On Another Type of Transform Called Rangaig Transform

Transcription

1 Ieraioal Joural of Parial Differeial Equaios ad Applicaios, 7, Vol 5, No, 4-48 Available olie a hp://pubssciepubcom/ijpdea/5//6 Sciece ad Educaio Publishig DOI:69/ijpdea-5--6 O Aoher Type of Trasform Called Ragaig Trasform Norodi A Ragaig *, Norhamida D Mior, Grema Fe I Pe~oal, Jae Lord Dexer C Filipias, Verie C Covico Deparme of Physics, Midaao Sae Uiversiy-Mai Campus, Marawi Ciy 97, Philippies *Correspodig auhor: azisorodip6@gmailcom Absrac A ew Iegral Trasform was iroduced i his paper Fudameal properies of his rasform were derived ad preseed such as he covoluio ideiy, ad sep Heaviside fucio I is prove ad esed o solve some basic liear-differeial equaios ad had succesfully solved he Abel's Geeralized equaio ad derived he Volerra Iegral Equaio of he secod kid by meas of Iiial Value Problem The Naural Logarihm (eg loge x l x ) has bee esablished ad defied by meas of modifyig he Euler Defiie Iegral based o he Ragaig's fomulaio Hece, his rasform may solve some differe kid of iegral ad differeial equaios ad i compees wih oher kow rasforms like Laplace, Sumudu ad Elzaki Trasform Keywords: Ragaig Trasform, Iegral Trasform, liear ordiary differeial fucio, Iegro-differeial equaio, Covoluio Theorem Keywords: ragaig rasform, iegral rasform, liear ordiary diereial fucio, iegro-diereial equaio, covoluio heorem Cie This Aricle: Norodi A Ragaig, Norhamida D Mior, Grema Fe I Pe~oal, Jae Lord Dexer C Filipias, ad Verie C Covico, O Aoher Type of Trasform Called Ragaig Trasform Ieraioal Joural of Parial Differeial Equaios ad Applicaios, vol 5, o (7): 4-48 doi: 69/ijpdea-5--6 Iroducio Oe of he mos effecive ools for solvig problems i physics ad egieerig is usig he rasform mehod o obai a soluio for a give parial differeial equaios or ordiary differeial equaio by meas of iverse rasformaio Amog hese Trasforms are he Fourier [], Laplace [], Hakel [3], Melli [3] Sumudu [4], ad Elzaki [5] These rasforms play a versaile role i solvig may problems of physical ieress which ca be described by iegro-differeial equaios wih a give appropriae iiial ad boudary codiios I hese pas years, applicabiliy of hese rasforms were show ad cosidered by may auhors [6,7,8,9,] I his paper, we ivesigae a ew rasform which is also based o he previous rasforms bu i aoher domai This Trasform akes aoher form i process of rasformig a equaio ad obais he soluio of he equaio wherei i follows he se of fucios H, a expoeial order defied as h: N,,, h Ne i, H i, () For he se i (), he arbirary cosa N mus be a fiie ad he cosas, ca be ifiie or ifiiely fiie Iroducig a ew rasform which is defied i (), Figure Schemaic diagram whe dealig wih he soluio of a problem usig iegral rasform

2 Ieraioal Joural of Parial Differeial Equaios ad Applicaios 43 h exp hd, () called Ragaig Trasform I his rasform, he variable facorizes he variable of he fucio h or i h is mapped io he of -space Applicabiliy ad efficiecy aoher sese, he fucio fucio of his ieresig rasform will be show i solvig liear differeial equaios such as some problems o Classical Physics Ragaig Trasform of Some Fucios For ay fucio o h H such ha he fucio h exiss The he Ragaig Trasform () is saisfied if he codiio for a fucio h ad is piecewise coiuous wih decreasig expoeial order Properies of his rasform are preseed i he subseque secios Defiiio Le deoes he Ragaig Trasform of h H The he followig heorems holds: Theorem If h()=, he is Ragaig Trasform yields ad for h exp d yields d 3 exp Now, he geeral form for h, we have! exp d Proof: Now, from he defiiio of he Ragaig Trasform, we have exp exp exp d exp!! Theorem 3 For expoeial fucio of he form exp(a), he exp a exp exp a d a Theorem 4 The Ragaig Trasform of he rigoomeric fucio such as cos() ad si() yields ad si cos respecively Proof: We ca wrie he rigoomeric fucio i erms of Taylor's series expasio si! The, obaiig he Ragaig Trasform of si() yields! si exp d ad applyig Theorem ad furher simplificaio yields si 3 Now, imposig he Biomial series expasio will resul o si cos Similarly, for Defiiio 5 If he fucio h H exiss such ha is -h derivaive also exiss i he se (), he le he Ragaig Trasform of he -h derivaive of h be deoed as holds he ex heorem Theorem 6 (Ragaig Trasform for derivaives) If h, h,, h H, he h k k k k h Proof: (By Mahemaical Iducio) We verify for ay saisfies he heorem So, for, h For =, we have h exp( ) h ( ) d, applyig iegraio by pars yields

3 44 Ieraioal Joural of Parial Differeial Equaios ad Applicaios h h Suppose, here exis a ieger j such ha h holds for j j r h Hece, j Leig r h j h r r r j j h j j i j i i h h i j j h j j i j i i h i This fiishes he proof Defiiio 7 Le he fucios so ha m ad h H The he ex Theorem holds he Ragaig Trasform of Iegrals deoed by Theorem 8 (Ragaig Trasform for Iegrals) If such ha m H m is defied as 3 m h T dt The, he Ragaig Trasform of m Proof: (By Mahemaical Iducio) We eed o verify ha Theorem 8 is saisfied for For, we ca obai m h For, we ge m (see Theorem 6 ad refer o i) To complee he proof, suppose here exiss a ieger j such ha j, for j, i is verified ha his is rue The for j j j m j j Hece, he Proof has bee doe by Mahemaical Iducio Noe ha for muliple Iegrals or aiderivaive of a fucio h(), we ca wrie is equivalece as 3 h T dt h d Defiiio 9 Cosider he fucios h ad r H The ex Theorem holds he Ragaig Trasform for he Covoluio Ideiy Theorem The Ragaig Trasform for he Covoluio Ideiy give by is defied as h T r T dt h* r h* r where ad h ad r respecively is he Ragaig Trasform for Proof: Muliplyig he Ragaig Trasform of he fucios h ad r ad o avoid complicaio, we assume ha he limis are fiie ad hus resulig o lim exp exp y r y dy a a ax xhxdx This formulaio is possible sice he iegrads are cosidered o decrease expoeially For he limi a, he iegrads are very complicaed o deal wih, hus o reduce his complicaio, le us assume a ieger ad T such ha xt, y T To verify hese assumpios, we make use of he Jacobia's Trasformaio for differeials x dxdy x T y ddt y T or simply dxdy ddt Subsiuig his expressio o he above equaio yields lim exp a a h T r T Corollary Le he fucios h T r T dtd h ad r T be i H The he Ragaig Trasform of he h aidiffereial of he Covoluio Ideiy is defied as h* r Furhermore, for ay ieger, we have he more geeralized Covoluio Ideiy defied by is Ragaig Trasform as g g g3 g G G G G 3

4 Ieraioal Joural of Parial Differeial Equaios ad Applicaios 45 Proof: The same procedure i Theorem 8 for he expressios i his Corollary The key o his proof is he associabiliy ad symmeric propery of he covoluio operaor Hece, Theorem 8 is he implicaio of Corollary Defiiio The dualiy relaio of he Ragaig Trasform o Laplace Trasform exiss if he fucio h H h is cosidered exiss such ha he fucio o chage iself hh H Theorem 3 (Dualiy relaio of Ragaig Trasform h exis over H, ad Laplace Trasform) If h ad he he relaio of Ragaig Trasform ad Laplace Trasform is F where F is he Laplace Trasform of h Proof: The Laplace Trasform of he fucio h is F exp h d followig he give rages of i () may exis ad hus modifyig he Laplace Trasform io F exp h d Therefore he dualiy relaio is saisfied M a be a Heaviside sep fucio Theorem 4 Le The is Ragaig Trasform akes he form M a exp a Proof: From he defiiio of he Ragaig Trasform (), we have M a exp a M a expm ad expd exp exp a a d exp d Oe proof of his ca be doe by he dualiy relaio of he Ragaig Trasform ad Laplace Trasform (Theorem 3) 3 Some Applicaio of Ragaig Trasform I his secio, we will prese a applicaio of Ragaig Trasform o some differeial equaios The followig examples are govered by liear differeial equaios wih iiial codiio wherei Ragaig Trasform is used i solvig some iiial value problems described by ordiary differeial equaios Example 3 Cosider he firs-order liear differeial equaio of he form df f d g, where g() is a ipu fucio Sol': Le he iiial codiio be f() = so ha we ca wrie he Ragaig Trasform for he firs-order derivaive as df d Obaiig Ragaig rasform will yield g Furher maipulaio, we ca obai he geeral soluio as g ( ) ad akig he iverse Ragaig Trasform leads o he soluio For g() =, we have he soluio f exp Example 3 For a RC-circui, he poeial o he capacior for is, he poeial o v Afer a log ime he capacior follows he firs-order liear differeial equaio of he form from Theorem 6, we have ad evaluaig yields v v, v v RC v RC v RC Iversig he Ragaig Trasform (usig Theorem 3) we have he poeial for he capacior for log ime RC v exp v Example 33 A bulle movig horizoally ad experieces a resisive force durig he moio The cosider he equaio of moio wih cosa k [] dv m kmv d

5 46 Ieraioal Joural of Parial Differeial Equaios ad Applicaios he fid he velociy a ay ime ad give ha V V Sol': This problem is almos he same wih he previous Example 3 Followig similar sep, we ge he Ragaig Trasformed equaio of moio as V k hus iversig, we obai he soluio V V exp k Example 34 Solve he differeial equaio dy y x dx usig he Ragaig rasform if he iiial codiio is y Sol': Agai, Usig he Theorem 6, we ge he Ragaig Trasform of he firs derivaive of he fucio as y hus he rasformed differeial equaio akes he form a 3 Furher maipulaio, we ge Usig he Previous heorems, we ge he soluio as 5 yx exp x x 4 4 Example 35 (Applicaio of Ragaig Trasform o Abel's Geeralized equaio) The Geeralized Abel's equaio is give by Where f T rt dt, f is assumed o be kow ad rt is ukow I his example, we will show he effeciveess of he Ragaig Trasform i his kid of equaio Paricularly, he validiy of he Ragaig Trasform o Covoluio Theorem will be show Takig he Ragaig Trasform for boh side of he equaio yields f T rt dt Noice ha he righ-had side of his expressio demosraes he Covoluio Ideiy give i Theorem h T T The we ca wrie he Ragaig if Trasform i accordace o he Covoluio Theorem as! f r r Rearragig his expressio yields f r! ad muliplyig he lef side of he expressio by!, we have!! f r!! f r!! By usig Theorem ad 8 we ge r T dt T f T dt!! he from [3], he facorial erm of his expressio ca be wrie as!! si Subsiuig his expressio o he previous equaio ad he differeiaig Hece, we have obaied he ukow erm of Abel's equaio as si d r d f T T dt Example 36 (Derivaio of Volerra Iegral Equaio of he Secod Kid Usig Ragaig Trasform) Le us cosider firs a Iiial Value Problem of he form h r h wih he codiio h, h Now, le h f he geig is Ragaig Trasform yields Simplifyig, we ge h f h f Usig Theorem 8 wih =, obaiig he iverse Ragaig Trasform of his expressio gives h f d f T dt

6 Ieraioal Joural of Parial Differeial Equaios ad Applicaios 47 Therefore, we ge he Volerra Iegral Equaio of he secod kid as f r T f T dt 4 Uiqueess of he Ragaig Trasform The Gamma Fucio is ypically well defied for, if <, I is hard o defie, sice i diverges for The Gamma fucio ca be defied by is secod deffiio kow as Euler's Defiie Iegral [3] exp d Logically, if he Gamma fucio happes o have a egaive value of, of course i is difficul o deal, bu wha we do is o creae a couer par of he Gamma fucio jus i case for o ake values from egaive If be he couer form of he Gamma fucio we le if < Aoher problem arises, ha is, how ca we make a equivalece form of, ad i order o do his, we eed o modify he Euler Defiie Iegral based o he formulaio of he Ragaig Trasform We have he proposed modificaio of he Euler Defiie Iegral based o he form of Ragaig Trasform as exp d, The reaso of his formulaio is resraiig he values of o become egaive Sice his formulaio is based o Ragaig Trasform, i is required ha his resul is applicable oly o Ragaig Trasform for < Cosider he Fucio h Ragaig Trasform as where <, he we have is, For which he Laplace Trasform of / is L Hece, his gives us he propery of a Naural Logarihm (Napieria) h l H The we Proposiio 4 For a fucio have obaied he Ragaig Trasform for Naural Logarihm defied as l, Proof: From he Iegral defiiio of aural logarihm, we have l d Recall, from Theorem 8 he Ragaig rasform of a give iegral for, we ge d d Acually, usig he Uiqueess of Ragaig Trasform, we have l Therefore, he Proposiio is saisfied Lemma 4 For he fucio ad d, h d r d, The, hier Ragaig Trasform akes he form h d d,, r d,, respecively Proof: (By Mahemaical Iducio) We will verify ha for is rue For, we have, which is rue (ca be verified by iegraig) Suppose here exis a ieger j such ha j, for j, i is ideed verified Now, we will show ha for all j is rue, ha is j j j, j j This fiishes he proof Similar proof ca be doe for d, d Example 43 Effeciecy of Uiqueess of he Ragaig Trasform I his followig examples, we wil goig o show he validiy of he Uiqueess of he Ragaig Trasform o some separable differeial equaio Cosider a differeial equaio of he form x y y dy where y So, he equaio ca be immediaely dx wrie as

7 48 Ieraioal Joural of Parial Differeial Equaios ad Applicaios dx dy x y Usig Theorem 8 we have x y Now, Usig he Uiqueess we ge l x ad iversig boh sides of he equaio, we ca obai is soluio as l x y For geeral soluio, we have l x C, wherec cosa y Example 44 Solve his differeial equaio y dx xydy by applyig he Ragaig Trasform Sol': Sice he equaio is separable we ca wrie i as dx ydy dy x y y Applyig he Ragaig Trasform for boh sides of he equaio we have x y, y 3 ad furher maipulaios, we have he soluio as 5 Coclusio l x l y y The deffiio of Ragaig Trasform ad is applicaio o he soluio of ordiary differeial equaio has bee preseed I was show ha he Ragaig Trasform has a clearer ad deeper coecio o Laplace Trasform However, here are cases ha Laplace Trasform ad Elzaki Trasform cao solve iegro-differeial equaios bu ca be solved i Ragaig Trasform Therefore, he Ragaig Trasform ca be used as a effecive ool i solvig iegro-differeial equaios Ackowledgemes The auhors would like o hak he Deparme of Physics for he suppor ad ideas i developig his paper The auhors would also like o hak he Midaao Sae Uiversiy-Mai Campus for he suppor exeded o his work Refereces [] E D Raivill ad P E Bedie, Elemeary Differeial Equaios, 6h ed, Macmillia Publishig Co, Ic, New York, pp 7-3, 98 [] L Debah, D Bhaa, Iegral Trasform ad hier Applicaio, d Ediio; Chapma ad Hall/CRC, 6 [3] G B Arfke ad H J Weber, Mahemaical Mehods for Physiciss, 6h ed, Elsevier Ic, pp 965-7, 5 [4] G K Waugala, Sumudu Trasform- a ew Iegral Trasform o Solve differeial Equaio ad Corollig Egierig, Mah Eg'g Iduc, vol 6, o, pp39-39, 998 [5] T M Elzaki, The New Iegral Trasform Elzaki rasform, Global Joural of Pure ad AppliedMahemaic, vol 7, o, pp57-64, [6] Zhag J, A Sumudu Based Algorim for Solvig Differeial equaios, Comp Sci J Moldova, vol 3, o 5, pp33-33, 9 [7] T Elzaki, S Elzaki, ad E Elour, O some applicaios of ew iegral rasform 'Elzaki Trasform', The Global Joural of Mahemaical Scieces: Theory ad Pracical, vol 4, o, pp5-3, [8] S Weera Koo, Applicaio of Sumudu Trasform o Parial Differeial Equaio I J Mah Educ Sc Tech, vol 5, o, pp 77-83, 994 [9] T M Elzaki ad S M Ezaki, O he coecios bewee Laplace ad ELzaki rasforms, Adv Theo ad Appl Mah, vol67 o 6, pp - [] T M Elzaki, S M Ezaki ad E MA Hilal, ELzaki ad Sumudu rasform for solvig some differeial equaios Global Joural of Pure ad Applied Mahemaic, vol 4, o 8, pp 67-73, [] M R Spiegel, S Lipschuz, J Liu, Mahemaical Hadbook ad Formulas ad Tables, Third Ediio, McGraw-Hill Comp 9 [] S T Thoro ST, J B Mario JB, Classical Dyamics of Paricles ad Sysems, Fifh Ediio, Academic Press, 4 [3] G Arfke, Mahemaical Mehods for Physicis, 4h ediio; Academic Press, New York, (985)