Simultaneous triple series equations

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  • 1. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org 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)
  • 2. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org 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 y ò0 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)
  • 3. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org 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 y ò0 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 ¥ etr (- x) x - y gi ( x)dx , y m +1 Gm ( 2 - b ) for b < (3.2) m +1 m +1 , a +b > - 1, D < y < ¥ . 2 2 90
  • 4. 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 y 2 m +1 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 y Dy ò 0 x a +b www.iiste.org 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 ò etr (- y)Lni (a + b - m2+1 ; y) . 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
  • 5. Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 www.iiste.org 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) . 92
  • 6. This academic article was published by The International Institute for Science, Technology and Education (IISTE). The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe. The aim of the institute is Accelerating Global Knowledge Sharing. More information about the publisher can be found in the IISTE’s homepage: http://www.iiste.org CALL FOR JOURNAL PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world. There’s no deadline for submission. Prospective authors of IISTE journals can find the submission instruction on the following page: http://www.iiste.org/journals/ The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. Printed version of the journals is also available upon request of readers and authors. MORE RESOURCES Book publication information: http://www.iiste.org/book/ Recent conferences: http://www.iiste.org/conference/ IISTE Knowledge Sharing Partners EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open Archives Harvester, Bielefeld Academic Search Engine, Elektronische Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial Library , NewJour, Google Scholar