A Tutorial Review on Fractal Spacetime and Fractional Calculus

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1 From the SelectedWorks of Ji-Huan He 2014 A Tutorial Review on Fractal Spacetime and Fractional Calculus Ji-Huan He Available at:

2 A Tutorial Review on Fractal Spacetime and Fractional Calculus Ji-Huan He International Journal of Theoretical Physics ISSN Volume 53 Number 11 Int J Theor Phys (2014) 53: DOI /s

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4 Int J Theor Phys (2014) 53: DOI /s A Tutorial Review on Fractal Spacetime and Fractional Calculus Ji-Huan He Received: 15 December 2013 / Accepted: 27 March 2014 / Published online: 25 June 2014 Springer Science+Business Media New York 2014 Abstract This tutorial review of fractal-cantorian spacetime and fractional calculus begins with Leibniz s notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie s mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given. Keywords Fractal spacetime Fractional differential equation Fractal derivative Q-derivative Cantor set El Naschie mass-energy equation E-infinity theory Hilbert cube Kaluza-Klein spacetime Zero set Empty set Fractal stock movement 1 Introduction Fractional calculus could be said to begin with the seminal work of Leibniz (July 1, 1646 November 14, 1716), who later authorized a very influential essay on monadology as the forerunner for Cantor sets and fractal. It is a generalization of the ordinary differentiation and integration to non-integer order. In a letter to L Hospital in 1695, Leibniz raised the following question: Can the meaning of derivatives with integer order be generalized to J.-H. He ( ) National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-ai Road, Suzhou , China Hejihuan@suda.edu.cn

5 Int J Theor Phys (2014) 53: derivatives with non-integer orders? To answer this question, we consider a continuous function, f(x), the first-order derivative is df (x) Df (x) = = f (x) dx (1) Now we define a new operator H defined as D = H 1/α, α < 1 (2) The problem arising becomes Hf (x) = D α f(x)=? (3) This is the original idea of the fractional calculus. Leibniz s calculus is quite different from that of Newton because Leibniz did not take the limit in his infinitesimal calculus. The derivative of f(x)with respect to x, in the sense of Leibniz s notation, is the standard part of the infinitesimal ratio: f (x) = st( f x ) = st(f(x 1) f(x 2 ) ) x 1 x 2 (4) instead of f (x) = lim x 0 y x. Leibniz s definition is very close to the definition of a fractal derivative (see Section 5) and q-derivative (see Section 6). In a fractal medium, the distance between x 1 and x 2 tends to infinity ( x ) even when x 1 x 2, and therefore Leibniz s work was nearer to fractal and Cantor sets which are the basis for fractional calculus. Although the fractional calculus was invented over three centuries ago, it only became a hot topic recently owing to the development of the fractal theory, computer science, effective analytical methods (e.g. the variational iteration method, the homotopy perturbation method) and its exact description of many real-life problems. 2 Dark Energy in Fractal Spacetime Before introducing the basic properties of fractional calculus, we would like to give a very brief introduction to recent findings on dark energy by El Naschie [1 8]. In recent years, based on Finkelstein s quantum sets [9, 10], fractal and fractional calculus have had quite a triumph in high energy physics, and all conceivable applications in science and technology as well, especially in as nanotechnology [11 15]. Some of the most fundamental theories can be explained with considerable ease and elegance using fractal and fractional calculus [16, 17]. Dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Despite the great success and undeniable brilliance of the standard model of high-energy physics, it is fair to say that it is by no means perfect. Now El Naschie [1 4] has changed the situation radically by using a simple concept of fractal-cantorian spacetime. According to E-infinity theory, Prof. El Naschie revealed that dark energy currently accounts for about 95.5 % of the total mass energy of the universe [1 4]. In mathematics there is something called quantum calculus (q-calculus, see Section 6 for detailed discussion) [18, 19]. This is deeper than fractional calculus. Ord [20, 21]uses his own kind of fractional calculus. The problem is taking the limit which falsifies nature because it gives a wrong impression of smooth differentiable geometry while nature is non smooth, non differentiable Cantorian fractal [16, 17].

6 3700 Int J Theor Phys (2014) 53: Einstein s mass-energy equation is derived under the assumption of absolute smooth spacetime. In reality spacetime is intrinsically discontinuous when it tends to a quantum scale [22, 23] and a Hilbert cube can excellently model the actual fractal spacetime [24]. To show discontinuity of spacetime, we consider a TV screen which is smooth at any observable scales. However, when the scale tends to a very small one, the surface becomes unsmooth and consists of many arrayed pixels. Time is also discontinuous when it is extremely small. A film gives 24 slips per second, this gives a continuous movement. However, for instance in the case of 10 slips per second, the movement becomes discontinuous. When spacetime becomes discontinuous, fractal theory can be adopted to describe various phenomena. According to El Naschie s E-infinity theory [16, 17], spacetime is a random Cantor set with Hausdorff dimension of instead of ln2/ln3, see Fig. 1. When we construct a Cantor set, whether deterministic or random, we end up with two Cantor sets. The first Cantor set is measure zero and we call it a thin Cantor set. What is left of the unit interval is on the other hand another Cantor set but it is measured one and any Cantor set which is not measure zero is called a fat Cantor set [25]. These two Cantor sets are the basis of KAM theorem and E-infinity theory [26]. The measure zero Cantor set represents the quantum particle. It is the zero set and its Hausdorff dimension is the golden mean, ϕ = ( 5 1)/2. The fat Cantor set measure one on the other hand is the empty set. Its Hausdorff dimension is the golden mean square, ϕ 2 = 1 ϕ. ThefatCantor set models the quantum wave but it also models quantum spacetime itself [4]. The average Hausdorff dimension of our spacetime is [16, 17] 4 + ϕ 3 = = 1 = (5) ϕ3 when it tends to quantum scale and φ 3 is the correction of continuous spacetime [16, 17]. This means 4 dimensional spacetime is a rough approximation and we generally require 5 dimensional Kaluza-Klein spacetime [27]. Consider a quasi-hausdorff hyper volume of the 5 dimensional Kaluza-Klein spacetime, which consists of the zero set of quantum particles, and the empty set of quantum waves [1 4]. The quantum particles are not occupied within a 1*1*1*1*1 hyper volume but a ϕ ϕ ϕ ϕ ϕ fractal hyper volume. That means quantum particles have energy [1 4] ϕ ϕ ϕ ϕ ϕ 1 E = lim v c mv2 = 1 2 ϕ5 mc 2 (6) This is El Naschie s meantime equation [1 4] predicting that % of the energy in the cosmos is the missing hypothetical dark energy. El Naschie s theory combines Newton s mechanics with quantum mechanics. Fig. 1 Cantor set with Hausdorff dimension of ln2/ln3

7 Int J Theor Phys (2014) 53: What is Fractional Calculus To reveal the basic idea of the fractional calculus in a layman way, we consider the following simplest equation y = x (7) The first-order derivative is y (1) = 1 (8) and the zero-th order derivative is defined as y (0) = x (9) Now the question is what it is when order is α(0 <α<1). Hereby y (α) can be explained the gradient of y on the fractal media (see Fig. 2), which might be continuous but nondifferential anywhere, and have typically self-similar patters, as illustrated in Fig. 3 for the stock movement, the price change may be nearly the same at different scales (see the red pattern in Fig. 3). Any a fractal medium has certainly a fractal boundary, however, a differential equation can not be established for a fractal boundary. Consider a stead flow in continuum mechanics, the mass conversation follows that ρu ds = 0 (10) S where u = (u, ν, w), ρ is the density. According to Gauss theory, (10) becomes (ρu)dv = 0 (11) V where S is the boundary of V. From (11) the governing equation for the mass conversation is obtained, which is (ρu) = 0 (12) or x (ρu) + y (ρν) + (ρw) = 0 (13) z Fig. 2 The fractional derivative of y = x. The curve is for α = 1/2

8 3702 Int J Theor Phys (2014) 53: Fig. 3 Fractal movement of stock. The above zigzag is the self-similar patter In the above derivation, we assume that the boundary (S) of the volume (V) is smooth enough, this is absolutely required in continuum mechanics. However, such an assumption is forbidden for many practical problems, e.g., porous flows [28 35], flows in carbon nanotube [36 38](Fig.4) and nanoscale flow and heat conduction and electric current [39 43]. Generally we call fractal materials as El Naschie materials, where fractal space or fractal time or fractal spacetime has to be adopted. Before introducing the basic properties of fractional calculus, we consider first an interesting experiment. Majumder et al. [36] found that liquid flow through a membrane composed of an array of aligned carbon nanotubes is 4 to 5 orders of magnitude faster than would be predicted from conventional fluid-flow theory, similar phenomena were observed by other researchers [37, 38]. Why does the fluid in nano-tubes flow extraordinary fast? We will give hereby a heuristical explanation using the basic idea of fractional calculus. Fig. 4 Fractal boundary of carbon nano-tube

9 Int J Theor Phys (2014) 53: Governing equations for continuum media are well established by using the Gauss divergence theorems: φdv = n φds (14) V V AdV = n AdS (15) V V AdV = n AdS (16) V Mathematically the above Gauss divergence theorems are invalid for fractal media, such as porous media and weaves. So the governing equations for non-continuum media should be derived by fractal approach or fractional derivatives [31 33]. We write the conversation of mass in continuum media in the form 1 ρurdl = Q (17) 2 where Q is the flow rate. For the continuum media, we have u = Q πr 2 (18) ρ For nano scale hydrodynamics, for example, the flow in carbon nanotubes, the perimeter of a section is of fractal [34]: l NH = kl α (19) where α is the fractal dimension of the perimeter, k is a constant depending upon α, k = 1 when α = 1. We re-write (17) in the form 1 Q = 2 ρurdl NH = 1 2 ρkur(dl)α = According to the fractal integral [34], (20) can be approximately calculated as V 1 2 ρkur(1+α) dθ α (20) Q 1 2 ρkur(1+α) (2π) α (21) So the velocity in discontinuous carbon nanotubes can be expressed as 2Q u NH = k(α)(2π) α ρr (1+α) (22) Comparing (22) with (18), we find u NH u 1 r α 1 (23) We consider a case as illustrated in Fig. 4, the fractal dimension can be calculated as α = ln 2 ln( = 1.26 (24) 3) In case r tends to be extremely small, saying tens of nanometers, we can predict a large value for u NH compared with its continuum partner, which agrees with Majumder et al. s experimental observation [36]; while at any observable scales, the zigzag boundary disappears, and it becomes a continuum model and α = 1.

10 3704 Int J Theor Phys (2014) 53: Definition of the Fractional Derivatives There are many definitions on fractional derivatives. A systematical study of various fractional derivatives is given by Yang [44 46]. Hereby we will introduce the basic properties of fractional derivatives by the variational iteration method [47 51]. The variational iteration method [47 51] has been shown to solve a large class of nonlinear differential problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions. In 1998, the method was first adopted to solve fractional differential equations [28], and now it has matured into a relatively fledged theory for various nonlinear problems [52 56]. A complete review on its development and its application is available in Refs. [57, 58]. We consider the following linear equation of n-th order u (n) = f(t) (25) By the variational iteration method [47 51], we can construct the following iteration formulation t u m+1 (t) = u m (t) + λ(u (n) m (s) f m(s))ds, (26) t 0 After identifying the multiplier, we have t u m+1 (t) = u m (t) + ( 1) n 1 [ ] t 0 (n 1)! (s t)n 1 u (n) m (s) f m(s) ds. (27) For a linear equation, from (27), we have the following exact solution t u(t) = u 0 (t) + ( 1) n 1 [ ] t 0 (n 1)! (s t)n 1 u (n) 0 (s) f(s) ds. (28) where u 0 (t) satisfies the boundary/initial conditions. Introducing an integration operator I n defined by I n f = t 1 [ ] t 0 (n 1)! (s t)n 1 u (n) 0 (s) f(s) ds = 1 t (s t) n 1 [f 0 (s) f(s)] ds (29) Ɣ(n) t 0 where f 0 (t) = u (n) 0 (t). Definition 1 We can define a fractional derivative in the form D α t f = Dα t d n dt n (I n f) = dn dt n (I n α f)= d n 1 Ɣ(n α) dt n t (s t) n α 1 [f 0 (s) f(s)] ds t 0 (30) For a continuous and differentiable f,wehave f 0 (t) = f(t 0 )+(t t 0 )f (t 0 )+ 1 2 (t t 0) 2 f 1 (t 0 )+ + (n 1)! (t t 0) n 1 f (n 1) (t 0 ) (31) If f 0 (t) is continuous but not differentiable anywhere, we have f 0 (t)=f(t 0 )+ (t t 0) Ɣ(1+α) f (α) (t 0 )+ (t t 0) 2 Ɣ(1+2α) f (2α) (t 0 )+ + (t t 0) n 1 Ɣ(1 + (n 1)α) f ((n 1)α) (t 0 ) (32) where f (nα) (t) = Dt α Dα t Dt α f(t). }{{} n times

11 Int J Theor Phys (2014) 53: Equation (30) is equivalent to d n t Dt α f = 1 Ɣ(n α) dt n (s t) n α 1 t 0 for continuous and differentiable case; d n t Dt α f = 1 Ɣ(n α) dt n (s t) n α 1 t 0 for continuous and nondifferentiable case. [ n 1 i=0 [ n 1 i=0 ] 1 i! (s t 0) i f (i) (t 0 ) f(s) ds (33) ] (s t 0 ) i Ɣ(1 + iα) f (iα) (t 0 ) f(s) ds (34) Definition 2 Keeping only the first term off 0 (s), we give another definition of fractional derivative in the form Dt α f = 1 d n t Ɣ(n α) dt n (s t) n α 1 [f(t 0 ) f(s)] ds (35) t 0 Hereby f can be continuous and possibly not differentiable anywhere. (27) is the variational iteration algorithm-i [50 52]. The variational iteration algorithm- II [50 52]is t u m+1 (t)=u 0 (t) ( 1) n 1 t 0 (n 1)! (s t)n 1 f m (s)ds=u 0 (t) ( 1)n t (s t) n 1 f m (s)ds Ɣ(n) t 0 Note: u 0 must satisfy the initial/boundary conditions. For a linear equation, we have u(t) = u 0 (t) ( 1)n Ɣ(n) t (36) t 0 (s t) n 1 f(s)ds (37) Definition 3 We can define another fractional derivative in the form Dt α f(t)= 1 d n t Ɣ(n α) dt n (s t) n α 1 f(s)ds (38) t 0 The above definitions of fractional derivative are introduced by the variational iteration method [46 51].There are other definitions. Definition 4 Caputo fractional derivative is defined as [44 46] D α x (f (x)) = 1 Ɣ(n α) x 0 (x t) n α 1 dn f(t) dt n dt (39) Definition 5 Riemann-Liouville fractional derivative reads [44 46] Dx α (f (x)) = 1 d n x Ɣ(n α) dx n (x t) n α 1 f(t)dt (40) Caputo derivatives are defined only for differentiable functions, while f can be a continuous (but not necessarily differentiable) function. Riemann-Liouville definition can be used any functions that are continuous but not differentiable anywhere, however, Dx α (f (x)) = 0 when f(x)is a constant. 0

12 3706 Int J Theor Phys (2014) 53: Definition 6 To overcome the shortcomings, Jumarie [59 62] suggested the following modification of Riemann-Liouville fractional derivative Dx α (f (x)) = 1 d n x Ɣ(n α) dx n (x t) n α 1 [f(t) f(0)]dt (41) 0 where f is a continuous (but not necessarily differentiable) function. Jumarie s fractional derivative meets the following simple rules: α c x α = 0, α x α [cf ]=c α f x α, α x β Ɣ(1 + β) = xα Ɣ(1 + β α) xβ α,β α>0 (42) The chain rule for all above derivatives is complex. Definition 7 Recently the local fractional derivative has attracted much attention due to its simple chain rule which is defined as [44 46] f (α) (X 0 ) = dα f(x) dx α α (f(x) (f x 0 )) = lim x x x=x0 0 (x x 0 ) α (43) where α (f (x) f(x 0 )) = Ɣ(1 + α) (f(x) f(x 0 )). The local fractional derivative has the same properties as given in (42), and the following simple chain rules: { ktimes }} { kα f(x) α x kα = x α α f(x) xα (44) α f(g(x)) x α = f (1) (g(x))g (α) (x) (45) 5 Fractal Derivative The fractal derivative can be categorized as a special local fractional derivative. There are also several definitions. Definition 8 Chen s definition is as follows [63, 64] du(x) dx D = lim u(x) u(s) s x x D s D, (46) whered is the order of the fractal derivative. This definition is much simpler but lacks physical understanding. Now we consider a fractal media illustrated in Fig. 5, and assume the smallest measure is L 0, any discontinuity less than L 0 is ignored, then the distance between two points of A and B in Fig. 5 can be expressed using fractal geometry. Definition 9 Hereby we introduce another fractal derivative in the form [65] Du Dx α = lim u(a) u(b) x L 0 kl α 0 (47)

13 Int J Theor Phys (2014) 53: Fig. 5 The distance between two points in a discontinuous spacetime where kis a constant, α is the fractal dimension and the distance between two points in a discontinuous space can be expressed as ds = kl α 0,wherek is a function of fractal dimension, k = k(α), and it follows that k(1) = 1. Definition 10 The fractal derivative can be defined alternatively as Du = Ɣ(1 + α) Dxα lim u(a) u(b) x=x A x B L 0 (x A x B ) α (48) Applications of the fractal derivative to fractal media have attracted much attention, for example it can model heat transfer and water permeation in multi-scale fabric and wool fibers [31 33]. Considering a heat conduction in a fractal medium we have the following equation using the fractal derivative T t To solve this equation, we introduce a transformation + D DT D α (k x D α x ) = 0 (49) s = x α (50) This transformation is similar to the fractional complex transform (see Section 7. As a result, (49) is converted to a partial differential equation which is T t + T (k s s ) = 0 (51) This equation can easily be solved by the advanced calculus. It should be pointed out that T is local continuous when the scale is larger than x = L 0,whereL 0 is the smallest measure.

14 3708 Int J Theor Phys (2014) 53: Q-Derivative Q-derivatives are part of so called quantum calculus [18, 19]. In quantum scales, spacetime becomes discontinuous and Leibniz s notation of derivative is especially useful in quantum calculus without limits. Definition 11 Let f(x)be a real continuousfunction.the q-derivative is defined as [66, 67] d q f(qx) f(x) f(x)=,x = 0, d q x (q 1)x 0 <q<1 (52) Hereby q originally stands for quantum q = e ih, (53) where h is Planck s constant. In the limit, as q goes to 1, (52) takes on the form of the derivative of classical calculus. We thus have the following q-leibniz product law d q d q x [g(x)f(x)]=g(qx) d q d q x [f(x)]+f(x) d q [g(x)]. (54) d q x and q-integration by parts b a g(qt) d q d q t f(t)d qt = f(t)g(t)/ b a b a f(t) d q d q t g(t)d qt. (55) Recently the application of the variational iteration methods [47 51] to q-difference equations has had much attention [66 68] and it still leaves much space for further development. Wu considered the following q-difference equation [66] dq 2u d dq 2 u = 0,u(0) = 1, t where the q-difference is defined as d q u d q t = u(qt) u(t) (q 1)t q u(0) = 1 (56) d q x Using the variational iteration method [47 51], Wu obtained the following iteration formulation [66] [ ] t d 2 u n+1 (t) = u n (t) + q 1 (q 2 q u n s tq) 0 dq 2s u n ds = 0 (58) This iteration algorithm is much better than that constructed by Liu [68], however it can be further improved. According to the variational iteration algorithm-ii [50, 51, 57], Wu s algorithm can be updated as If we begin with u n+1 (t) = u 0 (t) t 0 u 0 = 1 + (57) q 1 (q 2 s tq)u n ds = 0 (59) t [1] q! (60)

15 Int J Theor Phys (2014) 53: By (59), we have and u 1 = 1 + t [1] q! + t2 [2] q! + t3 [3] q! u n = 2 n +1 k=0 t k [k] q! where [n] q is the q-bracket. Equation (62) is exactly same as that obtained by Wu [66]. For any q-difference equations of second orders dq 2u dq 2 f(u)= 0 (63) t we have three iteration algorithms [50, 51, 57]: Variational iteration algorithm-i: t ( ) [ u n+1 = u n + q 1 q 2 dq 2 s tq u ] n(s) f(u n (s)) ds (64) 0 d q s Variational iteration algorithm-ii: t ( ) u n+1 = u 0 q 1 q 2 s tq f(u n (s))ds (65) Variational iteration algorithm-iii: u n+1 = u n + t 0 0 (61) (62) q 1 ( q 2 s tq) [f(un 1 (s)) f(u n (s)) ] ds (66) The variational iteration algorithm-ii is recommended for q-calculus to avoid unnecessary repeated calculation. Other iteration formulae for q-difference equations are summarized in Ref. [57]. 7 Analytical Approaches to Fractional Calculus There are various analytical methods for fractional calculus, among which the variational iteration method [28, 50, 56, 69 75], the homotopy perturbation method [57, 76 92], the fractional complex transform [93 96], Yang-Laplace transform [44 46], and Yang-Fourier transform [44 46] have received much attention. Hereby we illustrate the basic solution processes by the fractional complex transform, Yang-Laplace transform, exp-function method and variational iteration method. 7.1 Fractional Complex Transform The fractional complex transform was first proposed in 2010 [94] to convert fractional differential equations into ordinary differential equations. In previous applications [93 96], Jumarie s modification of Riemann-Liouville fractional derivatives was adopted. However, a counter-example was found [93], making the method much skeptical. The main problem for its applications is how to define the fractional derivative. Hereby we find that the previous demerit can be completely eliminated when the local fractional derivative [44 46]is used.

16 3710 Int J Theor Phys (2014) 53: Example Consider the local fractional differential equation in the form d α U 1 (x) dx α + dα U 2 (y) dy α = 0, 0 <α 1 (67) By the fractional complex transform [93 96] { X = (px) α Y = Ɣ(1+α) (qy)α Ɣ(1+α) where p and qare constants. (67) turns out to be the following ordinary differential equation p α du 1 (X) dx + du qα 2 (Y ) = 0 (69) dy Example 2 U 2 (x,y) x 2 + 2α U 1 (x,y) x α y α + 2α U 1 (x,y) y α x α Similarly by the fractional complex transform (68), we have (68) + 2α U 2 (x,y) y 2α = 0 (70) 2α U 1 (x,y) x 2α = 2 U 1 (X, Y ) α X X 2 x α + 2 U 1 (X, Y ) α Y Y 2 x α = p2α 2 U 1 (X, Y ) X 2. (71) 2α U 2 (x,y) y 2α = q α 2 U 2 (X, Y ) α Y Y 2 y α + qα 2 U 2 (X, Y ) α X Y 2 y α = q2α 2 U 2 (X, Y ) X 2, (72) 2α U 1 (x,y) y α x α = p α 2 U 1 (X, Y ) α Y X Y y α + pα 2 U 1 (X, Y ) α Y X Y x α = qα p α 2 U 1 (X, Y ) Y X, (73) 2α U 2 (x,y) α x y α = q α 2 U 2 (X, Y ) α X Y X x α + qα 2 U 2 (X, Y ) α X X X y α = pα q α 2 U 2 (X, Y ), (74) X Y Using the above relations, we convert (69) into the following one p 2α 2 U 2 (X, Y ) X 2 + p α q α 2 U 1 (X, Y ) + q α p α 2 U 2 (X, Y ) + q 2α 2 U 2 (X, Y ) X Y Y X Y 2 = 0 (75) 7.2 Exp-Function Method for Fractional Differential Equations The exp-function method [97 99] is routinely employed to search for solitary solutions for variousnonlinear equations [ ]. However, its application to fractional calculus is rare and primary. Zhang and his colleagues [100] suggested a fractional exp-function method with help of Mittag-Leffer function which is however elusive to non-mathematicians. In this paper we will suggest a standard solution procedure for fractional differential equations using the exp-function method. Consider a fractional nonlinear wave equation in the form [99] 2α u t 2α 2β u x 2β + u u3 = 0, 0 <α,β<1, (76) where α u/ t α denotes the local fractional derivative of order α with respect to t.

17 Int J Theor Phys (2014) 53: The first step to solve a fractional differential equation by the exp-function method is to convert the equation into its differential partner by the fractional complex transform [93 96]: t α T = Ɣ(1 + α), X = x β (77) Ɣ(1 + β) (76) becomes the Phi-four equation [99]: 2 u T 2 2 u X 2 + u u3 = 0 (78) (78) can be easily solved by the exp-function method [93 96]. Using a transformation ξ = kx + wt (79) (78) becomes an ordinary differential equation: (w 2 k 2 )u + u u 3 = 0. (80) Using the exp-function method [93 96], we assume that the solution of (80) can be expressed as u(ξ) = a 1 exp(ξ) + a 0 + a 1 exp( ξ) exp(ξ) + b 0 + b 1 exp( ξ), (81) where a i,b i (i = 1, 0, 1) are unknown parameters. Substituting (81)into(80), collecting terms of the same term ofexp(iη),wehave 1 C i exp(iξ) = 0. (82) A i Equating coefficients, C i to zero yields a series of linear equations C i = 0. (83) Solving the system of algebraic equations, we can identify parameters in (81). We write one solution here u(x, t) = exp( k Ɣ(1+β) xβ + exp( k Ɣ(1+β) xβ + where b 0 is a free parameter. 7.3 Yang-Laplace Transform w Ɣ(1+α) tα ) 1 4 b2 0 exp( w Ɣ(1+α) tα ) + b 0 + b2 0 4 exp( Ɣ(1+β) k xβ Ɣ(1+α) w tα ) Ɣ(1+β) k xβ Ɣ(1+α) w tα ). (84) The Yang-Laplace transform [44 46] is an effective mathematical tool for solving local fractional differential equations. Consider the following local fractional differential equation with initial condition y(t) t=0 = 0. By the Yang-Laplace transform, (85) becomes y α (t) + 2y(t) = E α ( t α ), 0 <α 1, (85) Solving y L,α s from (86), we have s α y L,α s + 2y L,α s = s (86)

18 3712 Int J Theor Phys (2014) 53: ys L,α (ω) = s α s α (87) The inverse Yang-Laplace transform of (87)gives f(x)= E α ( t α ) E α ( 2t α ) (88) Consider another local fractional differential equation given by α u y α + kα u = 0,y >0, 0 <α 1,u(0) = 1. (89) Taking the Yang-Laplace transform we have s α ũ (s) + k α ũ (s) u (0) = 0. (90) It is obvious that 1 ũ (s) = s α + k α. (91) The inverse Yang-Laplace transform results in the following solution ( u (y) = E α k α y α). (92) 7.4 Variational Iteration Method As early as 1998, the variational iteration method was shown to be an effective tool for factional calculus [28], afterwards, the method was routinely used to solve various fractional differential equations for many years (see the review articles [57, 58] for a detailed summarization). We first consider a nonlinear differential equation in the form u (n) (t) + f(u,u, u (n) ) = 0 (93) Its variational iteration formulation [47 51] can be readily obtained, which is u m+1 (t) = u m (t) + ( 1) n [ ] t 1 t 0 (n 1)! (s t)n 1 u (n) m (s) + f m (s) ds [ ] = u m (t) + ( 1)n t Ɣ(n) t 0 (s t) n 1 u (n) (94) m (s) + f m (s) ds where f m (s) = f m (u(s), u (s), u (n) (s)). Now we generalize (93) into its fractional partner: Dt α u + f = 0 (95) Its variational iteration formulation can be obtained as t u n+1 (t) = u n (t) + ( 1)α (s t) α 1 (D α u n (s) + f n (s))ds (96) Ɣ(α) 0 As an example, we consider the relaxation oscillator equation Dt α u + ωα u = 0, u(0) = 1, u (0) = 0, t>0, 0 <α<2, ω>0, (97) with the exact solutione α (( ωt) α )wheree α (( ωt) α )denotes the Mittag-Leffler function. By (96), we have the following iteration formulation u n+1 (t) = u n (t) + ( 1)α t (s t) α 1 (D α u n (s) + ω α u n (s))ds (98) Ɣ(α) 0

19 Int J Theor Phys (2014) 53: We begin with u 0 (t) = 1, by (96), we have ωα t α u 1 (t) = 1 Ɣ(1 + α), (99) u 2 (t) = 1 ωα t α Ɣ(1 + α) + ω2α t 2α (100) Ɣ(1 + 2α) and other components can be easily obtained, it is obvious that u n (t) rapidly tends to the exact solution for n tends to infinity. 7.5 Homotopy Perturbation Method for Fractional Calculus The homotopy perturbation method was originally proposed to nonlinear differential equations [76 79]. In 2007 Momani and Odibat adopted the method for fractional differential equations with great success [80] and now it is an effective method for fractional calculus [81 92]. In 2010, the Laplace transform and He s polynomials are used in the homotopy perturbation method [81]. The Laplace transform is a well-known mathematical tool for linear equations while it cannot deal with the nonlinear terms. In order to perform the inverse Laplace transform, He s polynomials [ ] are widely applied. At present the homotopy perturbation method coupled with the Laplace transform has been widely used to solve fractional differential equations [80 83, ]. Consider a following fractional differential equation: D α u(x, t) + Ru(x,t) + Nu(x,t) = q(x,t), (101) where D α is the fractional derivative,r is the linear operator, N represents the general non-linear differential operator and q(x,t) is continuous and exponential order function. Applying Laplace transform to (101), we have L { D α u(x, t) + Ru(x, t) + Nu(x,t) q(x,t) } = 0 (102) In order to solve (102), we construct a homotopy equation in the form [76 79] L { D α u + Ru q } + pl {Nu} = 0 (103) In order to make the solution process simple, He s polynomials are included [107, 108] Nu = p i H i (u) (104) i=0 where H i (u) are He s polynomials. The solution is expanded into a series of p in the form u = u 0 + pu 1 + p 2 u 2 + (105) The solution process is same as that by the classical perturbation method. It was Gondal and Khan [81] who took the first step to couple the homotopy perturbation method and the Laplace transform. Afterwards, the modified homotopy perturbation method coupled with the Laplace transform became a hot topic in analytical methods, see for examples, Refs. [82, 83, ]. The names for the modification of the homotopy perturbation method include 1) Laplace homotopy perturbation method [114], 2) homotopy perturbation transform method [83, 115], 3) He-Laplace method [116], 4) A coupled method of homotopy perturbation method and Laplace transform and others. All the above names mean the same technology and it is now widely applied for fractional calculus and it is recommended to the terminology by Mishra and Nagar [116] or the modified homotopy perturbation method.

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