A Note on Integral Transforms and Differential Equations

Transcription

1 Malaysia Joral of Mahemaical Scieces 6(S): -8 () Special Ediio of Ieraioal Workshop o Mahemaical Aalysis (IWOMA) A Noe o Iegral Trasforms ad Differeial Eqaios, Adem Kilicma, 3 Hassa Elayeb ad, Ma Rofa Ismail ¹Facly of Sciece, Uiversii Pra Malaysia, 434 UPM Serdag, Selagor, Malaysia Isie for Mahemaical Research, Uiversii Pra Malaysia, 434 UPM, Serdag, Selagor 3 College of Sciece, Kig Sad Uiversiy, PO Box 455, Riyadh 45, Sadi Arabia akilicma@prapmedmy, hgadai@ksedsa ad mrofa@prapmedmy ABSTRACT I his work a ew iegral rasform, amely Smd rasform was applied o solve liear ordiary differeial eqaios wih cosa coefficies wih covolio erms Frher i order o geerae a pde wih o cosa coefficies, he covolios were sed ad solios were demosraed Keywords: Smd rasform, differeial eqaios ad covolio INTRODUCTION I he lierare here are several works o he heory ad applicaios of iegral rasforms sch as Laplace, Forier, Melli, Hakel, o ame a few, b very lile o he power series rasformaio sch as Smd rasform, probably becase i is lile kow ad o widely sed ye The Smd rasform was proposed origially by Wagala (993) o solve differeial eqaios ad corol egieerig problems I Wagla (), he Smd rasform was applied for fcios of wo variables Some of he properies were esablished by Weerakoo (994, 998) I Asir (), frher fdameal properies of his rasform were also esablished Similarly, his rasform was applied o he oe-dimesioal ero raspor eqaio i Kadem (5) I fac i was show ha here is srog relaioship bewee Smd ad oher iegral rasform, see Kilicma e al () I pariclar he relaio bewee Smd rasform ad Laplace rasforms was proved i Kilicma ()

2 Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail Frher, i Elayeb e al (), he Smd rasform was exeded o he disribios ad some of heir properies were also sdied i Kilicma ad Elayeb () Recely, his rasform is applied o solve he sysem of differeial eqaios, see Kilicma e al () Noe ha a very ieresig fac abo Smd rasform is ha he origial fcio ad is Smd rasform have he same Taylor coefficies excep he facor, see Zhag (7) Ths if he = f = = F =! a, see Kilicma e al () Similarly, he Smd rasform seds combiaios, C(m, ), io permaios, P(m,) ad hece i will be sefl i he discree sysems The Smd rasform is defied by he formla over he se of ; F = S f = e f d, ( τ, τ ) τ j A f M, ad or, τ >, sch ha f < Me = if ( ) [, ) j Or prpose i his sdy is o show he applicabiliy of his ieresig ew rasform ad is efficiecy i solvig he liear ordiary differeial eqaios wih cosa ad o cosa coefficies havig he o homogeos erm as covolios Throgho his paper we eed he followig heorem which was give by Belgacem (7), where hey discssed he Smd rasform of he derivaives: a Malaysia Joral of Mahemaical Scieces

3 A oe o Iegral Trasforms ad Differeial Eqaios Theorem : Le ad le G ( ) be he Smd rasforms of he f ( ) The G ( k) = k k = G f () for more deails, see Belgacem (7) Sice he rasform is defied as improper iegral herefore we eed o discss he exisece ad he iqeess Theorem : Le f ( ) ad g( ) be coios fcios defied for ad have Smd rasforms, F ad G, respecively If F = G he f = g where is complex mber Proof: If α are sfficiely large, he he iegral represeaio of f by = π i α + i f α i e F d sice F = G almos everywhere he we have = π i α + i f α i e G d By sig Laplace rasform of he fcio f ( ) deoed by s F ( s) = e f ( ) d, ca be rewrie afer a chage of variable, w = s wih dw = sd w w dw F( s) e = f, s s from above relaio ad replace by s rasform as follow we obai he iverse Smd Malaysia Joral of Mahemaical Scieces 3

4 Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail α + i s ds f = e G g, π i = α i s s ad he heorem is prove I he ex heorem we sdy he exisece of Smd rasform as follows Theorem 3: (Exisece of he Smd rasform) If f is of expoeial order, he is Smd rasform S [ f ] ; = F is give by F = e f d, where i = + The defiig iegral for F exiss a pois = + i i η τ η τ he righ half plae η > k ad ζ > L Proof: Usig = + i ad we ca express F ( ) as η τ η F = f cos e d τ η i f si e d τ The for vales of >, we have η k K η η cos f e d M e d τ MηK η K 4 Malaysia Joral of Mahemaical Scieces

5 A oe o Iegral Trasforms ad Differeial Eqaios ad K η η si f e d M e d τ MηK η K which imply ha he iegrals defiig he real ad imagiary pars of F exis for vale of R e >, compleig he proof κ Noe: The fcio f o R is said o vaish below if here is a cosa c R sch ha f = for < c The se of fcios ha are locally iegrable ad vaish below will be deoed by loc + Mos of he fcios we shall be cocered i his paper vaish for < Theorem 4: Le λ > he f S ; () If f = loc + ad lim λ exiss, so does lim f + λ + f S ; ad we have lim lim f λ = λ + λ + Γ + () If f is Smd rasformable ad saisfies f ( ) = for < ad if f S f ; lim + λ also lim λ+ f S ; lim lim f + λ = λ + ( λ ) Γ + ad we have Malaysia Joral of Mahemaical Scieces 5

6 Proof: () Le f ρ > sch ha Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail α as This implies ha here are cosas A ad f A for ρ λ > This frher implies ha e f is iegrable for all > so ha we may wrie, if ; S f = e f d c ρ c ρ = e f d + e f d f = for < c, I is easy o see ha ime he firs erm o he righ of Eqaio λ+ () eds o zero as + The imes he secod erm o he λ+ righ of Eqaio () may be wrie as follows x λ x f ( x) ρ e f ( x) dx = ρ x e dx λ λ as +, f x eds o α x x λ ad sice i is boded i he rage of iegraio by he cosa A, we may apply he domiaed covergece heorem ad coclde ha λ x S f ρ x e αdx = αγ ( λ + λ+ ) as + which complee he proof () Le f λ β as + Sice his fcio boded i a eighborhood of zero he here are cosas B ad σ > sch ha f B for < < σ Usig a mehod similar o ha i he proof of λ heorem (3) we le f f H ( σ ) f = f H ( σ ) () = ad 6 Malaysia Joral of Mahemaical Scieces

7 The for doms [ f ] A oe o Iegral Trasforms ad Differeial Eqaios σ ; = + ; S f e f d S f () ad apply ( c S f ); Ae he we have σ ; S f Ke for some cosa K ad sfficiely large, frher S f ; λ+ as Also, by a similar argme o ha was sed i () imes he firs erm o he righ had side of λ+ eqaio () eds o β ( λ ) proof Γ + as, which complees he Now we le, f be a locally iegrable fcio o R We shall say ha f is coverge (raher ha iegrable) if here is a cosa k sch ha, for each, ω D, lim ω f ( ) d λ exiss ad eqal kω (where λ λ eds o ifiiy hrogh) real vales greaer ha zero The cosa k we shall deoe by f d Oher oaios migh also be sed, for example if f ( ) = for <, f ( ) d will also be wrie as f d f d = ω f d λ λ for ay D Tha meas lim ω =, see Ges (99) Proposiio : (Smd rasform of derivaive) ω sch ha, () Le f be differeiable o (, ) ad le f ( ) = for < Sppose The f L dom ( Sf ) dom( f ) ad ha f L loc S ( f ) = S f ; f ( ) + loc, Malaysia Joral of Mahemaical Scieces 7

8 () For dom S ( f ) Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail More geerally, if f is differeiable o fcio f ( ) = for c,, he < ad f Lloc he S f ; S f ; e = f ( c + ) for dom S ( f ) Proof: We sar by () as follows, he local iegrabiliy implies ha f ( c + ) exiss sice if x > c, = ( + ) ( + ) c c+ c+ x as x c + x f x f c f d f c f d Le dom S f ; If ω D o sig he iegraio by pars we have ω e f ( ) d ω e f d = λ c λ = lim e ω f d x c+ c λ lim e f x c ω = + λ x lim e ω ω f d x c+ c λ λ λ x lim e f ( x) x c ω = + λ e ω ω f d c λ λ λ c e f c ω λ The firs erm o he righ had side is give by ( + ) eds o e f ( c ) c c which ω + as λ The he secod erm is give by 8 Malaysia Joral of Mahemaical Scieces

9 A oe o Iegral Trasforms ad Differeial Eqaios ω λ λ λ e f d + e ω f d, which eds o + S ( f ) ω as λ We have hs proved ha for ay ω D, This implies ha e f lim ω e f d λ λ ω() = [ S( f ) f ( c+ )] is coverge, ha is, dom( S [ f ] ) ha c S [ f ; ] = S [ f ; ] e f ( c+ ) I order o prove par (), we js replace c by zero I geeral case, if f be differeiable o (, ) < a or > b ad f Lloc he, for all S f ( ); = S f ; a ;, ad a b wih a < b ad f ( ) = for e f ( a + ) + e f ( b ) I he ex example we se differeial eqaio ad Smd rasform Example: Le y ( ) = sih The Smd rasform of f = y is reqired We firs obai a differeial eqaio saisfied by y We have cosh( ) y = b Malaysia Joral of Mahemaical Scieces 9

10 Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail Hece Therefore ( ) y = cosh sih cosh d y d = y y +, > 4 If we ow wrie f = y we shall have, for ay, d [ ] f f f = + (3) d 4 Now by akig Smd rasform of eqaio (3) we have S f ; S f ; S f ; k = + 4 where k = lim [ f ] Clearly follows ha + f = ad k = lim cosh( ) =, + S f ; S f ; S f ; = + 4 Now o sig he proposiio () he lef had side of eqaio (4) becomes d S f ; S f ; = + S f ; d 4 by simplificaio we have F F 3 = + 4 (4) Malaysia Joral of Mahemaical Scieces

11 A oe o Iegral Trasforms ad Differeial Eqaios Tha is 4 3 F = C e Now by replacig by ad i order o fid he vale of C ' we apply s he formla f ( ) sih lim = lim = + + herefore he sice 3 4s F = C e s s 3 s F s lim = s 3 Γ s F s C C = e4s Γ Γ Γ 3 as s ad we have 3 π C = Γ, = Γ = 4s S sih ; π = e 3 s hs fially we obai Before we exed proposiio () o higher derivaives, we irodce he ak followig oaio: Le P( x) = be a polyomial i, k x x where a We defie k = ad M p x o be he marix of polyomials which is give by he followig marix prodc: Malaysia Joral of Mahemaical Scieces

12 Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail a a a 3 a a a M p ( x) = a 3 3 a x x x x a (5) x For each complex mber, M x defie a liear mappig of C io C i obvios way We shall wrie vecors y i C p as row vecors or colm vecors ierchageably, whichever is coveie, whe M p comped ad he marix represeaio by eqaio (5) of M he of corse y ms be wrie as a colm vecor i M x y a y = for ay p i i+ k k i= x k = x y is o be p x is sed, y = y, y,, y C If =, M p ( x ) is defied o be iqe liear mappig of { } C marix) I geeral, if > ad f is ierval (, ) ad If, ϕ a b wih a < b, we shall wrie, ϕ φ ( ( f a ) ( f ( a ) f ( a ) f ) ( a )) = io C (empy imes differeiable o a ; ; = +, +,, + C ( ( f b ) ( f ( b ) f ( b ) f ) ( b )) ; ; =,,, C a = we wrie ϕ ( f ; ) for ( f ) ( f ; a; ) = φ ( f ; a;) = C ϕ ;; If =, we defie I he ex we recall he Smd rasform of higher derivaives Proposiio : (Smd rasform of higher derivaives): Le f be imes differeiable o (, ) ad le f ( ) = for < Sppose ha The ( k ) f L loc for ad, for ay polyomial P of degree, f L loc ( ) k, doms f ; dom S f ; Malaysia Joral of Mahemaical Scieces

13 for A oe o Iegral Trasforms ad Differeial Eqaios ϕ (, ) P S y = S f + M y (6) ( ) dom S f ; I pariclar p S f = S f f ( ) ; ;,,, ϕ ( ; ) (7) (wih ( f ; ) ϕ here wrie as a colm vecor) For = we have Proof: S f S f f f = ( + ) ( + ) ; ; (8) We se idcio o The resl is rivially re if = ad he case = is eqivale o proposiio () Sppose ow ha he resl is re for some ad le + ak p( x) =, k x k = havig a degree + The firs wo saemes follow by pig z = f ad sig he idcio hypohesis ad proposiio () Now wrie P( x) = a + W ( x), where x ak+ W ( x) = k x The k = P D f = a f + W D z ad herefore ; = ; + M w ϕ ( z; ) S P D f a S f S W D z ; i a f + = a S f + W S f ; f ( + ) i i= k = i+ k+ ( k + ) ( ) Malaysia Joral of Mahemaical Scieces 3

14 Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail o sig eqaio (7) ad i he form of ( k ) ( k + ) z = f The above smmaio ca be wrie i + i + ( k ) ( k ( ) ) i k i k ( i a + f + = a f i + + ) ai f ( + i ) i= k = i= k = i= Ths + i+ = a f + i i= k = i+ k ( k ) ( ) a f ( ) a i f ( i ) i= = M p ϕ ( f ; ) W f ( + ) S P( D) f = a + W S f ; W f + M p ϕ f ; + W f + = P S f ( ); M p ϕ ( f ; ) I geeral, if f is differeiable o (, ) < a or b f > ad loc L he, for all a b wih a b, < ad f = for a S ; P D f = P S f M p e ϕ ( f ; b; ) y = si clearly y + y = ad p I pariclar, if we cosider f = yh, he also f + f = Ths Sice dom ( Sf ) coai wih = ad P( x) x ( D ) + f =, we have from eqaio (6) ad eqaio (7) = +, for >, = + S ( f ) 4 Malaysia Joral of Mahemaical Scieces

15 Sice ϕ ( y ) f f A oe o Iegral Trasforms ad Differeial Eqaios, =, =, Ths from above eqaio we ge S si H ; = + Le y be imes differeiable o (, ), zero o (,) followig eqaio der he iiial codiio ad saisfy he P( D) y = f g (9) ( ) y = y, y = y,, y = y ( k ) The y is locally iegrable ad Smd rasformable for k ad for every sch k, he Smd rasform of eqaio (9) give by eqaio (6) where P a a = a, a a a y a a3 a M p ϕ ( y, ) = 3 a a a y where * idicaes he sigle covolio I pariclar for, = we have a a a a y + a S y ( ); S ( f g )( ) = + a y I order o ge he solio of Eqaio (9), we akig iverse Smd rasform for Eqaio (6) as follows ( ) f g M (, ) P P ϕ P y = S + S y Malaysia Joral of Mahemaical Scieces 5 y ()

16 Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail we assme ha he iverse exis for each erms i he righ had side of Eqaio (9) Now, le s mliply he righ had side of Eqaio (9) by polyomial k = k= Ψ, we obai he o cosa coefficies i he form of Ψ P D y = f g () der he same iiial codiios sed above By akig Smd rasform for eqaio () ad sig he iiial codiio, afer arrageme we have! k F G S y ; = k P a a a y a a3 a + a 3 a + P a y by akig iverse Smd rasform we have y k! P a a a y a a3 a F G y = S + a 3 a + k P a y here we assme he iverse exis y () Now, if we sbsie Eqaio () io Eqaio () we obai he o homogeeos erm of Eaio () f g ad polyomial i he form of k k! k= Φ = 6 Malaysia Joral of Mahemaical Scieces

17 A oe o Iegral Trasforms ad Differeial Eqaios Ths we oe ha he covolio prodc ca be sed o geerae he differeial eqaios wih variable coefficies CONCLUSION I his sdy he applicaios of he Smd rasform o he solio of differeial eqaios wih cosa ad o-cosa coefficies have bee demosraed ACKNOWLEDGEMENT The ahors graeflly ackowledge ha his research was parially sppored by he Uiversiy Pra Malaysia der he Research Uiversiy Gra Scheme RU REFERENCES Asir, M A Frher properies of he Smd rasform ad is applicaios I J Mah Edc Sci Tech 33(3): Belgacem, F B M, Karaballi, A A ad Kalla, L S 7 Aalyical Ivesigaios of he Smd Trasform ad Applicaios o Iegral Prodcio Eqaios Mah Probl Egr 3: 3-8 Elayeb, H, Kilicma, A ad Fisher, B A ew iegral rasform ad associaed disribios I Tras Spec Fc (5): Ges, P B 99 Laplace rasform ad a irodcio o disribios New York: Ellis Horwood Kadem, A 5 Solvig he oe-dimesioal ero raspor eqaio sig Chebyshev polyomials ad he Smd rasform, Aalele Uiversiaii di Oradea Fascicola Maemaica :53-7 Kilicma, A, Elayeb, H ad Agarwal, P Ravi O Smd Trasform ad Sysem of Differeial Eqaios Absrac ad Applied Aalysis, Aricle ID 5987, doi: 55//5987 Kilicma, A ad Elayeb, H A oe o iegral rasforms ad parial differeial eqaios Applied Mahemaical Scieces 4(3): 9-8 Malaysia Joral of Mahemaical Scieces 7

18 Adem Kilicma, Hassa Elayeb & Ma Rofa Ismail Kilicma, A ad Elayeb, H O he applicaios of Laplace ad Smd rasforms Joral of he Frakli Isie, vol 347, o 5, pp Kilicma, A, Elayeb, H ad Kamel Ariffi Mohd Aa A Noe o he Compariso Bewee Laplace ad Smd Trasforms Bllei of he Iraia Mahemaical Sociey 37(): 3-4 Wagala, G K 993 Smd rasform: a ew iegral rasform o solve differeial eqaios ad corol egieerig problems I J Mah Edc Sci Techol 4(): Wagala, G K 998 Smd rasform a ew iegral rasform o solve differeial eqaios ad corol egieerig problems Mahemaical Egieerig i Idsry 6(4): Wagala, G K The Smd rasform for fcios of wo variables Mahemaical Egieerig i Idsry 8(4): 93-3 Weerakoo, S 994 Applicaios of Smd Trasform o Parial Differeial Eqaios I J Mah Edc Sci Techol 5(): Weerakoo, S 998 Complex iversio formla for Smd rasforms I J Mah Edc Sci Techol 9(4): 68-6 Zhag, J 7 A Smd based algorihm for solvig differeial eqaios Comp Sci J Moldova 5: Malaysia Joral of Mahemaical Scieces