Simultaneous Triple Series Equations. Associated With Laguerre Polynomials With Matrix Argument

Transcription

1 Sultaneous Trple Seres Equatons Assocated Wth Laguerre Polnoals Wth Matrx Arguent Kuldeep Naran School of Quanttatve Scences, UUM College of Arts and Scences, Unverst Utara Malasa, Sntok 0600, Malasa Abstract E-al: Integral and Seres equatons are ver useful n the theor of elastct, elastostatcs, dffracton theor and acoustcs. Partcularl these equatons are ver uch useful n fndng the soluton of crack probles of fracture echancs. In ths paper soluton of sultaneous trple seres equatons assocated wth Laguerre polnoals wth atrx arguent has been obtaned, whch arses n the Crack probles of Fracture Machancs. Kewords: Integral equatons, Seres equatons, Laguerre polnoals, Matrx arguent.. Introducton In the present paper, we have consdered the followng sultaneous trple seres equatons of the s a c nj n=s 0 j=s ( +b+ n) a n L ( : x) f (x), 0 x D, a = (. ) s b n=s 0 j=s c nj. + ( a+ n+ ) ( a+ b+ n) n) L n + ( a+ b-, ) = g ( x), D E, (.) s n=s 0 j=s c nj + ( a+ b- ) = Ln ( a, x) = h ( x), E, (.3) + a+ b > -,0 b where, II( a n) Ln ( a, x) = + F (- n, a+ + ), x) P ( a) 88

2 s Laguerre polnoal of atrx arguent, R (a ) > -, + R( n+ a) >, and II ( a) = ( a + ) - ( -)/4 ( a) = p ( a - ) = Õ n =0,,,., J =,,3, s, f(x), g(x) h(x) are known functons of non sngular atrx x of order ; a, b and c are known constant.b usng ultplng factor technque [ Srvastava], the unknown functon C nj s deterned.. Soe Useful Results () The followng ntegrals are requred fro Erdel et al wth atrx arguent a x ò0 - B- ( + )/ L ( a, xdx ) n + ( b ) ( a+ n+ ) a+ b = Ln ( a+ b; ) + ( a+ b+ n+ ) (.) + a > -, b > - and ò x- etr (- x) L ( a, xdx ) -b n = ( + - b) etr (- ) Ln ( a+ b - + ; ) (.) + + b <, a+ b > -. () The orthogonalt relaton Laguerre polnoal wth atrx arguent 89

3 a òx> 0 x etr (- x) Lp( a; x) Lq ( a; x) + ( a+ b+ ) = d + (p + ) pq (.3) a >- and d pq beng the kronecker delta. () The dfferental ula wth atrx arguent due to Erdel et al + a a- + Dx[ x L ( ; )] n a x = x Ln ( a- ; x) (.4) 3. Soluton Multpl eq. (.) b ( ) - a b + x - x and eq. (.) b x- etr( -x) and then ntegratng w.r.t. x -b over the range (0, ) and (, ) respectvel, on usng the result (.) and (.), we obtan n= 0 j= + ( a+ n+ ) ( a+ b+ n) a cnj Ln( a+ b; ) n) s åå = -a-b a b- ( + )/ ò0 x -x f( xdx ), (3.) ( b ) + b >, a >-, 0 < < D and + s ( a+ n+ ) ( a+ b+ n) + S S b cnj Ln( a+ b- ; ) n) n= 0 j= etr( ) - = ò etr (-x) x- b g ( xdx ) ( -b) +, (3.) + + b <, a+ b> -, D< <. 90

4 If we new ultpl eq.(3.) b b a+b, dfferentatng w.r.t. Y and usng the ula (.4), we fnd S S n= 0 j= + ( a+ n+ ) ( a+ b+ n) n) s b c nj. L n j= + ( ) - a - b s + ( a+ b- ; ) = å e. ( b) a b- (+ )/, Dò 0 x - x f( xdx ), (3.3) + 0 < < D, b > -, a >-, and e are the eleent of the atrx =,,... s. [ ][ ] b a - and The left- hand sde of eq.. (3.), (3.3) and (.) are dentcal and hence on usng the orthogonalt relaton (.3), we fnd the soluton of eq. (.), (.) and (.3) + a+ b > -,0 < b < + s ( n+ ) ( a+ b+ n) C nj = S d Bn ( a+ b; D) j= + ( a+ n+ )[ ( a+ b+ n)] (3.4) Where, n= 0,,...; j=,,3,...,s and d are the eleent of the atrx [ ] b - and s D + n ( a, b; ) = å (- ) n( a+ b- ; ) j= ( b) ò. 0 B D e etr L E a+ b- (+ )/ + F ( ) d+òetr( - ) Ln ( a+ b-, ) ( ) d D a+ b- ( + )/ + + ò Ln ( a+ b- ; ) H ( ) d + E ( ) -b (3.5) 0 a b- ( + )/ F ( ) = D òx - x f ( xdx ) (3.6) ( ) = g ( ) H ( ) =ò x- etr( - xh ) ( xdx ) - b 9

5 References T.W. Anderson ; (958), An Introducton to Multvarate statstcal Analss, John Wle and Sons, New York. A. Erdel, et al ; (953), Hgher Transcendental functons, Vol, II, Mc raw Hll Book co, Inc; New York. A. Erdel, et al ; (954),Tables of Integral Transs, Vol, II, Mc raw Hll Book co, Inc; New York. A.M. Matho and R.K. Saxena; (978), The H-functon wth Applcatons n Statstcs and other Dscplnes, Wle Eastern Lted, New Delh, Inda, H.M. Srvastava,; (969), Notces A. Math. Soc. 6, 568, (See also p. 57). H. M. Srvastava ; (969), Pacfc J. Math, 30 ( 969), H.M. Srvastava ; (970), J. Math. Anal. Appl. 3, (see also p.587). 9

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