The Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations
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1 Universal Jornal of Engineering Science ( 46-, OI.89/jes.. hp// The Linear, Nonlinear and Parial ifferenial Eqaions are no Fracional Order ifferenial Eqaions Ali Karci eparmen of Comper Engineering, Inon Universiy, Trkey Copyrigh by ahors, all righs reserved. Ahors agree ha his aricle remains permanenly open access nder he erms of he Creaive Commons Aribion License 4. Inernaional License Absrac The differenial eqaions were considered as fracional order differenial eqaions in lierare. Homoopy Analysis Mehod was sed o obain analyical solions of hese eqaions. We applied reverse processes o analyical solions of some fracional order differenial eqaions, and observed ha solions cold no saisfy he corresponding eqaions. e o his case, we proposed a new approach for fracional order derivaive and i was verified by sing his new approach ha any differenial eqaions canno be convered ino fracional order differenial eqaions so simply. Keywords erivaives, Fracional Calcls, ifferenial Eqaions, Fracional Order ifferenial Eqaions MSC Classificaion 6A Inrodcion ifferenial eqaions are applied o a large se of problems, de o his case, here are many sdies on differenial eqaions [,,,4,]. Some sdies are on convering differenial eqaions ino fracional order differenial eqaions. Fracional order derivaives and fracional order differenial eqaions have imporan roles in rheology, damping laws, diffsion process, ec [6,7,8,9,]. Parial differenial eqaions, nonlinear differenial eqaions and fracional order differenial eqaions reqire an effecive solions mehod. HAM is a candidae mehod for solving all parial differenial eqaions, nonlinear differenial eqaions and fracional order differenial eqaions []. ehghan and his friends ried o solve nonlinear parial differenial eqaions by sing HAM []. I is known ha hese solions shold saisfy he corresponding eqaions. We applied reverse process o eqaions and we observed ha any differenial eqaion canno be convered ino fracional order differenial eqaion by assming any order as fracional, since hose eqaions are saisfied by he crren condiions; changing he condiions will yield new eqaions. The deficiencies in hose mehods in lierare are coming from he fracional order derivaive definiions sed in he many sdies. The mehods involving deficiencies yield non-consisen resls. e o his case, here is a mehod proposed for fracional order derivaive in Karcı sdies [,6] and by sing his mehod concldes in wheher convering any differenial eqaions o fracional order differenial eqaions is valid or no. We saw ha convering any differenial eqaions ino fracional order differenial eqaions by changing order is no a valid mehod, since applicaions of he proposed mehod in Karcı s papers illsraed ha any differenial eqaions canno be convered ino fracional order differenial eqaions so simply. This paper is organized as follow. Secion described he applicaions of HAM o some famos differenial eqaions considered as fracional order differenial eqaions. Secion inclded resls of convering differenial eqaions ino fracional order differenial eqaions. Finally, Secion 4 finalized his paper.. HAM and Some Solions for Fracional Order ifferenial Eqaions Shijn [] proposed an analyic mehod in his Ph.. disseraion, and i was called as Homoopy Analysis Mehod (HAM. ehghan e al [] sed HAM o solve some fracional order differenial eqaions sch as Fracional KdV, Fracional K(,, Fracional Brgers and Fracional Cbic Bossnesq Eqaions. ehghan e al [] solved hese eqaions and we will apply hese solions o relaed eqaions wheher hey saisfy hese eqaions or no., ( n ( n ( n (, d, >
2 Universal Jornal of Engineering Science ( 46-, 47.. Fracional KdV Eqaion The mahemaical model of waves on he shallow waer srfaces is Koreweg de Vries eqaion (KdV eqaion, and i is a non-linear parial differenial eqaion whose solions can be eacly and precisely specified. The KdV eqaion can be considered as (, a ( (, b (, g(, Some researchers considered his as fracional order differenial eqaion. The fracional KdV eqaion [,4] is he firs eqaion o be handled in his paper. This eqaion is shown in Eq.. a, a ( (, b (, g(, Assme ha g(,, a- and b, he eqaion in Eq. will be obained., ( (, (, and his eqaion was solved by ehghan e al [] sing HAM []. The obained solion is seen in Eq.. 6,, 6 <, 6 ( 6 ( ( 6 (, ( The fncion symbols and, will be sed inerchange. The derivaives are 6 ( (4 ( 6 6, so, ( ( 6 ( 6 The parameers for fracional order derivaive are n and 6 ( / ( ( / ( 6 ( / ( 6 ( / d 6 d Sbsiing resls in Eq.4, Eq. and Eq.6 ino Eq. yields Eq ( / ( 6 This eqaion is no same as Eq.. When, 6, and 6 ( (6 (7 ( 6 6 ( 6 ( 6 The resls obained in Eq.7 and Eq.8 demonsraes ha any ordinary differenial eqaion, parial differenial eqaion or nonlinear differenial eqaion canno be convered o fracional order differenial eqaion wih he same coefficiens and same orders... Fracional K (, Eqaion A family of nonlinear KdV eqaions K(m,n is m n (, ( (, ( (,, m>, < n. Some researchers also considered his eqaion as fracional order differenial eqaions. The fracional order K (, eqaion is he ne eqaion o be handled in his paper []. ( (, ( (,,, (8,, and < (9 The solion of Eq.9 was obained by Mehdi and his friends and he obained solion is in Eq.., and The derivaives of hese solions are ( ( (, ( ( ( (, and finally ( ( ( ( The parameers for fracional order derivaive are n and d ( / ( 4 ( ( ( ( / d ( (/ The resls obained in Eq., Eq. and Eq. are sbsied in Eq.9, and he resl in Eq.4 is obained.
3 48 The Linear, Nonlinear and Parial ifferenial Eqaions are no Fracional Order ifferenial Eqaions ( ( (/ ( (4 The obained resl is no eqal o zero, so, eqaion is no saisfied. When,, and ( ( ( ( ( ( The Eq.4 and Eq. illsraes ha he ordinary differenial eqaion canno be convered o fracional order differenial eqaion by sbsiing any erm wih fracional order derivaive erm... Fracional Brgers Eqaion The mahemaical model of gas mechanics and raffic flows is Brgers' eqaion and his eqaion is a parial differenial eqaion from flid mechanics. This eqaion is also considered as fracional order differenial eqaion. The modified KdV eqaion [] is a Brgers eqaion ( (, ( (,,, and < (6 The solion of Eq.6 was obained by Mehdi and his friends as and (, (7, ( The derivaives of hese solions are ( (8 ( (9 The parameers are n and 4 (/ d ( / ( ( ( The resls obained in Eq.8, Eq.9 and Eq. are sbsied in Eq.6, and he resl in Eq. is obained. 4 (/ ( ( ( ( When, (, ( ( ( ( and ( The Eq. and Eq. illsraes ha he ordinary differenial eqaion canno be convered o fracional order differenial eqaion by sbsiing any erm wih fracional order derivaive erm..4. Fracional Cbic Bossinesq Eqaion The original Bossinesq eqaion is (, (, ( (, (,, and is fracional order is, (, ( (, (, ( The solion of Eq. was obained by Mehdi and his friends as and (, (4 (, ( The derivaives of hese solions are ( ( ( ( 6 4 (6 4 ( ( ( 4 ( (7 The parameers are n and / ( ( / ( 7 / ( / ( d (8 The resls obained in Eq., Eq.6, Eq.7 and Eq.8 are sbsied in Eq., and he resl in Eq.9 is obained. ( / 7 ( / ( 4 4 (9 ( ( / 7 ( / ( When,
4 Universal Jornal of Engineering Science ( 46-, 49 (,, ( 4 ( 4 ( ( ( ( and ( The Eq.9 and Eq. illsraes ha he ordinary differenial eqaion canno be convered o fracional order differenial eqaion by sbsiing any erm wih fracional order derivaive erm.. Conversion of ifferenial Eqaions o Fracional Order ifferenial Eqaions The secions.,.,. and.4 illsrae ha any differenial eqaions canno be convered ino fracional order differenial eqaions easily. The solions of Fracional KdV, Fracional K(,, Fracional Brgers and Fracional Cbic Bossinesq differenial eqaions by sing Homoopy Analyse Mehod demonsrae ha differenial eqaions canno be convered ino fracional order differenial eqaions. There will be wo imporan reasons for his case. Case. Assme ha n y n z n, n,,y,z Z. When n or here are solions for his eqaions, on he oher hand, when n> here is no solion for his eqaion. n can be considered as order of differenial eqaions, and he orders of differenial eqaions are all inegers. So, any differenial eqaions canno be convered ino fracional order differenial eqaions wiho changing he domain of variables of fncions. Case. The second reason is ha deficiencies in definiions of fracional derivaive. Insead of definiions for fracional order derivaive, a new definiion for fracional order derivaive [,6] can be sed o demonsrae ha differenial eqaions canno be convered ino fracional order differenial eqaions. efiniion [,6]. Assme ha f(r R is a fncion, R and L(. be a L Hospial process. The fracional order derivaive of f( can be rephrased as follow ( lim f f ( L h d lim dh h d(( h dh ( h f ( ( h ( f ( h f ( ( where is he order of differeniaion. The effec of his definiion can be illsraed on he Fracional KdV Eqaion, Fracional K(, Eqaion, Fracional Brgers Eqaion and Fracional Cbic Bossinesq Eqaion. a Fracional KdV Eqaion, ( (, (, and is solion 6 is,, 6 <. 6 The fracional order derivaives wih respec o new definiion (Eq. are, 6 6 ( (, 6 6 ( ( 6 ( 6 (, ( 6 (, so (,. The obained derivaives are sbsied in differenial eqaion (Eq. and he obained resl is, 6 ( (, (, 6 6 ( 6 ( 6 ( 6 6 ( 6 ( 6 ( 6 6 ( In order o saisfy his eqaion,. If, he differenial eqaion (Eq. is no a fracional order differenial eqaion. b Fracional K(, Eqaion, ( (, ( (, and is solion is,. The fracional order derivaives wih respec o new definiion (Eq. are, ( ( ( ( (, ( ( (, ( ( (, The obained derivaives are sbsied in differenial eqaion (Eq.9 and he obained resl is
5 The Linear, Nonlinear and Parial ifferenial Eqaions are no Fracional Order ifferenial Eqaions, ( ( (, ( ( (,. In order o saisfy his eqaion,. This means ha Fracional K(, Eqaion (Eq.9 is no a fracional order differenial eqaion. c Fracional Brgers Eqaion, ( (, ( (, and is solion is,. The fracional order derivaives wih respec o new definiion (Eq. are, ( ( (, ( ( ( ( (, ( (, ( The obained derivaives are sbsied in differenial eqaion (Eq.6 and he obained resl is, ( ( ( (, ( (. ( (, This means ha Fracional Brgers Eqaion (Eq.6 is no a fracional order differenial eqaion. d Fracional Cbic Bossinesq Eqaion ( and is solion is,., ( ( (, ( ( (, ( 4 ( 6 ( (, ( 4 ( The obained derivaives are sbsied in differenial eqaion (Eq.6 and he obained resl is ( 4 ( ( ( ( 6 4 There is no specific vale o saisfy his eqaion, so, Fracional Cbic Bossinesq Eqaion is no a fracional order differenial eqaion. 4. Conclsions This paper illsraed ha here are imporan poins for differenial eqaions. These poins can be lised as below. a HAM can be sed o solve fracional order differenial eqaions. The obained solion can be differeniaed and resl ms be sbsied in fracional order differenial eqaions. This process shold saisfy he eqaion. b ehghan e al issed his case for some differenial eqaions, and hey applied HAM o some differenial eqaions and obained heir solions. They also assmed ha hese differenial eqaions can be considered as fracional order differenial eqaions. c We applied fracional order derivaive mehod o hese solions and sbsied ino corresponding eqaions. This process did no saisfy he corresponding eqaions. These means ha any differenial eqaion canno be considered as fracional order differenial eqaions so simply. d We applied he proposed mehod in Karcı s papers [,6] o illsrae ha he differeniaion order ms be eqal o for any differenial eqaions, since hese eqaions were obained by applying differeniaion of order. REFERENCES [] F. Schwarz, Loewy ecomposiion of Linear ifferenial Eqaions, Bllein of Mahemaical Sciences, vol., pp9-7,. [] M.K. Yıldız, B. Karpz, Ö. Öcalan, Oscillaion of Nonlinear Neral elay ifferenial Eqaions of Second-Order wih Posiive and Negaive Coefficiens, Trkish Jornal of Mahemaics, vol., pp4-, 9. [] A. Aghajani, V. Roomi, An Oscillaion Theorem for Second-Order Nonlinear ifferenial Eqaions of Eler Type, vol.6, pp7-8,. [4] R.Z. Gmbaaliev, On Compleeness of Elemenary Generalized Solions of a Class of Operaor-ifferenial Eqaions of a Higher Order, Trkish Jornal of Mahemaics, vol., pp8-96, 9. [] M. Nadjafikhah, R.B. Chamazkoi, Caran Eqivalence Problem for Third-Order ifferenial Operaors, Trkish
6 Universal Jornal of Engineering Science ( 46-, Jornal of Mahemaics, vol.7, pp949-98,. [6] M.J. Ablowiz, P.A. Clarkson, Solions, Nonlinear Evolion Eqaions and Inverse Scaering, Cambridge Universiy Press, New York, 99. [7] K.S. Miller, B. Ross, An Inrodcion o he Fracional Calcls and Fracional ifferenial Eqaions, Wiley, New York, 99. [8] I. Podlbny, Fracional ifferenial Eqaions An Inrodcion o Fracional erivaives, Fracional ifferenials Eqaions, o Mehods of Their Solion and Some of Their Applicaions, Academic Press, New York, 999. [9] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fracional Inegrals and erivaives Theory and Applicaions, Yverdon, Gordon and Breach, 99. [] B.J. Wes, M. Bolognab, P. Grigolini, Physics of Fracal Operaors, Springer, New York,. [] L. Shijn, Homoopy Analysis Mehod A New Analyic Mehod for Nonlinear Problems, Applied Mahemaics and Mechanics, vol.9, pp97-96,998. [] M. ehghan, J. Manafian, A. Saadamandi, Solving Nonlinear Fracional Parial ifferenial Eqaions Using he Homoopy Analysis Mehod, Nmerical Mehods for Parial ifferenial Eqaions, OI./nm. [] A.M. Wazwaz, The Variaional Ieraion Mehod for Raional Solions for K(,, Brgers, and Cbic Bossinesq Eqaions, Jornal of Comping and Applied Mahemaics, vol.7, pp8-, 7. [4]. Kaya, I.E. Inan, Eac and Nmerical Traveling Wave Solions for Nonlinear Copled Eqaions Using Symbolic Compaion, Applied Mahemaics and Compaion, vol., pp77-787, 4. [] A. Karcı, A. Karadoğan, Fracional Order erivaive And Relaionship Beween erivaive And Comple Fncions, Mahemaical Sciences and Applicaions E-Noes, vol., pp44-4, 4. [6] A. Karcı, A New Approach For Fracional Order erivaive And Is Applicaions, Universal Jornal of Engineering Science, vol., pp-7,.
![Ross, An Inrodcion o he Fracional Calcls and Fracional ifferenial Eqaions, Wiley, New York, 99. [8] I.](/docs-images/48/23871064/images/page_6.jpg "Podlbny, Fracional ifferenial Eqaions An Inrodcion o Fracional erivaives, Fracional ifferenials Eqaions, o Mehods of Their Solion and Some of Their Applicaions, Academic Press, New York, 999. [9] S.G.")
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