Solution to Volterra Singular Integral Equations and Non Homogenous Time Fractional PDEs

Transcription

1 G. Math. Not Vol. No. Jauary 3 pp. 6- ISSN 9-78; Copyright ICSRS Publicatio 3 Availabl fr oli at Solutio to Voltrra Sigular Itgral Equatio ad No Homogou Tim Fractioal PDE A. Aghili ad H. Ziali Dpartmt of Applid Mathmatic Faculty of Mathmatical Scic Uivrity of Guila P.O. Bo- 8 Raht Ira arma.aghili@gmail.com homa_zialy@yahoo.com (Rcivd: 8-- / Accptd: 9--) Abtract I thi work th author implmtd Laplac traform mthod for olvig crtai partial fractioal diffrtial quatio ad Voltrra igular itgral quatio. Cotructiv ampl ar alo providd to illutrat th ida. Th rult rval that th traform mthod i vry covit ad ffctiv. Kyword: No-homogou tim fractioal hat quatio; Laplac traform; Voltrra igular itgral quatio. Itroductio I thi work th author ud Laplac traform for olvig Voltrra igular itgral quatio ad PFDE.

2 Solutio to Voltrra Sigular Itgral 7 Th Laplac traform i a altrativ mthod for olvig diffrt typ of PDE. Alo it i commoly ud to olv lctrical circuit ad ytm problm. I thi work th author implmtd traform mthod for olvig th partial fractioal hat quatio which ari i applicatio. Svral mthod hav b itroducd to olv fractioal diffrtial quatio th popular Laplac traform mthod [ ] [ ] [ 3 ] [ ] ad opratioal mthod [ ]. Howvr mot of th mthod ar uitabl for pcial typ of fractioal diffrtial quatio maily th liar with cotat cofficit. Mor dtaild iformatio about om of th rult ca b foud i a urvy papr by Kilba ad Trujillo []. Ataackovic ad Stakovic [5][6]ad Stakovic [] ud th Laplac traform i a crtai pac of ditributio to olv a ytm of partial diffrtial quatio with fractioal drivativ ad idicatd that uch a ytm may rv a a crtai modl for a vico latic rod. Oldham ad Spair I [3] ad [] rpctivly by rducig a boudary valu problm ivolvig Fick cod low i lctro aalytic chmitry to a formulatio bad o th partial Rima Liouvill fractioal with half drivativ. Oldham ad Spair [] gav othr applicatio of uch quatio for diffuio problm. K.Sharma t al i [8]driv a olutio of a gralizd fractioal Voltrra itgral quatio ivolvig K fuctio with th hlp of th Sumudo traform. Wy [] coidrd th tim fractioal diffuio ad wav quatio ad obtaid th olutio i trm of Fo fuctio.. Dfiitio ad Notatio Laplac traform of fuctio f ( t ) i a follow t L{ f ( t )} = f ( t ) d t : = F ( ). If L{ f ( t )} = F ( ) th t f ( t ) = F( ) d i c+ i ci L { F ( )} i giv by Whr F() i aalytic i th half- pla R( ) > c. For < o gt [5][6] C k ( k ) t = k = L{ D f ( t )} F ( ) f ().

3 8 A. Aghili t al. Thorm. (Effro Thorm [ 9 ]) Lt L{ f ( t )} = F ( ) ad L{ u ( t τ )} = U ( ) p( τ q ( )) ad aumig φ ( ) q( ) ar aalytic th o ha ( τ τ τ ) L f ( ) u ( t ) d = U ( ) F ( q( )). Eampl. Lt u aum that p( τ ) L{ u ( t τ )} = which lad to u t τ t W τ t U ( ) ( ) = ( ; ). = ad q ( ) = th o ha Th obtai w F ( ) L f ( τ ) W ( ; τ t ) dτ. t = Providd that th itgral i brackt covrg abolutly. Lik th Fourir traform th Laplac traform i ud i a varity of applicatio. Prhap th mot commo uag of th Laplac traform i i th olutio of iitial valu problm. Howvr thr ar othr ituatio for which th proprti of th Laplac traform ar alo vry uful uch a i th valuatio of crtai itgral ad i th olutio of fractioal igular itgral quatio of Voltrra typ. I th followig w may how om applicatio of itgral traform i valuatig crtai itgral. Lmma. Th followig rlatio hold tru - - p( co ϕ) dϕ = p( co ϕ) dϕ = I ( ) p( coϕ + y i ϕ ) dϕ = I( + y )

4 Solutio to Voltrra Sigular Itgral I ( co ϕ ) dϕ =. ( )!(!) k 3 k = k k Proof. I ordr to how th abov rlatio lt u itroduc th fuctio = g( ) p( co ϕ) dϕ w firt calculat th Laplac traform of g( ) a followig + L{ g( )} = p( ) d p( co ϕ) dϕ chagig th ordr of itgratio ad implifyig to obtai + L{ g( )} = dϕ p( + co ϕ) d = i ϕ = z At thi poit w itroduc th chag of variabl dϕ coϕ ad implifyig to gt dϕ dz L{ g( )} = = iz z + i coϕ z = Th valu of th compl itgral aftr uig Cauchy itgral formula i dz L{ g( )} = = = L{ I ( )} iz z + i z = - W ca rwrit th lft id of th quatio i th followig form + y ( coθ + y i θ ) + y + y I = dθ w itroduc a w variabl uch that + y i( + θ ) I d = θ y = i = co th + y + y ad agai itroducig th w variabl + θ = ϕ lad to + y i ϕ ϕ I = d = I ( + y ).

5 A. Aghili t al. It i obviou that if w t y = th w gt th rlatiohip ad. 3- Lt u dfi a w fuctio by th itgral = I( ) I ( co θ ) dθ takig Laplac traform of th abov rlatio w gt coθ coθ L{ I( )} = ( ) = dθ I( ). = O th othr had o ha th followig paio for modifid Bl fuctio of ordr zro k y I ( y ) = k k = ( k!) o w hav k I ( co θ ) dθ. = k k = ( k )!( k!) If w t = i th abov rlatio w gt th dird quatio. Lmma. Lt u aum that Show that th followig rlatio hold tru - f ( t ) = t t - F( ) = rf ( ) 3 - ta p( ta ) d = Ei( ) ( ( )) = ( ) = p{ (c + cc )}. L f t F d t whr Ei ( ) = dt i potial itgral ad th rlatio Ei ( ) = Γ( ) t for > i which Γ ( z ) i icomplt Gamma fuctio. Proof: - W hav = ( ) i θ F( ) dθ

6 Solutio to Voltrra Sigular Itgral ad coqutly applyig Bromwich' itgral w gt c+ i ( ) f t = dθ d i i θ t ( ) ( ) ci chagig th ordr of itgratio lad to f ( t ) = δ( t ) dθ. i θ Now w itroduc th w variabl t = w th w gt i θ f ( t ) =. t t - W tak Laplac traform of th abov quatio to obtai + t L{ f ( t )} = dt. t t Now w itroduc th w variabl t = u to gt + u { ( )} =. u + L f t du At thi poit lt u aum that + u I( ) = du I() = u + th u u u u ( ) = = + I du du u + u + = + I. Solvig th abov ODE w obtai I rf ad coqutly ( ) = ( ) F( ) = rf ( ).

7 A. Aghili t al. Lmma.3 If k ( ) ad ϕ( ) ar Laplac traformabl fuctio th w hav th followig rlatiohip + L k( t ) ϕ( t ) dt = K( ) Φ( ) whr K ( ) Φ( ) ar Laplac traform of fuctio k ( ) ϕ( ) rpctivly. Proof: S [ 9 ][ 7 ]. Solutio to Voltrra Sigular Itgral Equatio Laplac traform ca b ud to olv crtai typ of Voltrra igular itgral quatio. Problm. Lt u coidr fractioal Voltrra igular itgral quatio of th form c + λ D f ( ) = g( ) + k( t ) f ( t ) dt f ( ) = (.) i which k( t) = k( t) i th krl ad g ( ) i aumd to b a Laplac traformabl fuctio. Th (.) ha th formal olutio c+ i G( ) t f ( ) = { } d i λk( ) Solutio: Lt L( f ( )) = F( ) L( g( )) = G( ) L( k( )) = K( ) b th Laplac traform of f ( ) g( ) k( ) rpctivly th by uig Lmma.3 o gt th followig rlatiohip ci So o ca writ F( ) = G( ) + λk( ) F( ). G( ) F( ) = λk( ) (.) (.3) ad coqutly by Bromwich' itgral w gt th followig rlatio c+ i G( ) t f ( ) = { } d i λk( ) ci which ca b olvd by th u of Ridu thorm. Not that F() i aalytic i th half pla R > c. (.) Eampl.: Solv th followig igular itgral quatio c + λ (.5) D f ( ) = p( a ) + J ( ( t ) ) f ( t ) dt f ( ) =.

8 Solutio to Voltrra Sigular Itgral 3 Solutio: Laplac-traform of th abov itgral quatio lad to ad coqutly F( ) = + λ( ) F( ) + a (.6) λ Uig Bromwich' itgral yild F( ) = + a. (.7) + c+ i f ( ) = d. i (.8) ci ( a )( λ ) Now lt u coidr th ca: =.5 th rlatio (.8) bcom c+ i f ( ) = d. 3 i ci ( + a )( λ + ) (.9) So w may apply Laplac traform of covolutio of fuctio ad uig th fact that a( η) L = d { ; } η ( + a) (.) ad alo th followig rlatiohip L { } = L { ( ) + ( )...} = 3 3 λ λ 3 ad + λ k 3k k k = L { + ( ) λ } k= L { ( ) } ( ) ( ) ( ) I ( k ). k 3k 3k k k k k + λ = δ + λ 3k k = k = k (.)

9 A. Aghili t al. From rlatiohip (.9)-(.) o gt th formal olutio a follow a( η ) 3k k k f ( ) = { dη}*{ δ ( ) + ( ) λ ( ) I 3k ( k)} k k = which ca b calculatd a bllow w a( η w) 3k k k w η δ λ 3k w k = k f ( ) = { d }{ ( w) + ( ) ( ) I ( k( w))} dw(.) Problm. Solvig th ytm of fractioal igular itgral quatio of th form + c D φ( ) = g( ) λ k( t ) ψ ( t ) dt (.3) + c D ψ ( ) = h( ) + λ k( t ) φ( t ) dt with coditio φ( ) = ψ( ) =. Solutio: Multiplyig cod quatio by i ad addig to th firt quatio lad to c + D ( φ + i ψ ) = ( g + ih)( ) + i λ k( t)( φ + i ψ )( t) dt. (.) Now lt ( φ + iψ )( ) = ξ( )( g + ih)( ) = f ( ) iλ = γ th o ca rwrit th abov quatio i th form c + D ξ ( ) = f ( ) + γ k( t) ξ ( t) dt. (.5) At thi poit w ca apply prviou ampl to thi o a bllow. Takig Laplac traform of quatio (.5) lad to Φ ( ) = F( ) + γ K( ) Φ ( ) whr Φ( ) F( ) K( ) ar Laplac traform of fuctio ξ( ) f ( ) k( ) rpctivly. Hc o gt th followig rlatiohip

10 Solutio to Voltrra Sigular Itgral 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) G K H + λ H + λ K Φ = + i G λ ( K( )) + λ ( K( )) + whr G( ) H( ) ar Laplac traform of g( ) h( ) rpctivly. So o gt ɶ ( ) ( ) ( ) ( ) ( ) ( ) ( ) G K H ( ) λ H λ φ = + ψ = + K G. λ ( K( )) + ɶ λ ( K( )) + Ad fially by uig ivrio formula th olutio will b c+ i G + λk H d (.6) λ + ci ( ) ( ) ( ) φ( ) = i ( K( )) c+ i H( ) + λk( ) G( ) ψ ( ) = d. i λ ( K( )) + ci Eampl. coidr th followig ytm 3 + c D φ( ) = ( t ) dt ψ ( t ) + c D ψ ( ) = + ( t ) dt φ ( t ) th w hav H ( ) = G ( ) = K ( ) = o by uig rlatiohip (.6) o gt φ ( ) = δ ( ) ψ ( ) = co Problm.3 Lt u coidr fractioal Voltrra igular itgral quatio of th form c D φ( ) = f ( ) + λ l( t) φ( t) dt φ() =. (.7) Solutio: Aftr takig Laplac-traform of th abov itgral quatio ad implifyig o gt Φ ( ) = + F ( ) + λ ( γ + l ) (.8)

11 6 A. Aghili t al. i which γ.577 i Eulr cotat. Uig compl ivrio formula for th abov rlatio lad to c+ i F( ) φ( ) = d. (.9) + i + λ( γ + l ) ci Eampl.3 Coidr th followig fractioal igular itgral quatio c D φ( ) = + l( t) φ( t) dt φ() =. By uig quatio (.8) w gt th olutio a bllow Φ ( ) = = { } = 3/ 3 + γ + l γ + l + 3/ γ + l γ + l = { + ( )... } = 3 3/ 3/ γ + l γ + l γ + l = { + ( )... } = ( ). 5/ 3/ 3/ = At thi poit w may ivrt Φ ( ) aily by uig covolutio. Thrfor o ca fid that γ + l Γ(3 / ) whr l ɺɺ = = ɺɺ o w ca writ 3/ φ( ) = { + ( t) l tdt + ( ( t)l tdt)*l ) +...} Γ(3 / ) 3 Mai Rult 3/ l = ( t )(l t) dt = ( l ) Egirig ad othr ara of cic ca b uccfully modld by th u of fractioal drivativ. That i bcau of th fact that a ralitic modlig of phyical phomo havig dpdc ot oly at th tim itat but alo th prviou tim hitory. I thi ctio th author coidr crtai o-homogou tim fractioal hat quatio i a phrical domai that i a gralizatio to th problm which i 3

12 Solutio to Voltrra Sigular Itgral 7 tudid by Jorda ad Puri []. I thi work oly th Laplac traformatio i coidrd a a powrful tool for olvig th abov mtiod problm. Thi goal ha b achivd by formally drivig act aalytical olutio. 3. No-Homogou Tim Fractioal Hat Equatio i a Sphrical Domai Problm 3. Solv th o-homogou tim fractioal hat quatio c u( r t ) u( r t ) Dt u( r t ) = + λu( r t ) f ( t ) r r r r < t > < (3.) with th boudary coditio: lim r u( r t ) < u r ( t ) = ad th iitial coditio u( r ) = r <. Lt f ( t ) b Laplac traformabl fuctio. Solutio: Lt u itroduc a w variabl v( r t) = r u( r t). Th quatio (3.) bcom (3.) By takig th Laplac traform with rpct to variabl t of quatio (3.) ad boudary coditio w gt or (3.) with th boudary coditio lim V ( r ) = Vr ( ) V ( ) = r Solvig th abov quatio (3.) lad to v( r t ) Dt v( r t ) = λv( r t ) rf ( t ). r F( ) V( r ) = Acoh( r λ + ) + B ih( r λ + ) r. λ + Now w apply th boudary coditio to gt c V V( r ) = λv ( r ) rf( ) r V ( λ + ) V( r ) = rf( ) r (3.3) ih( r λ + ) F( ) V( r ) = r. ( λ + coh( λ + ) ih( λ + )) λ + (3.5) So by uig Bromwich' itgral w hav th followig rlatiohip

13 8 A. Aghili t al. c+ i ih( r λ + ) F( ) t v( r t) = d. i c i ( λ coh( λ ) ih( λ )) λ (3.6) To u th ridu thorm lt u aum that =.5 o rlatiohip (3.6) will b chagd to c+ i ih( r λ + ) F( ) t v( r t) = d. i (3.7) ci ( λ + coh( λ + ) ih( λ + )) λ + O ca that at λ + = i V ( r ) t ha impl pol at = = λ ad alo impl pol β or = ( λ + β ) whr ta β = β for =... By uig ridu thorm o gt t r λ η ih( r λ ) v( r t ) = f ( t η ) dη λ + λ coh λ i h λ i( r β ) = ( λ + β ) i β ( λ + β ) t ad coqutly th fial olutio i a bllow r ih( r λ ) i( rβ ) ( λ + β ) t u( r t ) = f ( t ) d +. t λ η η η λ r( λ coh λ ih λ ) r = ( λ + β )i β Not that if w t = th problm bcom th o-homogou hat quatio. Ca 3.. For f ( t ) = (homogou quatio) =.5 λ = w hav c u( r t ) u( r t ) Dt u( r t ) = + u( r t ) t > r < r r r Th th olutio i ih( r ) i( rβ ) ( + β ) t ( ) = r r = ( + β )i β u r t i which ta β = β for =... (figur).

14 Solutio to Voltrra Sigular Itgral 9 3 Cocluio Figur. I th prt papr th author implmtd th Laplac traform mthod for olvig fractioal igular itgral quatio. Thy alo coidrd crtai tim fractioal hat quatio which i a gralizatio to th problm ivtigatd i [] th problm of dyamic thrmo -latic tr i a phrical hll with fid boudari who ir urfac i ubjctd to a tp jump i tmpratur. W hop that it will alo bfit may rarchr i th dicipli of applid mathmatic mathmatical phyic ad girig. Rfrc [] A. Aghili ad H. Ziali Itgral traform mthod for olvig fractioal PDE ad valuatio of crtai itgral ad ri Itratioal Joural of Phyic ad Mathmatical Scic () () 7-. [] A. Aghili ad B.S. Moghaddam Laplac traform pair of - dimio ad a wav quatio Itr. Math. Joural 5() () [3] A. Aghili ad B.S. Moghaddam Multi-dimioal Laplac traform ad ytm of partial diffrtial quatio Itr. Math. Joural (6) (6) -. [] A. Aghili ad B.S. Moghaddam Laplac traform pair of -dimio ad cod ordr liar diffrtial quatio with cotat cofficit Aal Mathmatica t Iformatica 35(8) 3-. [5] T.M. Ataackovic ad B. Stakovic Dyamic of a vico-latic rod of Fractioal drivativ typ Z. Agw. Math. Mch. 8(6) () [6] T.M. Ataackovic B. Stakovic O a ytm of diffrtial quatio with fractioal drivativ ariig i rod thory Joural of Phyic A: Mathmatical ad Gral 37() () -5. [7] D.G. Duffy Traform Mthod for Solvig Partial Diffrtial Equatio Chapma & Hall/CRC (). [8] R.S. Dahiya ad M. Viayagamoorthy Laplac trafom pair of dimio ad hat coductio problm Math. Comput. Modllig 3() [9] V.A. Ditki ad A.P. Prudikov Opratioal Calculu i Two Variabl ad It Applicatio Prgamo Pr Nw York (96).

15 A. Aghili t al. [] A.A. Kilba ad J.J. Trujillo Diffrtial quatio of fractioal ordr: Mthod rult ad problm II Appl. Aal 8() () [] Y. Luchko ad H. Srivatava Th act olutio of crtai diffrtial quatio of fractioal ordr by uig opratioal calculu Comput. Math. Appl. 9(995) [] S. Millr ad B. Ro A Itroductio to Fractioal Diffrtial Equatio Wily Nw York. [3] K.B. Oldham ad J. Spair Th Fractioal Calculu Acadmic Pr Nw York (97). [] K.B. Oldham ad J. Spair Fractioal calculu ad it applicatio Bull. It. Polith. Iai. Sct. I (8) (3-) (978) 9-3. [5] I. Podluby Th Laplac Traform Mthod for Liar Diffrtial Equatio of Fractioal Ordr Slovak Acadmy of Scic Slovak Rpublic (99). [6] I. Podluby Fractioal Diffrtial Equatio Acadmic Pr Sa Digo CA (999). [7] G.E. Robrt ad H. Kaufma Tabl of Laplac Traform W.B. Saudr Co. Philadlphia (966). [8] K. Sharma R. Jai ad V.S. Dahakar A olutio of gralizd fractioal Voltrra typ itgral quatio ivolvig K -fuctio G. Math. Not 8() () 5-. [9] W. Schidr ad W. Wy Fractioal diffuio ad wav quatio J. Math. Phy. 3(989) 3-. [] B.A. Stakovic Sytm of partial diffrtial quatio with fractioal drivativ Math. Vik 3-(5) () [] P.M. Jorda ad P. Puri Thrmal tr i a phrical hll udr thr thrmolatic modl J. Thrm. Str () 7-7. [] W. Wy Th fractioal diffuio quatio J. Math. Phy. 7() (986)