APPLICATIONS OF THE MELLIN TYPE INTEGRAL TRANSFORM IN THE RANGE (1/a, )
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1 In. J. o heml Sene nd Applon Vol. No. Jnury-June 05 ISSN: APPLICATIONS OF THE ELLIN TYPE INTEGRAL TRANSFOR IN THE RANGE / S.. Khrnr R.. Pe J. N. Slunke Deprmen o hem hrhr Ademy o Enneern Alnd-405Pune Ind Eml-mkhrnr@ml.om Deprmen o hem A.S.&H.* R.G.I.T. Verov Andher Wumb-53 Ind Eml.perm@redml.om nd Deprmen o hem** Norh hrr Unvery Jlon hrhr Ind Eml.drjnlunke@ml.om Abr In h pper Lple operor re ued o olve he elln Type Inerl Trnorm whh n be ehnque or olvn boundry nd nl vlue problem.th Trnorm pplble n he nne nervl. Th work nend o undernd how Lple operor led o propere nd relon wh he elln Type Inerl Trnorm.The mn objeve o h work o nd he relon beween Lple Trnorm nd he elln Type Inerl Trnorm n/. The reul hve been moded by pplyn uble unon whh led o he reul o elln Type Inerl Trnorm n he nervl /.We llure he ue o h Trnormon by olvn he Cuhy derenl equon wh he help o lb. Keyword Lple Trnorm elln Trnorm Fne Trnorm Inerl Trnorm AS hem Clon
2 65D3 33C90 6D07 6D A60 44A35. INTRODUCTION The Lple Trnorm ued o udy he propere o elln Type Inerl Trnorm n he rne / o nny nd lo o how he vldy o propere lke Lner Propery Sln Propery Power Propery heorem lke Inveron Theorem Convoluon Theorem Prevl Theorem Fr nd Seond Shn Theorem. We obn he elln Type Inerl Trnorm o he nh order dervve o wh repe o nd Cuhy Lner Derenl equon L F '' ' 0 olved by un h Inerl Trnorm. The oluon o derenl equon rphlly preened by ATLAB.. PRELIINARY RESULTS Le be ven unon o whh dened or ll 0 nd prmeer hen Lple Trnorm L= 0 e d d Condern = lo. d= I =0 hen =/ nd = hen = rom L = / d / 4 - / = d denoed by 4 * - / Thu obned The elln Type Inerl Trnorm n /. >0. 3. PROPERTIES LINEAR PROPERTY The elln Type nne Inerl Trnormon Lner operor h or ny unon nd we hve 4 α + β-/ = α 4 -/ + β -/. Where α nd β re onn. Proo : 4 α + β-/ = / { } = d } d / = α 4 -/ + β 4-/ 4 α + β-/ = α 4 -/ + β -/ 3 / bscalling PROPERTY The elln Type Inerl Trnorm n / d
3 / 4 - / = d hen 4 b.- / = b - 4 p-b/ Proo The elln Type Inerl Trnorm n / / 4 - / = d / 4 b.- / = b. d Subun b=p we hve b / hen 4 b.- / = d = b - p p dp / 4 b.- / = d POWER PROPERTY The elln Type Inerl Trnorm n / / 4 - / = d b / = b - 4 p-b/ 4 hen 4 b - / = b 4 q- /b / b Proo The elln Type Inerl Trnorm n / Subun b = q; d = / 4 - / = d 4 b - / = b q b / dq nd I =/ hen q=/ b ; nd = hen q= 4 b - / = b b / q b b d hen q dq 4 b - / = b 4 q- /b / b 5 4 AIN RESULTS. INVERSION THEORE 3
4 4 Theorem: The elln Type Inerl Trnorm n / 4 - / = d / hen = d / 4 Proo: Aume h * - / reulr equon n he rp Re < r r o rel number o he -plne nd h 0<< - + where onn hen 4 - / = d / = d / d N N / 4 = d d N N / 4 0 = d N N / 4 / = d N N / / = d N N / 4 6 Le N nd ume h - / remn unbounded lm when Re hen he nerl on R.H.S o he equon 6 end. Hene = d / 4 7 b CONVOLUTION THEORE The elln Type Inerl Trnorm n / 4 - / = d / hen 4 - / = d / hen -/ = = d / 4 --/ = = d / 4
5 / = / 4 4 -u- /- d Proo The elln Type Inerl Trnorm n / 4 - / = d / hen 4 -- / = d / = / d / 4 - d ubun - =u hen d= -du nd =/ hen u=/- nd = hen u= = d / 4 / du u u = d / 4 / du u u = d / 4 4 -u- /- d 4 -- / = / 4 4 -u- /- d 8 PARSEVALS THEORE The elln Type Inerl Trnorm n / 4 - / = d / hen 4 - / = d / hen -/ = = d / 4 -/ = = d / / = d / / 4 4 Proo The elln Type Inerl Trnorm n /
6 6 4 - / = d / hen 4 - / = d / = / d / 4 - d = d / 4 / d 4 - / = d / / d DEFINITIONS Hevde Un Sep Funon For U-=H-= when > =0when < U- or H- known he Hevde Un Sep Funon. efirst SHIFTING THEORE I 4 - / = d / hen 4 n - / = 4 - +n/ Proo I 4 - / = d / hen 4 n - / = d n / = d n / = 4 - +n/ 4 n - / = 4 - +n/ 0 SECOND SHIFTING THEORE I 4 - / = d / hen 4 -- / = 4 u-/-b Proo
7 / I 4 - / = d 4 -bu-b-/ = / hen b U b d Subue u=- hen du=d =/ hen u=/-b nd = hen u= I U when >0 hen where kernel u+b 4 -bu-b-/ = 4 -bu-b-/ = b u b / b u b u U u du u du hen 4 -bu-b-/ = 4 u-/-b 5. DERIVATIVES 5. The elln Type Inerl Trnorm o r order dervve o w. r.. Suppoe h onnuou or ll 0 yn or ome vlue nd m nd h dervve whh peewe onnuou on every ne nervl n he rne o 0. Then he elln Type nne Inerl Trnorm o he dervve o e when > nd m or ome onn Proo: Condern he e when e or ll 0 onnuou or ll 0 hen nern by pr we e / 4 - / = ' d = - d =+ / / d - / / = / / = / - 5. The elln Type Inerl Trnorm o n h order dervve o w. r.. By pplyn o he eond-order dervve we obn 4 - / = / '' d 7
8 = / ' - ' d =- '/ ++ =+ =+ / =++ ' d / / / - / ' d 3 - d 3 d - / -+ / / - / = / -+ /- / 4 - / = / -+ /- / 3 Smlrly 4 - / = / / / - / 4 hen 4 n - / =+++--+n 4 - -n / -++ +n- n n / Th n h order dervve o w. r APPLICATIONS Conder Cuhy derenl equon L F = + / 4 - / = ' d 4 - / = = 4 - / / / '' d = / - - / - / 4 L - / = 4 - / - /- / I L =0 hen 4 L - / =0 4 - / - /- /=0 4 - / = /+ / 6 : The Cuhy Derenl Equon 5 8
9 I 4 - / = d / / = d 4 - / = / =0 hen 0 yy hen / - /- /= / = /+ / yy - /- / whh nde he hher order derenl equon wh onn oeen. The omplee oluon ven by 4 - / = /+ / 7 where onn re deermned by nl nd boundry ondon. 7. ATLAB PROGRAES lb Prormme or he equon L= 0 e d ym =''; %=''; =ep-.*; =.*; LT=n0n preylt LT = n*ep-* = 0.. In Inerl rom zero o nny o ep- d lb Prormme or he equon 4 - / = ' ym =.^+; =/; 3=''; 4=' '; 5=*3-*4; =*5 prey = /^+**-* = / + - 9
10 8. GRAPHICAL REPRESENTATION Conder he equon 4 - / = /+ / =^= + % =* % = hen == nd = = %y=/^**+ lb Prormme = = =; =0::0; y=.^-.** + ploy lbel'-' ylbel'y-' le'plo or =^' leend'plo or =0::0' y- 3 plo or = plo or =0:: b % =e^= + % =e^ % = hen ==e nd = =e %y=/^**+ lb Prormme ym e =.788 =.788 =0::0; y=.^-.** + ploy 0
11 lbel'-' ylbel'y-' le'plo or =e^ ' leend plo or =0::0 y- 6 4 plo or =e plo or =0:: % =lo+ % =/+ % = hen ==lo nd = =.5 % hen y= y=.^-.** + lb Prormme =0.693 =0.5 =0::0; y=.^-.** + ploy lbel'-' ylbel'y-' le'plo or =lo+' leend'plo or =0::0' y plo or =lo+ plo or =0:: Conder he equon 4 / = /+ / CONCLUSION We hve been een how he propere o he Lple operor hold ood or he elln Type Inerl Trnorm n /.The Cuhy Derenl Equon olved nd we hve ven rphl repreenon o he oluon o derenl equon by lb. I oberved h elln Type Inerl Trnorm n / ve beer reul hn oher Inerl Trnorm.
12 REFERENCES. Derek Nylor On elln Type Inerl Trnorm Journl o hem nd ehn 963 vol. No.. C. Fo Applon o elln Trnormon o he nerl equon J.. endez nd J. R.Nern On he ne Hnkel-Shwrz Trnormon o Drbuon Gn 988 vol.39 No. 4. In N. Sneddon The ue o Inerl Trnorm TH edon C. Fo Applon o elln Trnormon o Inerl Equon 3 rd rh934pp J.. endez nd J.R. NernOn he ne Hnkel Swhwrz Trnormon o DrbuonGn.Vol.39No A.H. Zemnn Generlzed Inel Trnormon Inerene Publon New York A.Z. Zemnn The Drbuonl Lple nd elln Trnormon J. SIA Vol.4. No.. Jn. 908 Prned n U.S.A. 9. Sohl Ahmed Le Yn nd Ljo Hnzo Shool o EC Unvery o SouhmponSO7 BJUK 0. Johnne Blmlen Hrmon Sum nd elln Trnorm Nuler phy B Pro.Suppl D.J.Bednhm Dmenonl Reulrzon nd elln ummon n Hh-Temperure Clulon Therol Phy The Blke lborory Imperl Collee Prne Conor RodSW7 BWUK.rX v:ph/000 NOV Anon De Sen' nd dvde Roheo 'A F elln nd Sle Trnorm Hndv Publhn Corporon EURASIP Journl on Advne n Snl Proen Vol.007Arle ID pe Do:055/007/ J.. endez ''A ed Prevl Equon And The Generlzed Hnkel rnormon ''Pro. o he ASVol.0Number 3rh J.J. Benor.T. Flore ''A Prevl equon nd enerlzed ne Hnkel Trnormon''Commen.h.Unv.rolne C.Fo''Inerl rnorm bed upon ronl Ineron'' Pro. Comb. Phl. So
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