APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS

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1 Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M ISSN APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS S.M.Khirnr, R.M.Pi*, J.N.Slun** Dprtmnt o Mthmti Mhrhtr Adm o Enginring, Alndi-45, Pun, Indi Mobil No.: ,F No.: , Dprtmnt o Mthmti (A.S.& H.* R.G.I.T. Vrov, Andhri (W, Mumbi-53, Indi Mobil No: , F.No nd Dprtmnt o Mthmti** North Mhrtr Univrit,Jlgon-Indi Mo.bil No Abtrt- In thi wor, th Lpl.Mllin intgrl trnorm i uitd in th ininit rgion [,;, ]..Thi wor whih i put orwrd to undrtnd th proprti nd rltion thorm,drivtiv nd pplition or th Lpl - Mllin Intgrl Trnorm in [,] to [, ]. To illutrt th dvntg nd u o thi intgrl trnorm,lpl qution in rtin orm, th on dimnionl wv qution nd hr low qution nd Cuh linr dirntil qution r olv b uing thi intgrl trnorm olvd t th nd. Solution o th dirntil qution r grphill rprntd b uing Mtlb. Ind Trm- Lpl trnorm, Mllin trnorm, Doubl Lpl trnorm,intgrl trnorm AMS Mthmtil Cliition : 44A,33C99 (, 47D3( T I. INTRODUCTION h Lpl- Bi Ltrl Lpl trnorm i ud to ind th Lpl - Mllin intgrl trnorm in th rng [,] to, [ ], proprti li Linr proprt, Sling proprt, Powr proprt, thorm li invrion thorm, onvolution thorm, prvl thorm, hiting thorm.b uing thi intgrl trnorm w obtin th rult o drivtiv w. r. t., nd w ind th gnrlizd rult o th n drivtiv o th untion (, w. r. t... Uing th drivtiv w olv th Lpl qution in Crtin orm, on dimnionl wv nd ht low qution. It olution id rprntd b uing tool o Mtlb. II. PRELIMINARY RESULTS Rltion o th Mllin Intgrl Trnorm with Lpl Trnorm Th Lpl trnorm o th untion ( o i dnotd b L[(, nd dind ( d, L[(,] whnvr thi intgrl i it or r> i th prmtr Th invr o th Lpl trnorm i dnotd b L [ (, L ( [ (, ] nd dind ( i i L[ (, dr Th Mllin intgrl trnorm o th untion o ( o i dnotd b M[(] nd i dind M[( r ( d whnvr thi intgrl i it or > prmtr. Th invrion o th Mllin intgrl trnorm i dnotd b M M [ ( [ ( ( ( nd dind i r i i M[ ( ] d

2 Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M ISSN Th Lpl-Bi Ltrl Lpl trnorm L[ z, L[ z, nd dind ( rz z ddz i dnotd b whnvr thi doubl intgrl i it or r> nd > r prmtr. Subtitut z -log( thn z nd i z thn d, dz -,i z- thn (, p dd, thn (,+ g(, (,+ m[g(, ( B. Sling Proprt Sond vr importnt proprt i tht th Lpl Mllin Intgrl Trnorm i Sling Proprt:, tht i or th untion (, who Lpl -Mllin intgrl trnorm it, thn onidr th qution L[ z, ( rz z ddz (, p dd,thn ( r ( d log( d( rz z z ddz r z ddz q, d z d p (q,r,, (3 (,d C. Powr Proprt Third vr importnt proprt i tht th Lpl Mllin Intgrl Trnorm i Powr Proprt:, tht i or th untion (, who Lpl -Mllin intgrl trnorm it, thn onidr th qution (, p dd, thn r t L[ z, r dd (, r, r (,t,p/ (4 Thi i th rltion in btwn Mllin Intgrl Trnorm nd Lpl trnorm or (, with prmtr r> nd > in th rng [,;, ]..Thi i dnotd b L M[ r, ] r dd L M M[ whr nd nd >,p>. III. PRELIMINARY PROPERTIES A. Linr Proprt A vr importnt proprt i tht th Lpl Mllin Intgrl Trnorm i linr oprtion thorm:, tht i or th untion (, nd g(, who Lpl-Mllin Intgrl Trnorm it nd r ontnt thn onidr th qution ( IV. MAIN RESULTS A. Invrion Thorm Th Lpl-Mllin Intgrl Trnorm i (, (, (i i i i i p p p B. Convolution Thorm Th Lpl-Mllin Intgrl Trnorm i (, dd, thn ddp p dd, thn (5

3 Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 3 ISSN (,g(t-, (i i i i i p t z p g( t, ddp p d d (6 C. Prvl Thorm (Orthogonlit Th Lpl-Mllin Intgrl Trnorm i (, (,g(, (i (7 i i i i p t z p p D. Shiting Thorm Th Lpl-Mllin Intgrl Trnorm i (, dd, thn g(, ddp p dd, thm Lm], (8 V. LAPLACE MELLIN INTEGRAL TRANSFORM OF DERIVATIVES Thorm: Suppo tht (, i ontinuou or ll t nd z tiing or om t, vlu, nd m nd h drivtiv whih i piwi ontinuou on vr init intrvl in th rng o t. nd z Thn b uing th Lpl- Mllin intgrl trnorm, th drivtiv o (, it whn > nd p> nd t (, m z or ll nd or om ontnt A. Lpl-Mllin Intgrl Trnorm o irt ordr prtil drivtiv o (t w.r.t. Th Lpl- Mllin trnormtion i ( p d[[ d[ (, ] p p p t (, whr - ( dd dd - ( p (, d - d] d] p (, d - (9 B. Lpl-Mllin intgrl trnorm o n th ordr prtil drivtiv o (, w.r.t. Lpl-Mllin Intgrl Trnorm o irt ordr prtil drivtiv o (, w.r.t. Th Lpl- Mllin Intgrl Trnorm i (, (, p dd p dd, thn (, p dd, thn p d d (, p dd p d[[ ] ( d]

4 Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 4 ISSN p d[ (, ( d] w. r. t. nd -ontnt. p d d,intgrt ( p dd - p (, d p d[[ ] ( d] p dd -(B uing DUIS p d[[ (, ( d] [ -] p (, d p d] (, (, (n (, - ( 3 - ( n - n ( Thi i th gnrlizd rult o h Lpl- Mllin Intgrl th Trnorm o n drivtiv o (,. VI. APPLICATIONS TO LIMIT A. On dimnionl wv qution i olvd b uing Lpl-Mllin Intgrl Trnorm. Th on dimnionl wv qution i whr ( t, z ( t, z Th Lpl- Mllin Intgrl Trnormtion i tht i (3 whr p (, d. + + p (, d p dd w. r. t. nd ontnt. p p d[[ d[[ (, p d ] ( ( d d] d], intgrt (, p dd, thn p (, d p d] p dd p (, d +

5 Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 5 ISSN whr p (, d [ (DUIS + ppling Lpl- Mllin Intgrl Trnorm (,- D D D - ( D,w hv to Lm (4 [ D (5 Thi i th ordinr dirntil qution w. r. t. CF C C nd PI whr CS C C p (, d (6 To illutrt th u o th Lpl-Mllin intgrl trnorm in olving th rtin prtil dirntil qution.w propo to ind th olution (, o th qution,tiing th boundr ondition Th initil nd boundr ondition r ( I thn (,, ( I thn (,. Th olution o th on dimnionl wvl qution i CS C C I thn (, C + C - C + C - I thn (, W gt C C C C C C - ( ( ( whr C -. nd C ( ( (7 p (, d Grphil Rprnttion, thn (,thn. Empl

6 -i Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 6 ISSN On Dimntionl.Wv Eqution lm[(t,z, i B. Lpl qution in Crtin orm i olvd b uing Lol-Mllin intgrl trnorm. Th on dimnionl wv qution i ( t, z ( t, z i Th Lpl- Mllin Intgrl Trnormtion i (, p dd +, thn Whr p (, d, thn (,- + ( D D + - (8 Thi i th ordinr dirntil qution o ond ordr in. It root r m i nd m i It Complmntr FuntionO.F. nd tht C o( C in( Prtiulr IntgrlP.I.-, th th Complt Solution i C o( C in( - (9. Empl whr p (, d To illutrt th u o th Lpl-Mllin intgrl trnorm in olving th rtin prtil dirntil qution.w propo to ind th olution (, o,tiing th boundr ondition An. ( I thn (,,( I thn (, Th olution o th prtil qution i givn b Lm [ C o( C in( ( I thn (,C-/ thn C / ( I thn (t,co(+cin(- / C o(-cot( o(+ ot( Th rquird olution i (, o(+[ o (+ ot(]in( - - ( whr p (, d Grphil Rprnttion

7 Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 7 ISSN ( lm[(, whr C C - p (, d 3. Empl To illutrit th u o th Lpl-Mllin Intgrl Trnorm in olving th On dimntionl Ht Slow Eqution i olvd b uing Lpl- Mllin Intgrl trnorm. Th on dimnionl qution i, whr Th Lpl- Mllin Intgrl Trnormtion i (, p dd, thn Lm [ whr, thn p (, d + D -, thn D D ( ( Thi i th ordinr dirntil qution in C.F. C P.I. - Th omplt olution i C + rtin prtil dirntil qution W propo to ind th olution tiing th boundr ondition It olution i (,p[ C C. - ( I thn (,, ( I thn '(,. Anwr. C ( I thn (,C+C- ( nd id thn (, thn ( ( C (( (, Whr Grphil Rprnttion + C C + (( p (, d (3., nd

8 Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M 8 ISSN VII. REMARKS B uing Lpl Mllin intgrl trnorm, olution o th Lpl qution in Crtin orm, on dimntionl wv qution nd Ht Flow qution r olvd nd it olution r rprntd grphill b uing tool o Mtl. VIII. CONCLUSION Rprnttion o olution o th dirntil qution b uing tool o Mtlb i th id i givn in thi ppr. REFERENCES [] Dr Nlor, On Mllin Tp Intgrl Trnorm Journl o Mthmti nd Mhni (963 vol., No. [] C.Fo, Applition o Mllin' Trnormtion to th intgrl qution,(933 [3] J.M.Mndz nd J.R.Ngrin, On th init Hnl-Shwrtz Trnormtion o Ditribution Gnit,(988, vol.39, No. [4] S. M. Khirnr, R. M. Pi, J. N. Slun,Bi Ltrl Lpl Mllin Intgrl Trnorm nd It Applition Imt. J. o Pur & Appl. Si. Th., (,, pp 4-6 [5] In N. Snddon,Th u o Intgrl TrnormTMH dition 974 [6] C. Fo Applition o Mllin Trnormtion to Intgrl Eqution, 3rd Mrh, 934,pp [7] A.H.Zmnin, Gnrlizd Intgrl Trnormtion, Intrin Publition, Nw Yor,(968 [8] A. H.. ZmninTh Ditributionl lpl nd Mllin Trnormtion J. SIAM Vol 4. No.. Jn.98 Pritd in U.S.A. [9] S. M. Khirnr, R. M. Pi, J. N. Slun, Stud O Th Sumudu Mllin Intgrl Trnorm nd It Applition IJMSEA, Vol. 4 N. IV ( t. Pp [] I.S. Rd, Th Mllin Tp Doubl Intgrl, Cmbridg, London. [] S. M. Khirnr, R. M. Pi, J. N. Slun, Rltion O Th Finit Mllin Intgrl Trnorm With Th Lpl nd Fourir Trnorm CES, Vol. 4,no. 6, [] S. M. Khirnr, R. M. Pi, J. N. Slun, Applition O Th Mllin Tp Intgrl Trnorm In Th Rng [, ], IJMSA, Vol., No., Jn. Pp -7.

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