The study of dual integral equations with generalized Legendre functions

Transcription

1 J. Mth. Anl. Appl. 34 (5) The study of dul integrl equtions with generlized Legendre funtions B.M. Singh, J. Rokne,R.S.Dhliwl Deprtment of Mthemtis, The University of Clgry, Clgry, Albert, Cnd TN-N4 Reeived Februry 4 Avilble online 7 Jnury 5 Submitted by H.M. Srivstv Abstrt Closed form solutions for dul integrl equtions involving generlized Legendre funtions s kernels re obtined. Conneted to these dul integrl equtions n ext solution for dul integrl equtions involving sine funtions s kernels is lso obtined. Properties of generlized Legendre funtions nd the inversion theorem for the generlized Mehler Fok trnsforms re used to obtin the solution of dul integrl equtions 4 Elsevier In. All rights reserved.. Introdution Dul integrl equtions involving Legendre funtions of imginry rguments nd trigonometril funtions hve been onsidered by. By mking use of the method of we obtin the solution of more generl type of dul integrl equtions involving generlized Legendre funtions in Setion 3 nd in Setion 4 we obtin solution of the dul integrl equtions involving trigonometril sine funtions. The inversion theorem for the generlized Mehler Fok trnsforms is used to find the solution of dul integrl equtions. The inversion theorem for the generlized Mehler Fok trnsforms hs been given by 3 * Corresponding uthor. E-mil ddress: dhli.r@shw. (R.S. Dhliwl). Deprtment of Computer Siene, The University of Clgry, Clgry, Albert, Cnd TN-N4. -47X/$ see front mtter 4 Elsevier In. All rights reserved. doi:.6/j.jm.4.9.5

2 76 B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) nd 4. The relevnt referenes for dul nd triple integrl equtions re given by 5. The nlysis is forml nd no ttempt hs been mde to justify hnges in the order of integrtions. It is worth mentioning tht the generlized Legendre funtion is very speilized se of the G-funtion nd lso the H -funtion nd tht dul integrl equtions involving more generl G- nd H -funtions hve lredy been solved by mny different tehniques (inluding, for exmple, the tehniques used by the present uthors) in 6. In Ref. 6, pp , the dul integrl equtions (3.3.) nd (3.3.) involving H -funtions hve been solved. By seleting prtiulr vlues for the onstnts in the H -funtions, in the dul integrl equtions (3.3.) nd (3.3.) we n obtin the dul integrl equtions involving generlized Legendre funtions nd then integrting with respet to y we obtin dul integrl equtions whih re different from the dul integrl equtions (9), () nd (8), (9) of this pper. The reson is tht the dul integrl equtions (9), () nd (8), (9) re formed by integrting with respet to the index of the generlized Legender funtion. Finlly we find tht the solution of the dul integrl equtions (9), () nd (8), (9) re not prtiulr se of the solution of the dul integrl equtions (3.3.) nd (3.3.) mentioned in the book of Srivstv et l. 6, pp Integrls involving generlized Legendre funtions nd some useful results In this setion we disuss some integrls involving the generlized Legendre funtion. We n esily find with help of the book 7, p. 33() ( ) sinh() µ µ osh() os(xτ) dτ = osh() osh(x) µ H( x), () µ< nd with the help of the book 8 nd fter some mnipultions we find tht ( ) 3/ + µ µ sinh() µ + i τ ) µ i τ ) sinh(f ) sin(x) osh() dτ = osh(x) osh() µ H(x ), () µ> nd H( ) denotes the Heviside unit funtion. Furthermore, = f, f>, nd P µ (osh()) is the generlized Legendre funtion defined in the book 9, +i τ p. 37.

3 B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) From, the generlized Mehler Fok trnsform is defined by ψ ( osh() ) = nd its inversion formul is F(τ)= fτ sinh(f τ) osh() F(τ)dτ (3) µ + i τ ) µ i τ ) osh() ψ ( osh() ) sinh() d. (4) Equtions () nd () re of the form (3). Mking use of the inversion formul given by Eq. (4) we find from Eqs. () nd () tht os(xτ) τ sin(xτ) τ = sinh(τf ) µ + i τ ) µ i τ ) ( µ) = x x sinh() +µ P µ osh() d +i τ, µ< osh() osh(x) +µ, (5) + µ) sinh() µ P µ osh() d +i τ, µ> osh(x) osh() µ. (6) Mking use of the inversion theorem for Fourier osine trnsforms, we get by using the results () nd () tht osh() = sinhµ () os(τs) ds µ), µ< osh() osh(x) +µ, (7) osh() = sinh() µ + µ) sinh(f τ) µ + i τ ) µ i τ ) sin(τs) ds, µ> osh(s) osh() µ. (8)

4 78 B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) Dul integrl equtions with generlized Legendre funtion kernels We now onsider the pir of equtions A(τ)P µ +i τ τa(τ)sinh(τf ) osh() dτ = f (), <<, µ >, (9) µ + i τ ) µ i τ ) P µ osh() dτ +i τ = f (), < <, µ <, () A(τ) is the unknown funtion to be determined. Multiplying Eq. (9) by + µ ) sinh() µ osh(x) osh() µ, integrting with respet to from to x, then differentiting both sides with respet to x nd mking use of the result (6), we find tht A(τ) os(xτ) dτ = d + µ ) dx x f ()sinh() µ d osh(x) osh() µ = F (x) (sy), x<. () Multiplying Eq. () by () µ ) sinh() +µ osh() osh(x) µ nd integrting both sides with respet to over x to, we find, by using Eq. (5) tht A(τ) os(xτ) dτ = f ()sinh() +µ d ( µ ) osh() osh(x) +µ x = F (x) (sy), <x<. () Mking use of the inversion theorem for Fourier osine trnsforms, we get from Eqs. () nd () tht A(τ) = F (x) os(xτ) dx + F (x) os(xτ) dx. (3) In some ses it is very useful to find the following expression: τa(τ)sinh(τf ) µ + i τ ) µ i τ ) P µ osh() dτ +i τ = G (), <<. (4) Integrting Eq. (3) by prts we find tht

5 A(τ) = τ B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) F () sin(τ) F () sin(τ) τ τ τ F (x) sin(xτ) dx F (x) sin(xτ) dx, <τ<, (5) prime denotes the derivtive with respet tot x. Substituting from Eq. (5) into Eq. (4), we find by mking use of the integrl () tht for µ <, { sinh() µ F () G () = + µ ) osh() osh() µ } F (x) dx osh(x) osh() µ { F () } F (x) dx +, osh() osh() µ osh(x) osh() µ <<. (6) We n esily find tht G () = = + A(τ)P µ +i τ sinh() µ µ ) osh() dτ F (x) dx osh() osh(x) +µ F (x) dx osh() osh(x) +µ, <<, µ <. (7) Now we onsider the following dul integrl equtions: τa(τ)p µ +i τ A(τ) sinh(τf ) osh() dτ = f (), <<, µ >, (8) µ + i τ ) µ i τ ) P µ osh() dτ +i τ = f (), < <, µ <. (9)

6 73 B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) We shll use the sme method of solving the dul integrl equtions (8) nd (9) s for the dul equtions (9) nd (). The finl result is A(τ) = F 3 (x) sin(xτ) dx + F 4 (x) sin(xτ) dx, () F 3 (x) = + µ ) x d F 4 (x) = ( µ ) dx sinh() µ f () d, () osh(x) osh() µ x sinh() +µ f () d. () osh() osh(x) +µ 4. Dul integrl equtions involving sine funtions kernels We shll now onsider the following dul integrl equtions: τ A(τ) sin(xτ) dτ = g(x), <x<, (3) A(τ) µ + i τ ) µ i τ ) oseh(f τ) sin(xτ) dτ = g (x), < x <, (4) <µ<. Differentiting both sides of Eq. (3) by x, we find tht A(τ) os(xτ) dτ = g (x), <x<, (5) prime denotes derivtive with respet to x. Multiplying both sides of Eq. (5) by sinh µ () ( osh() osh(x) +µ), µ) integrting with respet to x nd using Eq. (7) we find tht osh() A(τ) dτ = n (), <<, (6) n () = sinh µ () µ) g (x) dx. (7) osh() osh(x) +µ

7 B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) Multiplying Eq. (4) by µ osh(x) osh() sinh() µ + µ) nd integrting both sides with respet to x, between the limits nd, we find tht osh() A(τ) dτ = n (), < <, (8) n () = sinh() µ + µ) g (x) dx. (9) osh(x) osh() µ Equtions (6) nd (8) re of the form (3). Hene mking use of Eq. (4) we find tht A(τ) = τf sinh(f τ) µ + i τ ) µ i τ ) { n ()P µ osh() sinh() d +i τ + } n ()P µ osh() sinh() d. (3) +i τ Eqution (3) gives the solution of the dul integrl equtions (5) nd (4) but not of (3) nd (4). For the purpose of verifition substituting Eq. (3) into (3) nd interhnging the order of integrtion we find tht x sinh() µ n () d + µ) = g(x), <x<. (3) osh(x) osh() µ Substituting the expression for n () from Eq. (7) into Eq. (3), interhnging the order of integrtion nd using the following integrl: x u we find tht sinh() d = osh(x) osh() µ osh() osh(u) +µ os(µ), (3) g() =. Mking use of the ondition (33) we find tht Eq. (3) is solution of the dul integrl equtions (3) nd (4). If µ = then the dul integrl equtions (3) nd (4) redue to the following dul integrl equtions: (33)

8 73 B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) τ A(τ) sin(xτ) dτ = g(x), <x<, (34) A(τ) oth(f τ) sin(xτ) dτ = h(x), < x <, (35) h(x) = g (x). Mking use of Eqs. (7), (9), (3) nd (36) we n write the solution of the dul integrl equtions in the following form: A(τ) = τf tnh(f τ) + Ω = n () = ω = n () = Ω()P +i τ osh() sinh() d (36) ω()p +i τ osh() sinh() d, (37) g (x) dx, (38) osh() osh(x) h(x) dx. (39) osh(x) osh() When f =, the solution of the dul integrl equtions (34) nd (35) is given by, pp. 3 3 in Eq. (3.6) of his pper. Our solution nd Bbloin s solution re the sme nd they re only orret if g() =. This is the only new ondition whih should hve been imposed in the Bbloin solution. It is, however not mentioned in his pper. Tht ondition rises due to differentition of Eq. (3). Referenes A.A. Bbloin, The solution of some dul integrl equtions, Prikl. Mt. Mekh. 8 (964) 5 3, English trnsl. in Appl. Mth. Meh. 8 (965) B. Noble, The solution of Bessel funtion dul integrl equtions by multiplying-ftor method, Pro. Cmbridge Philos. So. 59 (963) P.L. Rosenthl, On generliztion of Mehler s inversion formul nd some of its pplitions, Ph.D. disserttion, Oregon Stte University, I.N. Sneddon, The Use of Integrl Trnsforms, MGrw Hill, New York, I.N. Sneddon, Mixed Boundry Vlue Problems in Potentil Theory, North-Hollnd, Amsterdm, H.M. Srivstv, K.C. Gupt, S.P. Goyl, The H -Funtions of One nd Two Vribles with Applitions, South Asin Publishers, New Delhi, 98.

9 B.M. Singh et l. / J. Mth. Anl. Appl. 34 (5) A. Erdelyi, Tbles of Integrl Trnsforms, vol., MGrw Hill, New York, F. Oberhettinger, Tbellen zur Fourier Trnsform, Springer-Verlg, Heidelberg, A. Erdelyi (Ed.), Tbles of Integrl Trnsforms, vol., MGrw Hill, New York, 954. W. Mgnus, F. Oberhettinger, R.P. Soni, Formuls nd Theorems for Speil Funtions of Mthemtil Physis, Springer-Verlg, Heidelberg, 966.