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1 Avilble online ww.inernionlejournl.om Inernionl ejournl Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 The Mellin Type Inegrl Trnform (MTIT in he rnge (, Rmhndr M. Pie Deprmen of Mhemi, (A.S.&H.* R.G.I.T. Verov, Andheri (W,Mumbi-5, Indi Emil.pierm@rediffmil.om Abr In hi pper Lple operor re ued o olve The Mellin Type inegrl rnform in he inervl o,a work whih i pu forwrded o undernd how Lple operor would led o properie, heorem,derivive nd ppliion o he Mellin Type Inegrl Trnform in hi rnge.the min view of my work i o give proedure from Lple Trnform h urn ou o be vlid for for he Mellin Type Inegrl Trnform in he rnge o.,where i poiive,.hi Inegrl Trnform,n be ehnique for olving boundry nd iniil vlue problem.. The reul hve been modified by pplying uible funion whih led o he reul in The Mellin Type Inegrl Trnform for he inervl o illure he dvnge nd ue of hi rnformion, ome imporn differenil equion hve been olved he end.he Mlb progrmme re given for he bi nd oher formule.the grphil onep i repreened by igning differen vlue o he prmeer by uing ool of Mlb whih give brigher view of ppliion o he Mellin Type Inegrl Trnform. Keyword: Lple rnform, Mellin rnform, Finie rnform, To AMS Mhemil Subje Clifiion A0,A5, A05, 9D5, F, A5, 5G5, A85 8
2 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie.Inroduion In he heory of Inegrl Trnform, Mellin Inegrl Trnform h preened dire nd yemi ehnique, for reoluion of erin ype of lil boundry nd iniil vlue problem.to be ueful he rnform mu be doped o he form of differenil operor o be elimined well o he rnge of inere nd oied boundry ondiion. Thi rnform i he exenion of he Mellin inegrl rnform nd hve imilr inverion formule. Thi rnform i uied o region bounded by he nurl oordine urfe of ylindril or pheril oordine yem nd pply o finie or infinie region bounded inernlly. Hiorilly,Riemnn (87 fir reognized he Mellin rnform in he fmou memoir on prime number, I explii formulion w given by Chen (89. Almo imulneouly Mellin (89, 90 gve n elbore diuion of he Mellin rnform nd i inverion formul In hi pper Lple Trnform onidered he bi definiion,,by uing repeive ubiuion,he Mellin Type Inegrl Trnform i derived. All properie, heorem, derivive nd ppliion of Lple Trnform re ified by he The Mellin Type inegrl rnform. in he rnge o..preliminry Reul: Le f(x be given funion of x whih i defined for ll x 0 nd i prmeer L[f(x= 0 e x f ( x dx Subiue x = log (./, dx= If x=0 hen = nd if x= hen =,hen from ( ( L [f(x = f (, denoed by M [f (,,, M [f (,-,, = f ( hu obined he Mellin Type inegrl rnform in he rnge o infiniy, ( 8
3 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie wih he new kernel nd new inegrl rnform.. Properie for MTIT in o..lineriy propery: Thi Mellin Type Inegrl Trnformion i Liner operion, h i for ny funion f( nd g( whoe he Mellin Type inegrl rnform exi in he rnge o infiniy nd α nd β re onn, hen M [α f(+ βg(,-,, = α M [ f(,-,, + βm [g(-,,, Proof : M [α f(+ βg(,-,, = { f ( g( } = f ( g( } = α M [ f(-,,, + βm [g(,-,,, M [α f(+ βg(-,,, = α M [ f(,-,, + βm [g(,-,,, ( Sling Propery: Conider The Mellin Type inegrl rnform in o, M [f (,-,, = f ( hen M[f (b,-,, = f ( b Subiuing b=p hen =p/b.=dp/b,if = hen p=b nd if = hen p=, we hve M [f (b,-,, = f ( b = b p ( b f ( p dp b =b p f ( p dp b 8
4 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie =b M[ f ( p,, b, M [f (b,-,, =b M[ f ( p,, b, (.. Oher Propery Conider he equion ( M f ( b b,-,, = f ( Subiuing b = q; = q b b dq nd if = hen q= b ; nd = hen q= M [f ( b,-,/,, = b q b f ( q dq b M [f ( b,-,, = b M [f (q,- /b, b, b (5. Min Reul... Inverion Theorem: for MTIT in o The MTIT i in o M [f (,-,, = f ( hen i inverion formul i f(= i M [ f (,,, d Proof: Aume h M [f (,,, i regulr equion in he rip Re( < r ( r o be rel number of he -plne nd h 0<<,-i + i,where i onn hen M [f (,-,, = f ( 8
5 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie = [ M[ f (,,, d i = M[ f (,,, d i / = M[ f (,,, dx i M [f (,-,, = M [ f (,,, d i Le N nd ume h M[f (,,, remin unbounded limi x, when Re( hen he inegrl on R.H.S of he equion ( end f(x.hene ( f(= i i i M [ f (,,, d... Convoluion Theorem: for MTIT in o (7 The MTIT i in o M [f (,-,, = f (,hn M [f ( g(x-,,, = M [ f (,,, i Proof Conider he equion M [f (,-,, = f ( M [g(x-u,-,-, d M [g (,-,, = g( hen i invere re M [f(,,, = f(x = M [ f (,,, d i 85
6 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie M [g(,,, = g(x = M [ g(,,, d i,hen M [f ( g(x-,,, = f ( g( x = M [ f (,,, d i g( x = M [ f (,,, i M [g(x-,-,, d M [f ( g(x-,, 0, = M [ f (,,, M [g(x-,-,, d (8 i... Prevl Theorem: for MTIT in o, If M [f (,-,, = f (,hn M [f ( g(-,,, = M [ f (,,, i Proof Conider he equion M [f (,-,, = f ( M [g(,-,, d M [g (,-,, = g( hen i invere re M [f(-,,, = f(x = M [ f (,,, d i M [g(-,,, = g(x = M [ g(,,, d i,hen 8
7 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie M [f ( g(x-,,, = f ( g( x = M [ f (,,, d i g( = M [ f (,,, d i M[g(,-,, d M [f ( g(x-,, 0, = M [ f (,,, i M[g(,-,, d (8..Definiion (Uhi Sep Funion If U(=H(=,when >0 =0,when <0, hen U( or H( i known he Uni Sep Funion (b Heviide Uni Sep Funion If U(-=H(-=, when > =0,when <,hen U(-( or H(- i known he Heviide Uni Sep Funion...5.Fir Shifing Theorem for MTIT in o If M [f (,-,, = f ( hen Proof M [ n f (,-,, = M [f (,- +,, If M [f (,-,, = f ( hen M [ n n (,-,, = f ( 87
8 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie n = f ( = M [f (,- +n,, M [ n f (,-,, = M [f (,-,,, (9... Seond Shifing Theorem for MTIT in o If M [f (,-,, = f ( hen M [f (-,,, = Proof If M [f (,-,, = f (, hen M [f(-bu(-b,-,, = f ( b U( b = f ( b M [f(-bu(-b,-,, = M [ f ( b,,, (0 5. MTIT for Derivive. 5.. MTIT of fir order Derivive of f( w.r.. in o Theorem: Suppoe h f ( i oninuou for ll 0 ifying ( for ome vlue nd m nd h derivive f whih i pieewie oninuou on every finie inervl in he rnge of 0. Then The Mellin Type inegrl rnform of he derivive f ( exi when > nd f ( m e for ll 0 for ome onn Proof: Conidering he e when f i oninuou for ll 0. Then on inegring by pr, hi follow M [f (,-,, = f ( hen 88
9 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie M [f (,-,, = f '( = [ f ( - ( f ( =(+ f ( - f ( = (+ M[f (,- -,, - f ( M [f (,-,, = (+ M [f (,- -,, - f ( ( ine f ( ifie f ( m derivive i obined. e nd hu The Mellin Type inegrl rnform for. 5.. MTIT of n h order Derivive f( w.r.. in o M [f (,-,, = f (, hen M [f (,-,, = f ''( = [ f '( - ( f '( =(+ f '( - f '( =(+[[ f ( - ( f ( - f '( =(+(+ f ( -(+ f(- f '( =(+(+ M [f (,- -,, -(+ f ( - f '( 89
10 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie M[f (,-,/, = (+ (+ M [f (,- -,, -(+ f ( - f '( imilrly M[f (,-,/, = (+ (+(+[f (,- -,, M[f (n -(+(+ f ( -(+ f '( - f ''( hen (,-,/, = (+ (+(+ (+n[f(r,- -n,, Thi i he genrerlixed n h Trnform in o. -(+(+ (+n- f '( -, ( n.. Appliion: for MTIT in o ( ( order derivive of f( by uing he Mellin Type Inegrl The Cuhy Liner differenil e quion i f ( f +f (+f(.. If M [f (,-,, = f ( hen M [ f (,-,, = f '( =[ f ( - ( f ( = f ( -f( = M [f (,-,, -f( M [ f (,-,, = M [f (,-,, -f( (5 nd M [ f,-,, = f ''( 80
11 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie = f ''( =[ f '( - ( f '( =- f '( -(-+ f '( = -f (+(- f '( =-f (+(- [[ f ( - ( f ( = -f (+ (- [- f ( - ( f ( = -f (-(- f ( -(-f( f ( =(-f (,-,, -(-f( f ( M [ f,-,, = M[f (,-,, -(-f( f ( ( If f ( f +f (+f( M [f(,-,, = f (,hen M [ f (,,, = f ( = [ f ''( f '( f ( = (-M [f(,-,, -(-f( f ( + M [f (,-,, -f(+ M [f (,-,, =( + M [f (,-,, -f-f ( 8
12 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie M [ f (,,, = ( + M[f (,-,, -f-f ( (7 If f ( =0 hen x f ( x xf ( x f ( x 0 x xx R.H.S of (7 give he vlue zero heb he rhs of (7 i zero ( + M [f (,-,, -f-f ( =0 x yy ( + M [f (,-,, =f(+f ([f(+f ( M [f (,-,, = [ f ( f '( ( (8 5.. Funion nd Reul for MTIT in o Sr.No. Fnion M [f(,-,, = f ( n n -( n e -(!(!( i e ( i(!(!( 8!( 0 5!( 5 in( -(!( 0 5!( 5 5 inh( -(!( 0 5!( 5 o( -(!(!( 8 7 oh( -(!(!( 8 8
13 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie 8 n( -( ( 5( 0 9 nh( -( ( 5( n in( - n [ n!( n 5!( n n inh( - n [ n!( n 5!( n 5 n o( 8 - n [ n!( n!( n n oh( 8 - n [ n!( n!( n n n( 0 - n [ n ( n 5( n 5 n nh( - n [ n ( n 0 5( n in ( -( ( 0( o ( -( in ( 8 n ( -( ( 5 5( 5 8
14 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie 9 inh ( -( ( 0( oh ( -( inh ( nh ( -( ( 5 5( 5 5 n in ( - n [ n ( n 0( n 5 5 n o ( - n [ n ( n 5( n 5 5 n n ( - n [ n ( n 5( n n inh ( - n [ n ( n 0( n 5 n oh ( -( inh ( 5 7 n nh ( - n [ n ( n 5( n 5 8 n( -( ( 0 5( 9 log(+ -( ( ( 8
15 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie 0 log(- -( ( ( ( -( ( ( ( -( ( ( e( -( ( 5 ( x -( log (log!( 5 logo( -( ( 5( loge( -( ( ( 5( 7 e in -( ( ( log 8 log( o -( ( 8( 9 log( n -( ( ( 0 e in -( ( ( loge( -( log 7. Grphil repreenion by uing ool of MATLAB: ( The Mellin Type Inegrl Trnform grph ploed beween x,y for vriou vlue of prmeer. Here he progrm h been hown wih one vlue of he prmeer. 85
16 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie Conider he equion M [f (, -,, = ( ( f ( f '( (d % f(x=x^= % f (x=*x ( f (+f ( % if =0 hen f=f(0=0,f=f (0=0 hen %y= ( f +f =0 % if = hen f=f(= nd f=f (= %y=/(^+*(*f+f Mlb Progrmme f=, f=, =; =0::0; y=(*f +f.*(^+ plo(,y xlbel('-xi', ylbel('y-xi', ile('plo for f(x=x^', legend('plo for =0::0', 8
17 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie plo for f(x=x^ plo for =0::0 y-xi xi (e % f(x=e^x= % f (x=e^x ( ( f (+f ( % if =0,=0hen f=f(0=,f=f (0= hen %y= ( f +f =0 % if = hen f=f(=e, nd f=f (=e %y=/(^+*(*f+f Mlb Progrmme ym e f=.788, f=.788, =0::0; y=(*f +f.*(^+ plo(,y xlbel('-xi', ylbel('y-xi', ile('plo for f(x=e^x ', legend('plo for =0::0', 87
18 y-xi Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie plo for f(x=e x plo for =0:: xi (f% f(x=log(+x %,f (x=/(+x % if ==0 hen f=f(=0,f=f'(=, %hen y=/(^+*(*f+f =0 %if = hen f=f(=log( nd f=f (=.5 % hen y=/(^+*(*f+f Mlb Progrmme f=0.9, f=0.5, =0::0; y=(*f +f.*(^+ plo(,y xlbel('x-xi', ylbel('y-xi', ile('plo for f(x=log(+x', legend('plo for =0::0', 88
19 y-xi Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie.5 plo for f(x=log(+x plo for =0:: x-xi 8. Conluion All he properie,heorem,derivive,ppliion,exmple of he MTIT re iefied. Thi Inegrl rnform give be reul for ehnil problem. 9. Referene: [. Derek Nylor, On Mellin Type Inegrl Trnform, Journl of Mhemi nd Mehni, (9 vol., No. [. C.Fox, Appliion of Mellin Trnformion o he inegrl equion,(9 [.J.M.Mendez nd J.R.Negrin, On he finie Hnkel-Shwrz Trnformion of Diribuion, Gni,(988, vol.9, No. [. In N. Sneddon,The ue of Inegrl Trnform,TMH ediion 97 [5.C.Fox Appliion of Mellin Trnformion o Inegrl Equion, rd Mrh,9,pp [.A.H.Zemnin,Generlized Inegl Trnformion,Ineriene Publiion, New York,(98 Trnformion, J. SIAM (8 A.Z.zemninThe Diribuionl lple nd Mellin Vol.. No.. Jn.908 Pried in U.S.A. [7. A.Z.zemninThe Diribuionl lple nd Mellin Trnformion, J. SIAM Vol.. No.. Jn.908 Pried in U.S.A. [8.Johnne Bliimlein,Hrmoni Sum nd Mellin Trnform, Nuler phyi B (Pro.Suppl.79 (999-8 [9.D.J.Bedinghm,Dimenionl Regulrizion nd Mellin ummion in High-Temperure Clulion,Theroil Phyi,The Blke lborory,imperil College, Prine Conor Rod,SW7 BW,UK. 89
20 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie rxiv:ph/000x NOV 000. [0.Anoni De Sen' nd divide Roheo',A F Mellin nd Sle Trnform,Hindvi Publihing Corporion,EURASIP,Journl on Advne in Signl Proeing,Vol.007,Arile ID 8970,9 pge Do:0,55/007/8970 [. J.M.Mendez,''A Mixwd Prevl Equion And The Generlized Hnkel rnformion'',pro.of he AMS,Vol.0,Number,Mrh 988. [. J.J.Benor,M.T.Flore,''A Prevl equion nd generlized Finie Hnkel rnformion, Commen.Mh.Univ.roline,, (99,7-8. [.C.Fox,''Inegrl rnform bed upon frionl Inegrion'',Pro.Comb.Phil.So.(9,59-. [. S. M. Khirnr, R. M. Pie, J. N. Slumke,. Appliion Of Finie Mellin Inegrl Trnform, In. J. Theoroil And Aoolied Phyi,Vpl., No.. (Nov.0 pp -78 [. S. M. Khirnr, R. M. Pie, J. N. Slumke, Appliion of The Mellin Type Inegrl Trnform In The Rnge (/,, In Inernionl Journl of Mhemil Siene nd Appliion (IJMSA Vol. No. (Jn 0 pp. -7 [5. S. M. Khirnr, R. M. Pie, J. N. Khirnr, Appliion Of Diribuionl Mellin Inegrl Trnform In The Rnge [0, Inernionl J.of Muli dipl. Adv in Engg(IJMRAE Vol.!, No.,Nov.009,pp - Reerh & 80
21 Inernionl Journl of Mhemil Siene, Tehnology nd Humniie 7 (0 8-8 Rmhndr M. Pie 8
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