FRACTIONAL VARIATIONAL ITERATION METHOD FOR TIME-FRACTIONAL NON-LINEAR FUNCTIONAL PARTIAL DIFFERENTIAL EQUATION HAVING PROPORTIONAL DELAYS

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1 S33 FRACTIONAL VARIATIONAL ITERATION METHOD FOR TIME-FRACTIONAL NON-LINEAR FUNCTIONAL PARTIAL DIFFERENTIAL EQUATION HAVING PROPORTIONAL DELAYS by Derya DOGAN DURGUN ad Ali KONURALP * Deparme of Mahemaics Faculy of Ars ad Scieces Maisa Celal Bayar Uiversiy Maisa Turkiye Origial scieific paper hps://doi.org/.98/tsci7669d Iroducio I his paper ime-fracioal o-liear parial differeial equaio wih proporioal delays are solved by fracioal variaioal ieraio mehod akig io accou modified Riema-Liouville fracioal derivaive. The umerical soluios which are calculaed by usig his mehod are beer ha hose obaied by homoopy perurbaio mehod ad differeial rasform mehod wih same daa se ad approximaio order. O he oher had o improve he soluios obaied by fracioal variaioal ieraio mehod residual error fucio is used. Wih his addiioal process he resulig approximae soluios are geig closer o he exac soluios. The resuls obaied by akig io accou differe values of variables i he domai are suppored by compared ables ad graphics i deail. Key words: modified Riema-Liouville derivaive proporioal delays ime-fracioal pde fracioal variaioal ieraio mehod Fracioal differeial equaios become a fudameal ool o udersad real life problems ad i is used a almos all disciplies. There are various sudies o fracioal differeial equaios []. Some of hese equaios are formed by replacig he posiive ieger order derivaives wih modified fracioal derivaives so i is aimed o fid ou wha he behavior is. I order o deermie he soluios ha guide he behavioral sae hose are solved umerically. The mos realisic models of ODE do o have aalyic soluios so ha he umerical ad approximaio mehods should be used i order o solve such problems []. The variaioal ieraio mehod (VIM) is also applied successfully o boh umerous liear ad o-liear fracioal order problems by may auhors [3-5]. Besides ha here are also several mehods used i order o solve he o-liear problems. Bu mos of he auhors claimed ha VIM ad he umerical resuls demosrae ha he VIM is relaively accurae ad also easily implemeed mehod such as Adomia decomposiio mehod differeial rasform mehod (DTM) ad some oher mehods [6 7]. O he oher had fracioal PDE ha appear i may physical pheomea are also sudied by some researchers usig some reaed ypes of VIM [8-3]. Parial fucioal-differeial equaios wih proporioal delays represe a paricular class of delay PDE ad also hese are solved by several auhors such as [4 5]. * Correspodig auhor ali.kouralp@cbu.edu.r

2 S34 O he oher had Polyai ad Zhurov [6] suggesed a mehod for cosrucig exac soluios o o-liear delay reacio-diffusio equaios. Addiioally Abazari ad Gaji [7] proposed -D DTM ad is reduced form o obai he soluio of PDE wih proporioal delay. The Sakar e al. [8] proposed homoopy perurbaio mehod (HPM) for umerical soluios of hese kids of special equaios. I his paper we examie o-liear fracioal PDE which have proporioal delays. Previously Ghaeai e al. [9] applied modified VIM o o-liear PDE Abazari ad Gaji [7] sudied hese kid of fracioal equaios by usig exeded DTM ad Sakar e al. [8] applied HPM akig io accou Capuo derivaive defiiio o aforemeioed equaios. Very recely Sigh ad Kumar [] has jus proposed o use a aleraive VIM cosiderig Capuo sese derivaive. The differeces of our sudy amog previous sudies are cosiderig modified Riema-Liouville derivaive operaor [] firsly improveme of he soluios wih 4 residual error fucio [] secodly so ha havig approximaely a leas imes more accurae daa ad fially obaiig semi-aalyic soluios ha is approximae soluios are fucios of x ad. Now le us cosider followig ime-fracioal PDE wih proporioal delays of he geeral form: D ( ) ( )D ( )...D ( ) ux = f xu pxq xu pxq xu pxq = () ( k subjec o he iiial codiios u ) ( x) = η k ( x) for k = m m< m+ ad m where ( x ) [] [] η k ( x) is a specified iiial fucio pi qj () for i j is a parameer describig he order of he ime fracioal derivaive ad ux () is he exac soluio. For fracioal iegrals he Riema-Liouville fracioal iegral defiiio ad for fracioal derivaives modified fracioal derivaive defiiio [ 3 4] are used i our approach. This defiiio is jus a modificaio o he defiiio of Riema-Liouville derivaive ad i is sricly equivale o he Capuo via Riema fracioal derivaive. The defiiios used are briefly iroduced: Defiiio. Riema-Liouville fracioal iegral of a coiuous fucio f : R R () x f() x wih respec o of order is: RL ( τ ) ( τ ) I f() x = f x dτ Γ( ) > () Defiiio. The modified Riema-Liouville fracioal derivaive is: D ( ) [ f( x ) f( x) ] d for τ τ τ < < D ( ) ( ) f x = Γ (3) D [D f ( x )] for < + where D deoes he h parial derivaive wih respec o while D is modified Riema Liouville derivaive of order []. Furhermore modified Riema-Liouville fracioal derivaive is sricly equivale o he Gruwald-Leikov fracioal derivaive [5] ad has valuable advaages accordig o boh sadard Riema-Liouville ad Capuo fracioal derivaives. For isace i is defied for arbirary coiuous (ca also be o-differeiable) fucios ad he fracioal derivaive of a cosa is equal o zero. If he fucio is o defied a he origi he fracioal deriva-

3 S35 ive will o exis. I order o overcome his maer Aagaa ad Secer [6] proposed o ake fiie par of he fracioal derivaive order operaor which is based o he cocep of fiie par approach of Esrada ad Kawal [7]. Wih his defiiio some impora properies ca be iroduced: fracioal iegraio of a fracioal derivaive local iegraio D ( ) = ( ) ( ) < I f x f x f x (4) RL RL I f( x ) = ( ) ( ) d = ( )(d ) Γ( ) τ f x τ Γ ( + ) f x τ where fracioal derivaive of compouded fucio is defied: τ τ (5) (d τ) Γ ( + )dτ < < (6) i he view of [ 3 4]. For compariso purposes especially we ake he derivaive order (] problem ha we have ow becomes: so he ux () = f x u( px q)d ( )...D ( ) xu p x q xu pmx qm m =... (7) subjec o he iiial codiio ux ( ) = η ( x). Process of fracioal VIM (FVIM) for fracioal PDE Accordig o sadard VIM heory which was firsly proposed by He [8] we shall regeerae a correced fucioal ha allows us o cosruc a ieraio formula i order o fid fixed poi of ha formula. Based o his srucure he FVIM has already bee preseed ad used by may auhors [9-3]. We are dealig wih problem of eq. (7) so is correced fucioal is wrie as i he form of: u + () x = u() x + { Dτ τ [ τ τ x τ ]} + λ( τ) u( x ) f x u( pxq )D u( pxq )... (d τ) Γ ( + ) (8) where λ( τ ) Lagrage muliplier ca be ideified opimally via variaioal heory i which f is resriced variaio so is δ f = cosequely. Makig eq. (8) saioary yields followig codiios: λ( τ) + = D λ( τ) = (9) ad i is easily udersood ha he rivial soluio of hose sysem is λ( τ ) =. Now our ieraio formula is: u + () x = u() x Γ ( + ) { D u x f[ x u pxq τ xu pxq τ ]} τ τ ( τ) τ ( )D ( )... (d τ ) ()

4 S36 u which () will have a fixed poi ux () akig io accou a special iiial approximae fucio x ha ca be freely chose if i saisfies he iiial ad boudary codiios of he problem. Approximae soluios are deermied: u ( ) x where lim u () x = ux () () Improveme of soluios obaied by FVIM Improveme wih residual error fucio I case he exac soluio of he problem is o kow or ca o be obaied aalyically i order o check he sesiiviy of approximae soluios obaied wih FVIM we will use he residual error fucio which allows us o approach he desired soluio ux () as u() x + e() x. We will correc he approximaed soluio u () x usig he residual error fucio e () x. Assume ha he h order approximae soluio u() x saisfies: D ( ) ( )D ( )...D u x f u pxq xu pxq xu( pmxq m) = g( x ) + Rx ( )() such ha a residual fucio remais as Rx () o he righ had side of eq. () where g() x is o-homogeous fucio removed from f. Sice ux () is he exac soluio of eq. (7) eq. () ca be also wrie: D { D u ( x ) f u ( p x q )D u ( p x q )...D u ( p x q ) m = g () x x x m Subracig eq. (3) from eq. () yields: { [ u x u x] D () () f u( px q)d ( )...D ( ) xu p x q xu pmx qm f u ( )D ( )...D ( ) p x q xu p x q xu pmxq m = R( x ) (4) Deoig by e () x he residual error fucio of u () x ad akig i cosideraio ha is a liear operaor we have he error differeial equaio wih homogeous iiial codiio: [ ()] D e x f ( u + )( )D ( + )( )...D ( + )( ) e pxq x u e pxq x u e pmxq m f u( )D ( )...D ( ) pxq xu pxq xu pmxq m = Rx ( ) subjec o e ( x ) =. Solvig his by a umerical mehod such as FVIM e () x is foud umerically herefore he soluio u() x is improved by addig ha erm. Numerical experimes I his secio ime-fracioal PDE wih proporioal delays of eq. (7) ha were solved by usig DTM [7] ad earlier by HPM [8] will be cosidered. } } (3) (5)

5 S37 Example The firs ime fracioal PDE is: ux () x = D xxux () + D xu x u + ux () < (6) wih iiial codiio ux ( ) = x ( x ) [] [] ad he exac soluio is u()= x xe whe = which was already solved by DTM i [7] HPM i [8]. The umerical daa are calculaed separaely for four cases i accordace wih he value of as follows: Case = So he equaio becomes firs order PDE wih respec o ad he ieraio formula for FVIM is cosruced: u+ ( x ) = u( x ) τ x τ Dτ u ( x τ ) D xxu ( x τ) D xu x u u ( x τ) (d τ) (7) + + Γ ( + ) ux akig iiial approximae fucio u ()= x x. While which will have a fixed poi () is icreasig approximae soluios of order are idicaed: () u x = x + Γ ( + ) 3 ( + ) Γ ( + ) u () x = x (9) Γ ( + ) Γ ( + ) [ Γ ( + )] Γ (3 + ) ad so o. From eqs. (8) ad (9) i is see ha he exac soluio u does o have a closed form. We also coclude ha because of o-lieariy of he problem. From u (.5 ) umerical experimes view he u (.5 ) umerical values of approximae u 3 (.5 ) u soluios for cerai () x combiaios chose iside he domai 4 (.5 ).5e are calculaed by usig our Mahemaica algorihm. Figure shows Figure. Firs four soluio of FVIM for = x =.5 firs four approximae soluio values () ad curves obaied by usig wih prese FVIM. From hose i is also see ha each u x soluio is geig closer o he exac soluio ha u () x. Compariso of our daa wih hose obaied oher wo mehods (DTM ad HPM) i [7 8] ca be see from ab.. While i [7 8] auhors foud Taylor series expasio of exac soluio wih opimized Lagrage muliplier FVIM gives oe commo ieraio formula ha geeraes successive approximae soluios wihou ay kow series forma. I he fourh approximaio (8)

6 S38 Table. Compariso fourh soluio of FVIM for = wih hose obaied by HPM ad DTM Exac soluio HPM ad DTM Abssolue x u u(x ) 4 (x ) u 4 (x ) error u 4 (x ) while he maximum error obaied wih FVIM is read from ab. as he maximum error obaied wih HPM ad DTM is read as hree imes bigger from ha of FVIM. Furhermore as i is expeced he miimum error occurs ear he origi i a subregio of [] [] ad is approximaely 7. I order o have closer umerical soluios o ux () residual mehod is goig o be applied o u () x obaied from FVIM. Accordig o secio Improveme of soluios obaied by FVIM error differeial equaio relaed wih eq. (6): x x [ e() x] D e() x D e x e D e x u x D xu4 x e e () x = R4() x D x x x 4 subjec o e ( x ) = where e() x ux () u() x = ad R() x is residue fucio which: u () x x D xxu() x D xu x u u() x R() x = Wih he same process i secio Process of fracioal VIM (FVIM) for fracioal PDE eq. () is solved fucioally he wih some values of ( x ) fig. is ploed ad ab. is obaied. Case =.9 =.8 ad =.7 Abssolue error HPM ad DTM u 4 (x ) [7 8] For =.9 wih modified Riema-Liouville fracioal derivaive defiiio he FVIM gives semi-aalyical soluios which are calculaed as fucios of x ad. These solu- () ()

7 S39 Figure. Firs four correced soluio obaied by FVIM for = ad exac soluio of eq. (6) e + u u (.5 ) + e 4 (.5 ) u (.5 ) + e 4 (.5 ) u 3 (.5 ) + e 4 (.5 ) u 4 (.5 ) + e 4 (.5 ).5e Table. Compariso of u( x ) 4 ad u4( x ) + e4( x ) improved approximae soluios of FVIM for = ad exac soluio of eq. (6) x Exac soluio u(x ) u 4 (x ) u 4 (x ) + e 4 (x ) Absolue error u 4 (x ) Absolue error u 4 (x ) + e 4 (x ) u u 4 [ ] for =.7 u 4 [ ] for =.8 u 4 [ ] for =.9 u 4 [ ] for = Figure 3. Forh order approximae soluios u( ) 4 x obaied by usig FVIM for = ad i eq. (6)

8 S4 Table 3. Improved approximae soluios u4 ( x ) + e( x ) of eq. (6) obaied by FVIM for = ad x u 4 (x ) + e (x ) u 4 (x ) + e (x ) u 4 (x ) + e (x ) u 4 (x ) + e (x ) for =.7 for =.8 for =.9 for = ios have several log erms hus i is o wrie here. Isead he obaied soluios for cerai values are give i fig. 3 ad ab. 3. Example The ime fracioal PDE is: ux () = D xxu x u x u() x < () wih iiial codiio ux ( ) = x ( x ) [] [] ad he exac soluio is ux () = xe whe = which was already solved by HPM i [8] ad DTM i [7]. The umerical daa are calculaed separaely i cases accordace wih he value of : Case = So he equaio becomes firs order PDE wih respec o ad he ieraio formula for FVIM is cosruced: u + () x = u() x τ τ Dτ u ( x τ) D xxu x u x u( x τ) (d τ) (3) Γ ( + ) which will have a fixed poi ux () akig iiial approximae fucio u ()= x x. While is icreasig approximae soluios of order are idicaed: () u x = x + Γ ( + ) (4)

9 S4 3 Γ + π Γ ( + ) Γ ( + ) Γ π Γ ( + ) Γ (3 + ) + 3 ( ) () = u x x ad so o. From eqs. (4) ad (5) i is see ha he exac soluio does o have a closed form. Followig ab. 4 shows obaied approximae soluios ad comparisos he values also obaied wih oher mehods. Table 4. Compariso fourh soluio of FVIM for = wih hose obaied by HPM ad DTM x Exac soluio HPM ad DTM Absolue Absolue error HPM u ν(x) 4 (x ) u 4 (x ) error u 4 (x ) ad DTM u 4 (x ) I order o improve semi aalyic soluios u () x i. e. o ge i closer o exac soluio we will apply residual mehod proposed i secio Improveme of soluios obaied by FVIM. Accordig o his secio error differeial equaio relaed wih eq. (): D [ e () x ] D e x e x + u x + x 4 + e x e u = R x subjec o e ( x ) = where R (x ) is residue fucio ad i is: ( ) x Dx 4 x 4( ) u () x x D xxu() x D xu x u u() x = R() x (7) Wih he same process i secio Process of fracioal VIM (FVIM) for fracioal PDE eq. (6) is solved he wih some values of ( x ) fig. 4 is ploed ad ab. 5 is give. (5) (6)

10 S4 Table 5. Compariso of ) 4 u4( x ) + e4( x ) improved approximae soluios for = ad exac soluio of eq. () x x e u 4 (x ) u 4 (x ) + e 4 (x ) Absolue error of Absolue error of u 4 (x ) u 4 (x ) + e 4 (x ) Example 3 e + u Fially cosider ime fracioal PDE: u() x x x = D xxu D xu D xu(x) ux () < (8) 8 wih iiial codiio ux ( ) = x ( x ) [] [] which was already solved by DTM i [7] ad HPM i [8]. Is exac soluio is ux () = xe whe =. The umerical daa are calculaed separaely a cases i accordace wih he value of as follows: Case = Figure 4. Firs four correced soluio obaied by FVIM for = ad exac soluio of eq. () So he equaio becomes firs order PDE wih respec o ad he ieraio formula for FVIM is cosruced: + Γ ( + ) u ( x ) = u( x ) D u( x τ ) τ u (.5 ) + e 4 (.5 ) u (.5 ) + e 4 (.5 ) u 3 (.5 ) + e 4 (.5 ) u 4 (.5 ) + e 4 (.5 ).5 e

11 S43 x τ x τ D xxu D xu + D xu( x τ) + u( x τ) (d τ) 8 (9) which will have a fixed poi ux () akig iiial approximae fucio u ()= x x. While is icreasig approximae soluios of order are foud: () u x = x (3) Γ ( + ) a x a x x x a u ( x ) = x + ( a ) + ( a ) ( a ) 4 ( a ) + Γ + Γ + Γ + Γ + xγ a+ 3a + πa Γ Γ 3 ( a) ( a) (3) ad so o. From eqs. (3) ad (3) i is see ha he exac soluio does o have a closed form ab. 6. Table 6. Compariso fourh soluio of FVIM for = wih hose obaied by HPM ad DTM x Exac soluio u(x ) Prese mehod u 4 (x ) HPM ad DTM u 4 (x ) Absolue error u 4 (x ) Absolue error HPM ad DTM u 4 (x ) Now i order o approximae soluios u () x of FVIM ge closer o exac soluio we will apply residual mehod. Accordig o secio Improveme of soluios obaied by FVIM error differeial equaio is: x x x x D [ ] e( x ) D xe D xe D xe D xu

12 S44 x x D xe D xu + D xe( x) + e( x) = R( x ) (3) 8 subjec o e ( x ) = where R () x is residue fucio ad i is: u () x x x D xxu D xu D xu() x u() x R() x + + = (33) 8 Wih he same process i secio Process of fracioal VIM (FVIM) for fracioal PDE eq. (3) is solved fucioally he fig. 5 is ploed wih some values of ( x ) ad he obaied values are give as i ab. 7. e + u u (.5 ) + e 4 (.5 ) u (.5 ) + e 4 (.5 ) u 3 (.5 ) + e 4 (.5 ) u 4 (.5 ) + e 4 (.5 ).5 e Figure 5. Firs four correced soluio obaied by FVIM for = ad exac soluio of eq. (8) Table 7. Compariso of ) 4 u4( x ) + e4( x ) improved approximae soluios for = ad exac soluio of eq. (8) x Exac soluio u 4 (x ) u 4 (x ) + e 4 (x ) Absolue Absolue error error u 4 (x ) u 4 (x ) + e 4 (x ) Coclusios I his paper ime-fracioal PDE wih proporioal delays are cosidered ad heir semi-aalyical soluios are obaied by usig FVIM composed of modified Riema-Liouville ype derivaive. For < < sice he exac soluios of hree es problems are o kow he residue error fucio is iroduced addiioally. Wih he aid of esimaed error

13 S45 fucio i is also showed by figures ad ables ha he FVIM mehod yields sesiive values o he exac soluios or esimaed errors of problems. Cosiderig FVIM he series soluios are foud by usig he iiial codiios oly. So he cosecuive erms is rasferrig he daa o ex erm ad his is a sigifica advaage of he FVIM. If a exac soluio exiss for he equaio i ca also be see ha he series soluio coverges o he closed form soluio. O he oher had i is observed from ables ha are idicaed for each case ha FVIM provides early exac soluios o problems wih he same approximaio order ad same iiial daa. Ackowledgme The auhors hak Maisa Celal Bayar Uiversiy Faculy of Ars ad Scieces ad Maisa Celal Bayar Uiversiy Applied Mahemaics ad Compuaio Research Ceer for heir parial suppor. Refereces [] Podluby I. Fracioal Differeial Equaios Academic Press New York USA 999 [] Baleau D. e al. Fracioal Calculus: Models ad Numerical Mehods World Scieific Sigapure 6 [3] Odiba Ζ. Μ. Momai S. Applicaio of Variaioal Ieraio Mehod o Noliear Differeial Equaios of Fracioal Order Ieraioal Joural of Noliear Scieces ad Numerical Simulaio 7 (6) pp [4] Kouralp A. e al. Numerical Soluio o he Va Der Pol Equaio wih Fracioal Dampig Physica Scripa 9 (9) T [5] He J. H. Wu X. H. Variaioal Ieraio Mehod: New Developme ad Applicaios Compuers & Mahemaics wih Applicaios 54 (7) 7 pp [6] Momai S. e al. Algorihms for Noliear Fracioal Parial Differeial Equaios: A Selecio of Numerical Mehods Topological Mehods i Noliear Aalysis 3 (8) pp. -6 [7] Molliq Y. e al. Variaioal Ieraio Mehod for Fracioal Hea-ad Wave-Like Equaios Noliear Aalysis: Real World Applicaios (9) 3 pp [8] Jafari H. Jassim H. K. Local Fracioal Variaioal Ieraio Mehod for Solvig Noliear Parial Differeial Equaios wihi Local Fracioal Operaors AAM (5) pp [9] Wu G. C. Lee E. W. M. Fracioal Variaioal Ieraio Mehod ad Is Applicaio. Physics Leers A 374 () 5 pp [] Yag X. J. e al. Local Fracioal Variaioal Ieraio Mehod for Diffusio ad Wave Equaios o Caor Ses Rom. Jour. Phys. 59 (4) - pp [] Ibis B. Bayram. M. Approximae Soluio of Time-Fracioal Advecio-Dispersio Equaio Via Fracioal Variaioal Ieraio Mehod The Scieific World Joural 4 (4) ID76973 [] He J. H. A Tuorial Review o Fracal Spaceime ad Fracioal Calculus. I. J. Theor. Phys. 53 (4) pp [3] Yag X.-J. Baleau D. Fracal Hea Coducio Problem Solved by Local Fracioal Variaio Ieraio Mehod Thermal Sciece 7 (3) pp [4] Ghaeai H. e al. Modified Variaioal Ieraio Mehod for Solvig a Neural Fucioal-Differeial Equaio wih Proporioal Delays Ieraioal Joural of Numerical Mehods for Hea & Fluid Flow () 8 pp [5] Bhrawy A. H. Zaky M. A. Numerical Algorihm for he Variable-Order Capuo Fracioal Fucioal Differeial Equaio Noliear Dyamics 85 (6) 3 pp [6] Polyai A. D. Zhurov A. I. Exac Soluios of Liear ad No-Liear Differeial-Differece Hea ad Diffusio Equaios wih Fiie Relaxaio Time Ieraioal Joural of No-Liear Mechaics 54 (3) Sep. pp. 5-6 [7] Abazari R. M. Gaji M. Exeded Two-Dimesioal DTM ad Is Applicaio o Noliear PDEs wih Proporioal Delay Ieraioal Joural of Compuer Mahemaics 88 () 8 pp [8] Sakar M. G. e al. Numerical Soluio of Time-Fracioal Noliear PDEs wih Proporioal Delays by Homoopy Perurbaio Mehod Applied Mahemaical Modellig 4 (6) 3-4 pp [9] Ghaeai H. e al. Modified Variaioal Ieraio Mehod for Solvig a Neural Fucioal-Differeial Equaio wih Proporioal Delays Ieraioal Joural of Numerical Mehods for Hea & Fluid Flow () 8 pp

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