New Solutions of Three Nonlinear Space- and Time-Fractional Partial Differential Equations in Mathematical Physics

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1 Commun. Theor. Phys. 6 (014) Vol. 6 No. 5 November New Solutions of Three Nonlinear Space- Time-Fractional Partial Differential Equations in Mathematical Physics YAO Ruo-Xia ( ) 1 WANG Wei ( ) 1 CHEN Ting-Hua ( ) 1 1 School of Computer Science Shaanxi Normal University Xi an China Information Education Technology Center Xi an University of Finance Economics Xi an China (Received March ; revised manuscript received June ) Abstract Motivated by the widely used ansätz method starting from the modified Riemann Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations new types of exact traveling wave solutions to three important nonlinear space- time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new first reported in this paper. PACS numbers: 04.0.Jb Key words: modified Riemann Liouville derivative fractional complex transformation nonlinear space- time-fractional partial differential equations traveling wave solution 1 Introduction In the late 17th century the creator of modern calculus Leibniz made some remarks on the meaning possibility of fractional derivative of order α = 1/. However a rigorous investigation was first fulfilled by Liouville in a number of papers from 183 to 1837 where he defined the first preform of an operator of fractional integration. The concept of fractional calculus is by no means new which has more than 300 years history has been widely studied. [1 3] Fractional calculus has been applied to almost every field of science such as electrical engineering electrochemistry biology biophysics bioengineering signal image processing mechanics physics control theory. In the last few decades scientists engineers realized that differential equations with fractional derivative provided a kind of natural framework for the discussion of various kinds of real problems modeled by the aid of fractional derivative. In general the fractional analogues are obtained via changing the classical time derivative by a fractional one which can be Riemann Liouville Caputo Hadamard or another one. For example the time fractional advection-dispersion equation could be obtained from the stard advection-dispersion equation by replacing the first order derivative in time by a fractional derivative in time of order α (0 < α < 1). [4] As valuable mathematical tools the great use of fractional calculus special functions in important physical mathematical problems has attracted more more attention of physicists mathematicians in solving the problems coming from mathematics physics electronics astrophysics other scientific fields. Recently it turns out that many models are formulated in terms of fractional derivatives. Fractional differential equations (FDEs) can be considered as the generalization form of the differential equations. We know that most of the FDEs do not have exact analytical solutions. However it is worthy to be mentioned that in the sense of fractional Caputo or Riemann Liouville derivatives the solutions of the FDEs are analytical. Different from classical or integer-order derivative there are several kinds of definitions for fractional derivatives which are generally not equivalent with each other. Furthermore the Riemann Liouville derivative has certain disadvantages in dealing with real-world phenomena with fractional differential equations. Therefore one may use one of the modifications of some fractional differential operators. In the following we introduce the definitions of the Caputo the Riemann Liouville the Jumarie s modified Riemann Liouville derivatives first. Then based on the method given in [5] we study the nonlinear space time-fractional Korteweg devries Burgers (KdVB) equation to obtain its new exact traveling wave solutions. As a matter of fact the exact traveling wave solutions to the space- time-fractional Burgers KdV equations are obtained simultaneously due to the relations among of them. Specially in Case 6 of the Subsec. 3. we point out that the solutions to the space- time-fractional KdV equation given in [6] were actually not the solutions of it. Preliminaries The Riemann Liouville fractional derivative the Caputo derivative their modification versions play important roles in many areas of science engineering Supported by National Natural Science Foundation of China under Grant Nos the Fundamental Research Funds for the Central Universities of GK GK rxyao@hotmail.com; rxyao@snnu.edu.cn c 014 Chinese Physical Society IOP Publishing Ltd

2 690 Communications in Theoretical Physics Vol. 6 mathematics. Some definitions of the above several fractional derivatives their properties are given below. As usual we use Γ(z) to represent the Gamma function which is an extension of the factorial function to complex real number arguments defined by Γ(z) = 0 e t t z 1 dt Re(z) > 0. RLD α axf(x) = dα f(x) d(x a) α = 1 Γ(n α) For all z with Re(z) > 0 the formula Γ(z + 1) = zγ(z) holds. For positive integer n one has Γ(n) = (n 1)!. Riemann Liouville fractional derivative of order α of a real function f(x) i.e. a generalization of the Cauchy formula for repeated integration to arbitrary order is given by [7] d n x dx n a where x > a n Z +. The Caputo derivative of fractional order α of function f(x) is defined as [8 9] CD α axf(x) = D α n ax d n dx n f(x) = 1 Γ(n α) x a f(τ) dτ n 1 α < n (1) (x τ) α n+1 f (n) (τ) (x τ) α n+1 dτ n 1 < α < n Z+. () In Eqs. (1) () when α = n Daxf(x) α = (d n /dx n )f(x). An alternative to the Riemann Liouville definition of fractional derivative of order α is defined by For positive α the following equality holds f (α) (x) = dα f(x) 1 dx α = Γ( α) f (α) (x) = D α x f(x) = (f (α 1) (x)) = { 1 Γ(1 α) x 0 f(x) f(0) dτ α < 0. (3) (x τ) α+1 d x dx 0 (x τ) α (f(τ) f(0))dτ 0 < α < 1 (f (n) (x)) α n n α < n + 1 n 1 which is the Jumarie s modified Riemann Liouville derivative. [10 11] Furthermore we list some known properties of the modified Riemann Liouville derivative just for reference. Dx α x γ = α x γ Γ(γ + 1) = xα Γ(γ α + 1) xγ α γ > 0 (u(x)v(x)) (α) = α (u(x)v(x)) x α = u (α) (x)v(x) + u(x)v (α) (x) (f[h(x)]) (α) = α f[h(x)] x α = f h[h(x)]d x α h(x) = f h[h(x)]h (α) (x) (5) which are direct consequences of the equality d α x(t) = Γ(1 + α)dx(t) where h : [a b] R is a function of class C n+3 on [a b] f is a function of class C n on h[a b] D (α n) ± [f (n) (h(x))] exists. 3 Method Solutions In this section we aim to present the solving procedure of traveling wave solutions to nonlinear space- timefractional partial differential equations (PDEs). The key idea lies in using variable transformations together with the Jumarie s modified Riemann Liouville derivative its properties to derive new traveling wave solutions to the space time-fractional KdVB Burgers KdV equations. Consider the nonlinear space- time-fractional KdVB equation [1] α u t α + u u β x β + η β u x β + ν 3β u = 0 x3β t > 0 0 < α β 1 (6) where local fractional partial derivative of high order is defined as [13] k times {}}{ kα f(x) α x kα = x α α f(x). (7) xα Besides (7) local fractional derivative of high order is defined as k times {}}{ f (kα) = D (α) x D (α) x f(x). Equation (6) has been used to model the propagation of waves on an elastic tube filled with a viscous fluid. [14] For Eq. (6) if set η = 0 it can be treated as the space- time-fractional KdV equation α u t α + u u β x β + ν 3β u = 0. (8) x3β (4)

3 No. 5 Communications in Theoretical Physics 691 If set ν = 0 it is the space- time-fractional Burgers equation where t > 0 0 < α β Method α u t α + u u β x β + η β u = 0 (9) xβ In this subsection we illustrate the procedure for solving fractional PDEs to obtain their exact traveling wave solutions. Step 1 Convert Fractional PDE to ODE Consider the following complex variable transformation [15] u(t x y...) = U(ξ) ξ = L tα Γ(α + 1) + K xβ Γ(β + 1) + M yγ Γ(γ + 1) + (10) where K L M are non-zero arbitrary constants which allow us to transform monomial appearing in (6) like ( β / x β )u(x t) to form K (d /dξ )U(ξ). In detail it is obtained after the following direct computations by properties given in (5). β u ( β β u ) ( = xβ x β x β = β x β (U Dxξ(x β t)) = = β x β K U β! ) = = K U. (11) Γ(1 + β) Then using the above complex variable transformation after integrating once to reduce Eq. (6) to an ordinary differential equation (ODE) of u(t x) = U(ξ) with the form C + LU + 1 KU + η K U + νk 3 U = 0 (1) where C is an integral constant ξ = L tα Γ(β + 1). Step Introduce an Auxiliary ODE to Determine Solution Expression of PDE First we suppose that Eq. (1) has solution with the form k U(ξ) = a i ψ i a k 0 (13) i= k where the positive integer k = that is determined using the homogeneous balancing method. [16 18] This value k is also referred to the order of a pole for the solution of Eq. (1). The function ψ is referred to ψ(ξ) = G /G G(ξ) satisfies the following linear ODE equation Furthermore using the relation of ψ(ξ) = G /G Eq. (14) is changed into G = µg λg µ λ are constants. (14) Solving Eq. (14) yields the following solution [( G(ξ) = Aexp λ + λ 4 µ ) ] [( ξ + Bexp λ where A B are integral constants. Then we get ψ = µ ψ λψ. (15) λ 4 µ ) ] ξ (16) ψ(ξ) = 1 A(λ λ 4 µ) e (ξ/) λ 4 µ + B(λ + λ 4 µ) e (ξ/) λ 4 µ A e (ξ/) λ 4 µ + B e (ξ/). (17) λ 4 µ Equation (16) can be written in terms of hyperbolic function cosh as ( ( G(ξ) = A cosh (λ λ 4µ)ξ ) ( + cosh ξ λ 4µ ( ( + B cosh (λ + λ 4µ)ξ ) ( ξ λ 4µ + cosh which is equivalent to ( ( (λ λ 4 µ)ξ ) ( (λ λ 4 µ)ξ )) G(ξ) = A cosh sinh + ξλ + 1 ) )I Iπ + ξλ + 1 ) )I Iπ (18)

4 69 Communications in Theoretical Physics Vol. 6 Then Eq. (17) can be written as ( + B cosh ( (λ + λ 4 µ)ξ ) sinh ( (λ + λ 4 µ)ξ )). (19) ψ(ξ) = 1 P sinh((ξ/) λ 4 µ) + Q cosh((ξ/) λ 4 µ) (A + B) cosh((ξ/) λ 4 µ) + (A B) sinh((ξ/) λ 4 µ) (0) where P = (A + B) λ 4 µ (A B)λ Q = (A B) λ 4 µ (A + B)λ. Step 3 Determine the Unknowns To proceed it is time to determine the unknowns K L η ν a i (i = 0; ±1; ±). Substituting Eqs. (13) (15) into (1) using the relation of (15) to eliminate terms with derivatives. Then collecting all terms with the same order of ψ together setting each coefficient of ψ in the obtained polynomial to be zero then one can obtain a system of algebraic equations of the unknowns. Solving the linear algebraic system to determine the unknowns or find the relationships between them. Step 4 Put out the Solution Once the unknowns are at h the traveling wave solutions to space- time-fractional PDEs can be obtained immediately using (10) (13) (17) or (0). 3. Solutions As mentioned in Step 3 of the Subsec. 3.1 several types of solutions under various parameter constraints are obtained. Case Solution 1 ν 0. In this case four solitary wave solutions of the space- time-fractional KdVB Eq. (6) are obtained. The first set of the unknowns reads as a 1 = 1K ν( λ λ 4 µ) 1ν K a = η = 5ν K λ 4µ C = 7 K5 ν µ (λ 3 µ) + 6 K3 ν λ λ 4 µ(1 ν K µ + a 0 ) + 1 K 3 a 0 ν µ + 1 K a 0 L = K ( 6 ν K λ λ 4 µ + 1 ν K µ + a 0 ) a = a 1 = 0. (1) From the ansatz (13) together with (1) we get U(ξ) = a 0 1 ν K (λ + λ 4 µ) 1 ν K ψ(ξ) ψ (ξ) () where K is an arbitrary constant. When λ > µ we get the first traveling wave solution of the space- time-fractional Eq. (6) with the form u(x t) = P 1[cosh( ξ λ 4 µ)] + P sinh( ξ λ 4 µ) cosh( ξ λ 4 µ) + P 3 (3) [(A + B) cosh( ξ λ 4 µ) + (A B) sinh( ξ λ 4 µ)] ξ = K(6 ν K λ λ 4 µ + 1 ν K µ + a 0 ) t α Γ(α + 1) + K xβ P 1 = 1λνK [ (A + B ) λ 4µ (A B )λ ] + 4(3A B )νµk + a 0 (A + B ) P = 1λ νk [(A B ) λ 4 µ (A + B )λ] + (3A + B )4νµK + a 0 (A B ) P 3 = 1νµK (B 3A)(B + A) 6λ νk (A B) λ 4 µ + 6 λ νk (A + AB B ) a 0 (A B). (4) Solution The second set of the unknowns is a 1 = 1 K ν( λ + λ 4 µ) 1 ν K a = η = 5 ν K λ 4 µ C = 7 K5 ν µ (λ 3 µ) 6 K3 ν λ λ 4 µ(1 νk µ + a 0 ) + 1K 3 a 0 νµ + 1 Ka 0 L = K(6 ν K λ λ 4 µ 1 ν K µ a 0 ) a = a 1 = 0. (5)

5 No. 5 Communications in Theoretical Physics 693 From (13) together with (5) we get U(ξ) = a 0 1 ν K (λ λ 4 µ) ψ(ξ) where K is an arbitrary constant. When λ > µ the traveling wave solution of this case of Eq. (6) is Solution 3 where 1 ν K ψ (ξ) (6) u(x t) = (P 1 P )e ( λ+ λ 4 µ)ξ + AB[6ν K (λ µ λ λ 4 µ) + a 0 ] e ξ λ [ ] Aexp[( λ + λ 4 µ )ξ] + Bexp[( λ λ 4 µ (7) )ξ] ξ = K(6 ν K λ λ 4 µ 1 ν K µ a 0 ) t α Γ(α + 1) + K xβ P 1 = A [6 ν K (λ µ λ λ 4 µ) + a 0 ] P = B [6 ν K (λ 6 µ + λ λ 4 µ) a 0 ]. The third set of the unknowns is a 1 = 1 5 (η + η K ν µ)µ K a = 1 ν K µ C = (88 µ K3 ν η ν 3 K 5 µ η + 5K a 0ν η a 0 µ K 3 ν η + 1K a 0 η 3 )P 5ν(η + P ) + ν µ (900 a 0ν η + 65 ν a η 4 )K 3 5ν(η + P ) + η a 0 (5 a 0 ν + 1 η )K 5ν(η + P ) µ3 ν 5 K 7 + (1008 ν 3 µ η a 0 ν 4 µ )K 5 ν(η + P ) L = P K(6 η3 + 5 a 0 ν η K ν µ η) 5ν(η + P ) ν 4 K 5 µ + (900 ν µ η a 0 ν 3 µ)k 3 + (6 η a 0 ν η )K 5 ν(η + P ) λ = 1 η + η η K ν µ K ν µ 5 ν K(η + a 0 = a 0 a 1 = a = 0. (8) η K ν µ) From (13) together with (8) we get the following solution expression When U(ξ) = a 0 1 η Kµ + 1 µ K η K ν µ 5 P = η K ν µ. (9) 1 ψ(ξ) 1 ν K µ µ < λ/ = 1 η + η η K ν µ K ν µ 10 ν K(η + η K ν µ) the traveling wave solution of this case of Eq. (6) is of the form 1 ψ (ξ). A Q 1 e ( λ+q)ξ + Q e ξ λ B Q 3 e (λ+q)ξ u(x t) = 5 [A( λ + Q) e (( λ+q)/) ξ B(λ + Q) e ((λ+q)/) ξ ] (30) ξ = L tα Q 1 = 5 a 0 λ + 1 Kµ η Q 1 Kµ P Q 1 Kµ η λ + 1 Kµ P λ + 10 ν K µ + 5 a 0 λ Q + 10 a 0 µ (31) Q = 8 ABµ ( 5 a 0 6 Kη λ + 6 KP λ + 60 ν K µ) Q 3 = 5 a 0 λ 5 a 0 λ Q + 1 Kµ P λ 1 Kµ η Q + 1 Kµ P Q 1 Kµ η λ + 10 a 0 µ + 10 ν K µ (3) Q = λ 4 µ (33)

6 694 Communications in Theoretical Physics Vol. 6 K is an arbitrary constant L P are given in (8) (9) respectively. Solution 4 where The fourth set of the unknowns is a 1 = 1 ( η + η K ν µ)µ K a = 1 ν K µ 5 C = (88 µ K3 ν η ν 3 K 5 µ η + 5K a 0ν η a 0 µ K 3 ν η + 1K a 0 η 3 )P 5ν( η + P ) + ν µ (900 a 0ν η + 65 ν a η 4 )K 3 5ν( η + P ) + η a 0 (5 a 0 ν + 1 η )K 5ν(η + P ) µ3 ν 5 K 7 + (1008 ν 3 µ η a 0 ν 4 µ )K 5 ν( η + P ) L = P K(6 η a 0 ν η K ν µ η) 5 5ν( η + P ) ν 4 K 5 µ + (900 ν µ η a 0 ν 3 µ)k 3 + (6 η a 0 ν η )K 5 ν( η + P ) λ = 1 η η η K ν µ K ν µ 5 ν K( η + a 0 = a 0 a 1 = a = 0 (34) η K ν µ) P = η K ν µ. The rest part of this solution is omitted here which is almost the same with solution 3 only with some differences of sign. Case ν = 0 Three solutions of this case i.e. three traveling wave solutions of the space- time-fractional Burgers Eq. (9) are obtained. Solutions 1 The first set of the unknowns of this case is a 1 = η Kµ a 0 = L + λ K η K C = 1 L λ K 4 η + 4 η K 4 µ K a = a 1 = a = 0. (35) From (13) together with (35) we get the solution equation When λ > µ the solution is of the form L K are arbitrary constants U(ξ) = η Kµ ψ(ξ) L + λ K η. K P 1 e (ξ/) λ 4 µ + P e (ξ/) λ 4 µ u(x t) = (36) K(P 3 e (ξ/) λ 4 µ + P 4 e (ξ/) λ 4 µ ) ξ = L tα P 1 = Bη K (4 µ λ λ λ 4 µ) + BL( λ λ 4 µ) P = Aη K (4 µ λ + λ λ 4 µ) + AL( λ + λ 4 µ) P 3 = B(λ + λ 4 µ) P 4 = A(λ λ 4 µ). Of course it can be written in terms of hyperbolic functions as u(x t) = (P 1 + P ) cosh( ξ λ 4µ) (P 1 P ) sinh( ξ λ 4µ) K(P 3 + P 4 ) cosh( ξ λ 4µ) (P 3 P 4 ) sinh( ξ λ 4µ). (37) Solution The second set of the unknowns of this case is a 1 = η K a 0 = λ K η L K C = 1 L λ K 4 η + 4 η K 4 µ K a = a 1 = a = 0. (38) From (13) together with (38) we get the solution expression U(ξ) = λ K η L K + η K ψ(ξ).

7 No. 5 Communications in Theoretical Physics 695 When λ > µ the traveling wave solution is of the form u(x t) = (P 1 + P ) cosh( ξ λ 4µ) (P 1 P ) sinh( ξ λ 4µ) K[(A + B) cosh( ξ λ 4µ) + (A B) sinh( ξ λ 4µ)] (39) where ξ = L t α /Γ(α + 1) + K x β /Γ(β + 1) L K are arbitrary constants P 1 = B(L + η K λ 4 µ) P = A(L η K λ 4 µ). Solution 3 The third set of the unknowns is a 1 = η Kµ a 1 = η K C = 1 K( a η K µ) L = a 0 K a 0 = a 0 a = a = 0. (40) From (13) together with (40) we get the following solution expression equation U(ξ) = η Kµ 1 ψ(ξ) + η K ψ(ξ) + a 0. When λ > µ the traveling wave solution is u(x t) = D(cosh( µξ) + sinh( µξ)) µ[(a B ) cosh( µξ) + (A + B ) sinh( µξ)] ξ = a 0K t α D = a 0 µ(a B ) 4 η Kµ(A + B ). Case η = 0 Three solutions of the case η = 0 i.e. three traveling wave solutions of the space- time-fractional KdV Eq. (8) are obtained. Solutions 1 The first two sets of the unknowns are a 0 = νk3 λ + 8νK 3 µ ν K 6 λ 4 8ν K 6 λ µ + 16ν K 6 µ + KC K L = K(ν K 5 λ 4 8 ν K 5 λ µ + 16 ν K 5 µ + C) 1 ν K 1 λν K a = a 1 = a = a 1 = 0. (41) From (13) together with (41) we get the solution expression with the form U(ξ) = 1 λν K ψ(ξ) 1 ν K ψ(ξ) + a 0. When λ > µ we have u(x t) = ± P (A + B ) cosh( ξ λ 4 µ) ± P (A B ) sinh( ξ λ 4 µ) cosh( ξ λ 4 µ) + Q (4) K [(A + B) cosh( ξ λ 4 µ) + (A B) sinh( ξ λ 4 µ)] ξ = K(ν K 5 λ 4 8 ν K 5 λ µ + 16 ν K 5 µ + C) t α P = (D ± 4 ν K 3 µ ν K 3 λ ) Γ(α + 1) + K xβ Q = B A( 40 ν K 3 µ ± D + 10 ν K 3 λ ) (A + B )(±4 ν K 3 µ + D ν K 3 λ ) D = K(ν K 5 λ 4 8 ν K 5 λ µ + 16 ν K 5 µ + C). Soluiton 3 The third solution is obtained from the following results. C = 1 ( 8 λ µ ν + λ 4 ν + 16 ν µ )K a 0 = (λ ν + 8 µ ν)k L K L K (43) a 1 = 1 λ µ ν K a = 1 ν K µ. (44)

8 696 Communications in Theoretical Physics Vol. 6 Then the solution expression is of the form U(ξ) = λ ν K µ ν K 3 + L K 1 λ µ ν K 1 ψ(ξ) 1 ν K µ 1 ψ(ξ). When λ > µ the third solution of the space- time-fractional KdV equation is obtained immediately u(x t) = where ξ = L tα (P 0 + E) cosh(ξ λ 4 µ) + (P 0 E) sinh(ξ λ 4 µ) + Q K[(P + P 1 ) cosh( 1 ξ λ 4 µ) + (P P 1 ) sinh( 1 ξ λ 4 µ)] (45) P 0 = [( 8 µ ν K 3 λ + Lλ + λ 3 ν K 3 )h λ 4 ν K 3 16 ν K 3 µ Lλ + 1 µ ν K 3 λ + 4 Lµ]A P 1 = (λ + λ 4 µ)b P = (λ λ 4 µ)a E = 4 LB µ LB λ h + 1 µ ν K 3 B λ LB λ λ 4 ν K 3 B 16 ν K 3 µ B λ 3 ν K 3 B h + 8 µ ν K 3 B λ h Q = 160 ν K 3 µ A B + 40 µ ν K 3 Aλ B 8 L A Bµ. (46) It is easy to check that the obtained ODE (35) in [6] was not a correct one to the space- time-fractional KdV equation. Hence the solutions reported there did not belong to the solution family of the space- timefractional KdV equation. 4 Summary The Modified Jumarie s Riemann Liouvulle fractional derivative a fractional variable transformation are used to deal with the fractional operator in the KdVB equation which is solved successfully to yield new exact traveling wave solutions. Furthermore the exact traveling wave solutions to the space- time-fractional Burgers KdV equations are naturally simultaneously obtained under various values of the parameters appearing in the space- time-fractional KdVB equation (6). All the solutions reported above have been verified using the symbolic computation system Maple. References [1] R. Hilfe Applications of Fractional Calculus in Physics World Scientific River Edge NJ USA (001). [] K.B. Oldham J. Spanier the Fractional Culculus Academic Press San Diege London (1999). [3] J. Sabatier O.P. Agrawal J.A. Machado Advancesin Fractional Calculus: Theoretical Developments Applications in Physics Engineering Springer New York (007). [4] F. Liu V.V. Anh I. Turner P. Zhuang J. Appl. Math. Comput. 13 (003) 33. [5] M.L. Wang X.Z. Li J.L. Zhang Phys. Lett. A 37 (008) 417. [6] Khaled A. Gepreel Omranb Saleh Chin. Phys. B 1 (01) [7] I. Podlubny Fractional Differential Equations Academic Press San Diego Calif USA (1999). [8] M. Caputo Elasticitae Dissipazione Zanichelli Bologna (1969). [9] R. Gorenflo F. Mainardi Fractinal Calculus: Integral Didfferential Equations of Fractional Order Fractional Calculus in Continum Mechanics Springer Verlag Wien New York (1997) 3. [10] G. Jumarie J. Comput. Appl. Math. 51 (006) [11] G. Jumarie Appl. Math. Lett. (009) 378. [1] Q. Wang Appl. Math. Comput.18 (006) [13] X.J. Yang Advanced Local Fractional Calculus ItsApplications World Science Publisher New York (01). [14] R.S. Johnson J. Fluid Mech. 4 (1970) 49. [15] Z.B. Li J.H. He Meth. Comput. Appl. 15 (010) 970. [16] M.L. Wang Phys. Lett. A 199 (1995) 169. [17] M.L. Wang Phys. Lett. A 13 (1996) 79. [18] M.L. Wang Z.B. Li Y.B. Zhou Journal of Lanzhou University 35 (1999) 8.