The Linear, Nonlinear and Partial Differential Equations Are Not Fractional Order Differential Equations

Universal Journal of Engineering Science 3(3): 46-51, 2015 http://www.hrpub.org DOI: 10.13189/ujes.2015.030302

The Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations

Ali Karci

Department of Computer Engineering, Inonu University, Turkey

Abstract The differential equations were considered as equations are satisfied by the current conditions; changing fractional order differential equations in literature. the conditions will yield new equations. Homotopy Analysis Method was used to obtain analytical The deficiencies in those methods in literature are coming solutions of these equations. We applied reverse processes to from the fractional order derivative definitions used in the analytical solutions of some fractional order differential many studies. The methods involving deficiencies yield equations, and observed that solutions could not satisfy the non-consistent results. Due to this case, there is a method corresponding equations. Due to this case, we proposed a proposed for fractional order derivative in Karcı studies new approach for fractional order derivative and it was [15,16] and by using this method concludes in whether verified by using this new approach that any differential converting any differential equations to fractional order equations cannot be converted into fractional order differential equations is valid or not. We saw that converting differential equations so simply. any differential equations into fractional order differential equations by changing order is not a valid method, since Keywords Derivatives, Fractional Calculus, Differential applications of the proposed method in Karcı’s papers Equations, Fractional Order Differential Equations illustrated that any differential equations cannot be converted MSC Classification: 26A33 into fractional order differential equations so simply. This paper is organized as follow. Section 2 described the applications of HAM to some famous differential equations considered as fractional order differential equations. Section 3 included results of converting differential equations into 1 Introduction fractional order differential equations. Finally, Section 4 finalized this paper. Differential equations are applied to a large set of problems, due to this case, there are many studies on differential equations [1,2,3,4,5]. Some studies are on 2. HAM and Some Solutions for converting differential equations into fractional order differential equations. Fractional Order Differential Fractional order derivatives and fractional order Equations differential equations have important roles in rheology, Shijun [11] proposed an analytic method in his Ph.D. damping laws, diffusion process, etc [6,7,8,9,10]. Partial dissertation, and it was called as Homotopy Analysis Method differential equations, nonlinear differential equations and (HAM). Dehghan et al [12] used HAM to solve some fractional order differential equations require an effective fractional order differential equations such as Fractional solutions method. HAM is a candidate method for solving all KdV, Fractional K(2,2), Fractional Burgers and Fractional partial differential equations, nonlinear differential equations Cubic Boussnesq Equations. and fractional order differential equations [11]. Dehghan and Dehghan et al [12] solved these equations and we will his friends tried to solve nonlinear partial differential apply these solutions to related equations whether they equations by using HAM [12]. satisfy these equations or not. It is known that these solutions should satisfy the corresponding equations. We applied reverse process to α Dt u(x, t) equations and we observed that any differential equation t 1 n−α −1 cannot be converted into fractional order differential = (t −t ) u (n) (x,t )dt , α > 0 Γ(n −α ) ∫ equation by assuming any order as fractional, since those 0 Universal Journal of Engineering Science 3(3): 46-51, 2015 47

2.1. Fractional KdV Equation α 2 Dt u − 3(u )x + u xxx 216x − 216x (8) The mathematical model of waves on the shallow water = + + 0 = 0 surfaces is Korteweg–de Vries equation (KdV equation), and (1− 36t)2 (1− 36t)2 it is a non-linear partial differential equation whose solutions can be exactly and precisely specified. The KdV equation The results obtained in Eq.7 and Eq.8 demonstrates that can be considered as any ordinary differential equation, partial differential equation or nonlinear differential equation cannot be u (x,t) + a u 2 (x,t) + bu (x,t) = g(x,t) converted to fractional order differential equation with the t ( )x xxx same coefficients and same orders. Some researchers considered this as fractional order differential equation. The fractional KdV equation [13,14] is the first equation to be handled in this paper. This equation is 2.2. Fractional K (2,2) Equation shown in Eq.1. A family of nonlinear KdV equations K(m,n) is a 2 + + = (1) m n Dt u(x,t) a(u )x (x,t) bu xxx (x,t) g(x,t) u (x,t) + (u ) (x,t) + (u ) (x,t) = 0, t x xxx m>0, 1 < n ≤ 3. Assume that g(x,t)=0, a=-3 and b=1, the equation in Eq.2 Some researchers also considered this equation as will be obtained. fractional order differential equations. The fractional order K α 2 Dt u(x,t) − 3(u )x (x,t) + u xxx (x,t) = 0 (2) (2,2) equation is the next equation to be handled in this paper [13]. and this equation was solved by Dehghan et al [12] using α 2 2 HAM [11]. The obtained solution is seen in Eq.3. Dt u(x,t) + (u )x (x,t) + (u )xxx (x,t) = 0, 6x 36x 2 u(x,0)=x, and 0<α≤1 (9) = 2 = u(x,t) , |36t|<1, u (x,t) 2 (3) 1− 36t (1− 36t) The solution of Eq.9 was obtained by Mehdi and his friends and the obtained solution is in Eq.10. The function symbols u and u(x,t) will be used interchange. The derivatives are x x 2 u(x,t) = and u 2 (x,t) = (10) −216x 1+ 2t (1+ 2t)2 − 2 = 3(u )x 2 (4) (1− 36t) The derivatives of these solutions are 6 0 u = ⇒ u = = 0 , so, u =0 (5) 2 2x x xx xxx (u )x = (11) (1− 36t) (1− 36t) (1+ 2t)2 The parameters for fractional order derivative are n=1 and 2 2 = 1 (u )xx 2 , and finally α = (1+ 2t) 2 2 0 216x t dt (u ) = = 0 (12) Dα u = xxx + 2 t ∫ 2 (1 2t) Γ(1/ 2) 0 t −t (1− 36t ) The parameters for fractional order derivative are n=1 and 432 t −t 216x t 2 t −t dt = − + (6) 1 2 Γ ∫0 3 α = Γ(1/ 2)(1− 26t) (1/ 2) (1− 36t ) 2 438x t = α 2x t dt Dt u = − Γ ∫0 2 (1/ 2) Γ(1/ 2) t −t (1+ 2t ) Substituting results in Eq.4, Eq.5 and Eq.6 into Eq.2 yields  t t  2 t −t 4 t −t Eq.7.  +  2 (1+ 2t )2 (1+ 2t )2  = −  0 0  (13) 438x t 216x Γ − + 0 ≠ 0 (7) (1/ 2)  t dt  Γ 2 + 2 (1/ 2) (1− 36t)  ∫ 2   0 t −t (1+ 2t )  This equation is not same as Eq.2. When α=1, 12x t α 216x = − Dt u(x,t) = and Γ(1/ 2) (1− 36t)2 The results obtained in Eq.11, Eq.12 and Eq.13 are substituted in Eq.9, and the result in Eq.14 is obtained. 48 The Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations

α 2 2 + + = α −x Dt u (u )x (u ) 0 = When α=1, Dt u(x,t) 2 and 12x t 2x (14) (1+ t) ⇒ − + + 0 ≠ 0 Γ(1/ 2) (1+ 2t)2 1 Dα u + (u 2 ) − (u) = 0 t 2 x xx The obtained result is not equal to zero, so, equation is not (22) − x x α −2x ⇒ + − 0 = 0 satisfied. When α=1, Dt u(x,t) = and 2 2 (1+ 2t)2 (1+ t) (1+ t)

α 2 2 The Eq.21 and Eq.22 illustrates that the ordinary D u + (u ) + (u )= 0 t x differential equation cannot be converted to fractional order − 2x 2x (15) ⇒ + + 0 = 0 differential equation by substituting any term with fractional (1+ 2t)2 (1+ 2t)2 order derivative term. The Eq.14 and Eq.15 illustrates that the ordinary differential equation cannot be converted to fractional order 2.4. Fractional Cubic Boussinesq Equation differential equation by substituting any term with fractional The original Boussinesq equation is order derivative term. 3 utt (x,t) − u xx (x,t) + 2(u (x,t))xx − u xxxx (x,t) = 0, and its fractional order is 2.3. Fractional Burgers’ Equation 2α 3 Dt u(x,t) − u xx (x,t) + 2(u (x,t))xx − u xxxx (x,t) = 0 (23) The mathematical model of gas mechanics and traffic The solution of Eq.23 was obtained by Mehdi and his flows is Burgers' equation and this equation is a partial friends as differential equation from fluid mechanics. This equation is 1 also considered as fractional order differential equation. The u 3 (x,t) = 1 and 3 (24) modified KdV equation [13] is a Burgers’ equation u(x,t) = (x + t) x + t α 1 2 Dt u(x,t) + (u )x (x,t) − (u)xx (x,t) = 0, and 0<α≤1 (16) The derivatives of these solutions are 2 −1 2 The solution of Eq.16 was obtained by Mehdi and his = ⇒ = u x 2 u xx 3 (25) friends as (x + t) (x + t) 2 −6 24 2 x u = ⇒ u = and u (x,t) = (17) xxx 4 xxxx 5 (26) x 2 + + u(x,t) = (1+ t) (x t) (x t) 1+ t 3 −3 3 12 (u )x = ⇒ (u )xx = (27) The derivatives of these solutions are (x + t)4 (x + t)5

2 2x (u )x = (18) 1 (1+ t)2 The parameters are n=1 and α = 3

1 0 −2 / 3 t u = ⇒ u = = 0 (19) 2α 1 (t −t ) dt x xx D u = − 1+ t 1+ t t ∫ 2 Γ(1/ 3) 0 (x +t ) 1 (28) The parameters are n=1 and α = 7 t −2 / 3 2 = 15Γ(1/ 3) x 2x t dt Dα u = − t ∫ 2 The results obtained in Eq.25, Eq.26, Eq.27 and Eq.28 are Γ(1/ 2) 0 ( t −t )(1+t ) (20) substituted in Eq.23, and the result in Eq.29 is obtained. 4x t 2α − + 3 − = = Dt u u xx 2(u )xx u xxxx 0 Γ −2 / 3 (1/ 2) 7 t 2 ⇒ − The results obtained in Eq.18, Eq.19 and Eq.20 are 15Γ(1/ 3) x (x + t)3 substituted in Eq.16, and the result in Eq.21 is obtained. 24 24 (29) + − 5 5 α 1 2 (x + t) (x + t) Dt u + (u )x − (u)xx = 0 2 7 t −2 / 3 2 (21) ⇒ − ≠ 0 4x t x Γ 3 ⇒ + − ≠ 15 (1/ 3) x (x + t) 2 0 0 Γ(1/ 2) (1+ t) When α=1, Universal Journal of Engineering Science 3(3): 46-51, 2015 49

2 a) Fractional KdV Equation: 2α = α α = Dt u(x,t) Dt (Dt u(x,t)) 3 and α 2 (x + t) Dt u(x,t) − 3(u )x (x,t) + u xxx (x,t) = 0 and its solution 6x D 2α u − u + 2(u 3 ) − u = 0 is u(x,t) = , |36t|<1. t xx xx xxxx 1− 36t 2 2 24 24 (30) ⇒ − + − = 0 The fractional order derivatives with respect to new + 3 + 3 + 5 + 5 (x t) (x t) (x t) (x t) definition (Eq.31) are The Eq.29 and Eq.30 illustrates that the ordinary α −1 α  6x  216x differential equation cannot be converted to fractional order Dt u(x,t) :     2 differential equation by substituting any term with fractional  t(1− 36t) (1− 36t) order derivative term. − 3 u 2 (x,t) : 216x ( )x − 2 (1− 36t) 3. Conversion of Differential Equations u (x,t) : 6 6 x = to Fractional Order Differential 1− 36t (1− 36t) Equations u (x,t) = 0 u (x,t) = 0 xx so xxx . The sections 2.1, 2.2, 2.3 and 2.4 illustrate that any The obtained derivatives are substituted in differential differential equations cannot be converted into fractional equation (Eq.2) and the obtained result is order differential equations easily. The solutions of Fractional KdV, Fractional K(2,2), Fractional Burgers’ and Dα u(x,t) − 3 u 2 (x,t) + u (x,t) = 0 t ( )x xxx Fractional Cubic Boussinesq differential equations by using α −1 Homotopy Analyse Method demonstrate that differential  6x  216x 216x   − + =   2 2 0 0 equations cannot be converted into fractional order  t(1− 36t) (1− 36t) (1− 36t) differential equations. There will be two important reasons α −1 for this case.  6x  216x 216x n n n + ⇒   = Case 1. Assume that x +y =z , n,x,y,z∈Z . When n=1 or 2   2 2  t(1− 36t) (1− 36t) (1− 36t) there are solutions for this equations, on the other hand, when α −1 n>2 there is no solution for this equation. n can be considered  6x  ⇒   = 1 as order of differential equations, and the orders of  t(1− 36t) differential equations are all integers. So, any differential equations cannot be converted into fractional order In order to satisfy this equation, α=1. If α=1, the differential equations without changing the domain of differential equation (Eq.2) is not a fractional order variables of functions. differential equation. Case 2. The second reason is that deficiencies in b) Fractional K(2,2) Equation: definitions of fractional derivative. Instead of definitions for α D u(x,t) + (u 2 ) (x,t) + (u 2 ) (x,t) = 0 and its fractional order derivative, a new definition for fractional t x xxx x order derivative [15,16] can be used to demonstrate that solution is u(x,t) = . differential equations cannot be converted into fractional 1+ 2t order differential equations. The fractional order derivatives with respect to new Definition [15,16]. Assume that f(x):R→R is a function, definition (Eq.31) are α∈R and L(.) be a L’Hospital process. The fractional order α −1 derivative of f(x) can be rephrased as follow α  x  2x − 2xα Dt u(x,t) : −   =   2 α −1 α +1 lim  f α (x + h) − f α (x)   t(1+ 2t) (1+ 2t) t (1+ 2t) (α ) =   f (x) L α α  h → 0  (x + h) − x  2 (u )x (x,t) : 2x α α d(f (x + h) − f (x)) (31) (1+ 2t)2 lim dh = α α 2 h → 0 d((x + h) − x ) (u )xx (x,t) : 2 2 dh (1+ 2t) where α is the order of differentiation. 2 (u )xxx (x,t) = 0 The effect of this definition can be illustrated on the Fractional KdV Equation, Fractional K(2,2) Equation, The obtained derivatives are substituted in differential Fractional Burgers’ Equation and Fractional Cubic equation (Eq.9) and the obtained result is Boussinesq Equation. 50 The Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations

α 2 2 equation (Eq.16) and the obtained result is Dt u(x,t) + (u )x (x,t) + (u )xxx (x,t) = 0 − 2xα 2x D 2α u − u + 2(u 3 ) − u = 0 + + 0 = 0 ⇒ α = 1. t xx xx xxxx t α −1 (1+ 2t)α +1 (1+ 2t)2 1 2 24 6 ⇒ − − + + ≠ 0 2α −1 + 2α +1 (x + t)3 + 5 + 4 In order to satisfy this equation, α=1. This means that t (x t) (x t) (x t) Fractional K(2,2) Equation (Eq.9) is not a fractional order There is no specific α value to satisfy this equation, so, differential equation. Fractional Cubic Boussinesq Equation is not a fractional c) Fractional Burgers’ Equation: order differential equation. 1 Dα u(x,t) + (u 2 ) (x,t) − (u) (x,t) = 0 and its t 2 x xx x 4. Conclusions solution is u(x,t) = . + 1 t This paper illustrated that there are important points for The fractional order derivatives with respect to new differential equations. These points can be listed as below. definition (Eq.31) are a) HAM can be used to solve fractional order differential

α −1 equations. The obtained solution can be differentiated α  x  − x xα Dt u(x,t) :   = − and result must be substituted in fractional order   2 α +1  t(1+ t)  (1+ t) (1+ t) differential equations. This process should satisfy the equation. 1 2x x 2 = b) Dehghan et al issued this case for some differential (u )x (x,t) : 2 2 2 2(1+ x) (1+ x) equations, and they applied HAM to some differential equations and obtained their solutions. They also 1 (u) x (x,t) = assumed that these differential equations can be 1+ t considered as fractional order differential equations. − (u) (x,t) = 0 c) We applied fractional order derivative method to these xx solutions and substituted into corresponding equations. The obtained derivatives are substituted in differential This process did not satisfy the corresponding equation (Eq.16) and the obtained result is equations. These means that any differential equation 1 cannot be considered as fractional order differential Dα u(x, t) + (u 2 ) (x, t) − (u) (x, t) = 0 t 2 x xx equations so simply. d) We applied the proposed method in Karcı’s papers xα x − + + = [15,16] to illustrate that the differentiation order must α +1 2 0 0 (1+ t) (1+ t) be equal to 1 for any differential equations, since these xα x equations were obtained by applying differentiation of ⇒ = ⇒ α = 1. + α +1 + 2 order 1. (1 t) (1 t) This means that Fractional Burgers’ Equation (Eq.16) is not a fractional order differential equation. d) Fractional Cubic Boussinesq Equation: 2α 3 REFERENCES Dt u − u xx + 2(u )xx − u xxxx = 0 and its solution is [1] F. Schwarz, “Loewy Decomposition of Linear Differential 1 u(x,t) = . Equations”, Bulletin of Mathematical Sciences, vol.3, x + t pp:19-71, 2013. 2α −1 [2] M.K. Yıldız, B. Karpuz, Ö. Öcalan, “Oscillation of Nonlinear 2α  1  −1 1 Dt u(x,t) :   = − Neutral Delay Differential Equations of Second-Order with   2 α − 2α +1  t(x + t)  (x + t) t 2 1 (x + t) Positive and Negative Coefficients”, Turkish Journal of Mathematics, vol.33, pp:341-350, 2009. − 2 (u)xx (x,t) : − 3 [3] A. Aghajani, V. Roomi, “An Oscillation Theorem for (x + t) Second-Order Nonlinear Differential Equations of Euler Type”, vol.36, pp:273-280, 2012. 2(u 3 ) (x,t) = 24 xx (x + t)5 [4] R.Z. Gumbataliev, “On Completeness of Elementary Generalized Solutions of a Class of Operator-Differential 6 Equations of a Higher Order”, Turkish Journal of − (u) xxx (x,t) = Mathematics, vol.33, pp:383-396, 2009. (x + t)4 [5] M. Nadjafikhah, R.B. Chamazkoti, “Cartan Equivalence The obtained derivatives are substituted in differential Problem for Third-Order Differential Operators”, Turkish Universal Journal of Engineering Science 3(3): 46-51, 2015 51

Journal of Mathematics, vol.37, pp:949-958, 2013. Mechanics, vol.19, pp:957-962,1998. [6] M.J. Ablowitz, P.A. Clarkson, “Solitons, Nonlinear Evolution [12] M. Dehghan, J. Manafian, A. Saadatmandi, “Solving Equations and Inverse Scattering”, Cambridge University Nonlinear Fractional Partial Differential Equations Using the Press, New York, 1991. Homotopy Analysis Method”, Numerical Methods for Partial Differential Equations, DOI:10.1002/num. [7] K.S. Miller, B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations”, Wiley, New [13] A.M. Wazwaz, “The Variational Iteration Method for York, 1993. Rational Solutions for K(2,2), Burgers, and Cubic Boussinesq Equations”, Journal of Computing and Applied Mathematics, [8] I. Podlubny, “Fractional Differential Equations: An vol.207, pp:18-23, 2007. Introduction to Fractional Derivatives, Fractional Differentials Equations, to Methods of Their Solution and [14] D. Kaya, I.E. Inan, “Exact and Numerical Traveling Wave Some of Their Applications”, Academic Press, New York, Solutions for Nonlinear Coupled Equations Using Symbolic 1999. Computation”, Applied Mathematics and Computation, vol.151, pp:775-787, 2004. [9] S.G. Samko, A.A. Kilbas, O.I. Marichev, “Fractional Integrals and Derivatives: Theory and Applications”, [15] A. Karcı, A. Karadoğan, “Fractional Order Derivative And Yverdon, Gordon and Breach, 1993. Relationship Between Derivative And Complex Functions”, Mathematical Sciences and Applications E-Notes, vol.2, [10] B.J. West, M. Bolognab, P. Grigolini, “Physics of Fractal pp:44-54, 2014. Operators”, Springer, New York, 2003. [16] A. Karcı, “A New Approach For Fractional Order Derivative [11] L. Shijun, “Homotopy Analysis Method: A New Analytic And Its Applications”, Universal Journal of Engineering Method for Nonlinear Problems”, Applied Mathematics and Science, vol.1, pp:110-117, 2013.