k ' ( xn )
f n ( xn ) . Using (1.10), 1.17), (1.18), (1.19) (1.20) and (1.21), we respectively obtain
i n S n S ( S ) x k (dS ) a' F (1 S ) k k k ' h n (2 i )n i kk n k 1 h 1 hk ( x ), 0 ( x ) a , k ' 1,2,3,.......n. 1k ' n n 1
i n S n % ( S ) x k (dS ) b' F (1 S ) S % k k k ' h n (2 i )n i kk n k 1 h 1 hk ( x ), a ( x ) b , k ' 1, 2,3,.......n. 2k ' n n
(1.24)
1
(1.25)
i n S n ' k % % S ( S ) x ( dS ) k b ' Fh (1 Sn ) kk n k k n (2 i) i k 1 h 1 hk (x ) , b (x ) c, k ' 1,2,3,.......n. 3k ' n n
(1.26)
i n S n S % ( S ) x k (dS ) c' F (1 S ) k k k ' h n (2 i)n i kk n k 1 h 1 hk ( x ), c ( x ), k ' 1,2,3,.......n. 4k ' n n
(1.27)
1
1
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are given and we
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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
M M k k k k (c j S k ) (l j Sk ) n j 1 n j 1 int o , q q k 1 k k 1 k k k ( d j N Sk ) ( f j N Sk ) j 1 j 1 k k N N k k k k ( d j Sk ) ( f j Sk ) n j 1 n j 1 int o p k 1 pk k 1 k k k (c j N S k ) (l j M Sk ) k k j 1 j 1 to make the first transformation in (1.24) replace
x ' s by v ' s
and then multiply both sides by
c kk ' l k 1 k M M xk l 1 k k M n x v .[v ] k (dv ) , k k k k k 1 0
evaluate the inner integral with the help of Erdelyi [ 2 ] to obtain
N M 1 k (c kk ' S ) k (d k S )(l k S ) i j k J k M k S n n n j 1 k j 1 [S ] .x k (dS ) a ' F (1 S ) kk k hk ' h n n p q (2 i) h 1 k 1 k k k kk ' i S ) (d S ) (c k jN k j 1 j M k j 1 k 1 c kk ' kk ' l k 1 l k 1 M x c k k M M M n [ xk ] k [v ] k (dv ) J [x v ] k (x ) , k k k k M 1k ' n k 1 (c kk ' l k ) k 0 M M k k
similarly applying the operator J j successively for j M k 1,.............,1 ; we have
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M N k k k k (l j Sk ) (d j Sk ) S n i n k n j 1 j 1 [ S ] x ( dS ) . a' F (1 S ) kk p k k hk ' h n n q (2 i) i h 1 k 1 k k k kk ' (c j M S k ) ( d j N S k ) j 1 j 1 k k n dhk' ' J1.J 2 ......J M 1k ' ( xn ) , k ' 1,2,.......; n, 0 ( xn ) a k h 1
'
'
(1.28)
'
where the matrix [ d hk ' ]= [ bhk ' ] [ ahk ' ]-1. also applying
J *j operator successively for j qk ,..,1 ; ; we obtain
M N k k k k S ) ( l S ) ( d j k j k n i n n j 1 j 1 [S ] x sk (dS ) a' F (1 S ) kk k hk ' h n q (2 i)n i h 1 k 1 pk k k k (1.29) ( c S ) ( f S ) k jN k j 1 j M k j 1 k n ' J * J * .......J * J J ............J ( x ) , 0 ( x ) a, (k ' 1,2,......, n), dhk n Mk' ' 1 2 q 1 2 1k ' n k h 1
Now applying the operators
R j and R*j to (4.1.27) for j Nk ,..,1 ; and for j pk , pk 1..,1 ; respectively , we have
M N k k k k (l j Sk ) (d j Sk ) n i S n n j 1 j 1 [S ] x k .(dS ) a' F (1 S ) kk k hk ' h n n p q (2 i) i h 1 k 1 k k k k (c j M S k ) ( f j N S k ) j 1 k k j 1
n ' * * * l R R .......R hk ' 1 2 p k h 1
Where
lhk '
, ( x ) c, (k ' 1,2,......., n)
R R ............R ' ( x ) Nk 1 2 3k ' n
' ' are the element of the matrix bhk ' chk '
n
1
, on setting
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(1.30)
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n d ' J * J *.........J * J J .......J ( x ) , 0 ( x ) a, q 1 2 M 1k ' n n h 1 hk ' 1 2 k k K (x ) ( x ) , a ( x ) b, n 2k ' n n ( x ) , b ( x ) c, 3k ' n n n ' R* R*.........R* R R .......R l ( x ) , c (x ) ' 1 2 p 1 2 N 4k ' n n hk h 1 k k
The equations (1.25), (1.26), (1.29) and (1.30) transformed into one with a common kernel can be written as
M N k k k k (l j Sk ) (d j Sk ) i S n n j 1 j 1 [S ] x k (dS ) kk k q (2 i)n i k 1 pk k k k (c j M S k ) ( f j N S k ) j 1 j 1 k k n . a' F (1 S n ) K x n hk ' h h 1
(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
p (1 n a j m Skk ) n n i j 1 f (x ) t h n hk ' n m (2 i ) k ' 1 i (n a j Skk ) j 1 q p k k k (1 c k S ) (1 f S ) jM k jN k S n j 1 j 1 k k x k (dS ) K (1 S ) k k n M N k 1 k k k k (l j 1 Sk ) (1 d j Sk ) j 1 j 1 where
thk '
(1.32)
h 1,2,3,......., n .
are the element of the matrix
bhk' '
1
and M
K ( xn ) K (Sn ) . 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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n p,0;( p ),(q ) n n f ( x ) t n G n hk ' 0 m p,0;( p M ),(q N ) k 1 n n n n (1 n a ),(1 n a ); m 1, p m n n n (1 c )( 1 l ); (x u ) K (un ){(duk )} 1 M , p M n n k 1 n n n n n (1 f ),(1 d ) 1 N , q N n n n
(1.33)
written out in full form (1.33) becomes
a p, 0;( p ), (q ) n n n ( x u ) f (x ) t n G h n hk ' m p , 0;( p M ), ( q N ) n n 0 n n n n k' 1 n n ' * * * ( x ) ( du ) d J J .......J J .........J ' 1 2 q 1 M 1k ' n k k 1 h 1 hk k k b p, 0;( p ), (q ) n n n ( x u ) ( x ) (du ) n G m p, 0;( M p ), ( N q ) n n 2k ' n k k 1 n n n n a c p, 0;( p ), (q ) n n n ( x u ) ( x ) (du ) n G m p, 0;( M p ), ( N q ) n n 3k ' n k k 1 n n n n b p, 0;( p ), (q ) n n ( x u ) n G m p, 0;( M p ), ( N q ) n n n n n n c
n ' * * * l R R .......R ' 1 2 p h 1 hk k
R1 R2 .........RN k
, h 1, 2,3,......, n
n ' ( xn ) (duk ) 4k k 1
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) . www.scirj.org © 2013, Scientific Research Journal
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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 Jiwaji University , 10 , 56 -65 (1986) . [7] Reed , I. S. , The Mellin type of double integrals , Duke Math . J , 11 , 565 (1944) .
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G( xn ) , Journal of