Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013

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Simultaneous Triple Series Equations Associated With Laguerre Polynomials With Matrix Argument Kuldeep Narain School of Quantitative Sciences, UUM College of Arts and Sciences, University Utara Malaysia, Sintok – 06010, Malaysia E-mail: kuldeep@uum.edu.my 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 this paper solution of simultaneous triple series equations associated with Laguerre polynomials with matrix argument has been obtained , which arises in the Crack problems of Fracture Machanics. Keywords: Integral equations, Series equations , Laguerre polynomials, Matrix argument.

1. Introduction In the present paper, we have considered the following simultaneous triple series equations of the form ¥

s

n=0

j =1

¥

s

n=0

j =1

S Sa

ij

S S

bij cnj .

Lni (a + b ¥

s

n=0

j =1

S S

for

G m (a + b + ni)

cnj

Lni (a : x) = fi (x), 0 £ x £ D,

m +1 )G m (a + b + ni) 2 II m (a + b + ni)

G m (a + ni +

m +1 , y ) = gi ( x), D £´£ E , 2

(1.2)

cnj G m (a + b - m + 1) = Lni (a , x) = hi ( x), E £´£ ¥ ,

a +b >

2

m +1 - 1,0 £ b £1 2

where,

Ln (a , x) =

(1.1)

II m (a + n) m +1 1F1 (-n, a + ), x) P m (a ) 2 88

(1.3)

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013

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is Laguerre polynomial of matrix argument , for

R (a ) > - 1,

R (n + a ) > 1, and II m (a) = G m (a + m

G m (a) = p m ( m-1)/4 Õ (a i =1

m +1 ) 2

i -1 ) 2

n =0,1,2,……………………., J = 1,2,3,……………s , f(x), g(x) h(x) are known functions of non – singular matrix x of order m ; aij , bij and cij are known constant .By using multiplying factor technique [ Srivastava], the unknown function C nj is determined . 2 . Some Useful Results (i) The following integrals are required from Erdelyi et al with matrix argument

ò 0y

y

a

y-x

B -(m +1)/2

Ln (a , x)dx

m +1 ) a +b 2 = y Ln (a + b ; y ) m +1 G m (a + b + n + ) 2 G m ( b ) G m (a + n +

for

a > - 1, b >

¥

ò x- y y

= Gm (

for

-b

m +1 -1 2

(2.1)

and

etr (- x) Ln (a , x)dx

m +1 m +1 - b ) etr (- y) Ln (a + b ; y) 2 2

b<

m +1 m +1 ,a + b > -1 . 2 2

(ii) The orthogonality relation for Laguerre polynomial with matrix argument

89

(2.2)

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013

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a

ò x >0 x etr (- x) Lp (a ; x) Lq (a ; x)

m +1 ) 2 d pq m +1 ) G m (p+ 2

G m (a + b +

=

for a > - 1 and

d pq

(2.3)

being the kronecker delta .

(iii) The differential formula with matrix argument due to Erdelyi et al a

Dx [ x Ln (a ; x)] = x

a-

m +1 2

Ln (a -

m +1 ; x) 2

(2.4)

3. Solution

Multiply eq. (1.1) by

y-x

b - ( m+1) 2

and eq. (1.2) by

x- y

-b

etr (- x) and then integrating w.r.t. x

m +1 ) G m (a + b + ni) 2 Lni (a + b ; y ) = II m (a + b + ni )

G m (a + ni +

s

åå

a

(0, y) and ( y, ¥) respectively, on using the result (2.1) and (2.2), we obtain

over the range

¥

x

aij cnj

n = 0 j =1

-a - b

y a b - ( m +1)/2 ò 0y x y - x fi ( x)dx , Gm (b )

b>

for

(3.1)

m +1 , a > -1, 0 < y < D 2

and

¥

s

S S bij cnj

n = 0 j =1

=

m +1 ) G m (a + b + ni) m +1 2 Lni (a + b ; y) II m (a + b + ni) 2

G m (a + ni +

etr ( y ) -b ò ¥y etr (- x) x - y gi ( x)dx , m +1 Gm ( 2 - b )

for b <

(3.2)

m +1 m +1 , a +b > - 1, D < y < ¥ . 2 2

90

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013

If we new multiply eq.(3.1) by by y

¥

s

S S bij cnj

for

a

, differentiating w.r.t. Y and using the formula (2.4), we find

m +1 m +1 ( ) -a - b ) G m (a + b + ni) s 2 y + 1 m 2 . Lni (a + b . ; y ) = å eij 2 Gm (b ) II m (a + b + ni) j =1

G m (a + ni +

n = 0 j =1

Dy ò 0y x

a +b

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y-x

b -(m +1)/2,

0 < y < D, b >

fi ( x)dx ,

(3.3)

m +1 - 1, a > -1, and eij are the element of the matrix [bij ][aij ]-1 and 2

i = 1, 2,.....s. The left- hand side of eq.. (3.2), (3.3) and (1.2) are identical and hence on using the orthogonality relation (2.3), we find the solution of eq. (1.1), (1.2) and (1.3) for

a +b >

m +1 - 1, 0 < b < 1 2

m +1 ) G m (a + b + ni ) 2 Cnj = S dij Bni (a + b ; D) (3.4) j =1 m +1 2 )[G m (a + b + ni)] G m (a + ni + 2 G m (ni +

s

Where ,

n = 0,1, 2.......; j = 1, 2,3,.......,s and d ij are the element of the matrix [bij ]-1 and s

Bni (a , b ; D) = å eij j =1

D

1 etr (- y )Lni (a + b - m2+1 ; y ) . ò Gm (b ) 0

m +1 , y )Gi ( y )dy 2 ¥ m +1 1 a + b - ( m +1)/2 + ò y Ln (a + b ; y ) H ( y )dy m +1 E 2 Gm ( - b) 2 E

Fi ( y )dy + ò etr (- y ) y

a + b - (m +1)/2

D

y

Fi ( y ) = Dy ò x

a

y-x

-b

etr (- x)hi ( x)dx

0

b - ( m +1)/2

Ln (a + b -

fi ( x)dx

(3.6)

Gi ( y) = gi ( y) ¥

H i ( y) = ò x - y y

(3.5)

91

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References T.W. Anderson ; (1958) , An Introduction to Multivariate statistical Analysis, John Wiley and Sons, New York . A. Erdelyi , et al ; (1953) , Higher Transcendental functions, Vol, II, Mc Graw Hill Book co, Inc; New York. A. Erdelyi , et al ; (1954) ,Tables of Integral Transforms , Vol, II, Mc Graw Hill Book co, Inc; New York . A.M. Mathoi and R.K. Saxena; (1978) , The H-function with Applications in Statistics and other Disciplines, Wiley Eastern Limited, New Delhi , India, 96-132 . H.M. Srivastava,; (1969) , Notices Am. Math. Soc. 16 , 568, (See also p. 517) . H. M. Srivastava ; (1969) , Pacific J. Math , 30 ( 1969) , 525 -27 . H.M. Srivastava ; (1970) , J. Math . Anal . Appl. 31, 587-94 (see also p.587) .

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