The Yang-Laplace Transform for Solving the Ivps with Local Fractional Derivative

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 386459, 5 pages http://dx.doi.org/10.1155/2014/386459

Research Article The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative

Chun-Guang Zhao,1 Ai-Min Yang,2,3 Hossein Jafari,4 and Ahmad Haghbin4

1 Department of Mathematics, Handan College, Handan, Hebei 056004, China 2 College of Science, Hebei United University, Tangshan 063009, China 3 College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China 4 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47415-416, Iran

Correspondence should be addressed to Ai-Min Yang; aimin [email protected]

Received 25 October 2013; Accepted 7 November 2013; Published 8 January 2014

Academic Editor: Abdon Atangana

Copyright © 2014 Chun-Guang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The IVPs with local fractional derivative are considered inthis paper. Analytical solutions for the homogeneous and nonhomogeneous local fractional differential equations are discussed by using the Yang-Laplace transform.

1. Introduction fractional derivative was presented [30]. The Yang-Laplace transform structured in 2011 [22]wassuggestedtodealwith In recent years, the ordinary and partial differential equations local fractional differential equations31 [ , 32]. The coupling havefoundapplicationsinmanyproblemsinmathematical method for variational iteration method within Yang-Laplace physics [1, 2]. Initial value problems (IVPs) for ordinary transform for solving the heat conduction in fractal media and partial differential equations have been developed by was proposed in [33]. some authors in [3–6]. There are analytical methods and In this paper, our aim is to use the Yang-Laplace trans- numerical methods for solving the differential equations, form to solve IVPs with local fractional derivative. The struc- such as the finite element method [6], the harmonic wavelet ture of the paper is as follows. In Section 2, some definitions method [7–9], the Adomian decomposition method [10–12], and properties for the Yang-Laplace transform are given. the homotopy analysis method [13, 14], the homotopy decom- Section 3 is devoted to the solutions for the homogeneous position method [15, 16], the heat balance integral method and nonhomogeneous IVPs with local fractional derivative. [17, 18], the homotopy perturbation method [19], the varia- Finally, conclusions are presented in Section 4. tional iteration method [20], and other methods [21]. Recently, owing to limit of classical and fractional differential equations, the local fractional differential equations 2. Yang-Laplace Transform have been applied to describe nondifferentiable problems for In this section we show some definitions and properties for the heat and wave in fractal media [22, 23], the structure the Yang-Laplace transform. relation in fractal elasticity [24], and Fokker-Planck equa- The local fractional integral operator is defined as tion in fractal media [25]. Some methods were utilized to [22, 23, 26–33] solve the local fractional differential equations. For example, the local fractional variation iteration method was used 1 𝑏 to solve the heat conduction in fractal media [26, 27]. 𝐼 (𝛼)𝑓(𝑥) = ∫ 𝑓(𝑡)(𝑑𝑡)𝛼 𝑎 𝑏 Γ (1+𝛼) The local fractional decomposition method for solving the 𝑎 local fractional diffusion and heat-conduction equations was 𝑗=𝑁−1 (1) 1 𝛼 considered in [28, 29]. The local fractional series expansion = lim ∑ 𝑓(𝑡𝑗)(Δ𝑡𝑗) , Γ (1+𝛼) Δ𝑡 → 0 method for solving the Schrodinger¨ equation with the local 𝑗=0 2 Abstract and Applied Analysis where Δ𝑡𝑗 =𝑡𝑗+1 −𝑡𝑗, Δ𝑡 = max{Δ𝑡0,Δ𝑡1,...,Δ𝑡𝑗,...}, 3. IVPs with Local Fractional Derivatives [𝑡𝑗,𝑡𝑗+1], 𝑗 = 0,...,𝑁−1, 𝑡0 =𝑎,𝑡𝑁 =𝑏,isapartitionof the interval [𝑎, 𝑏]. In this section we handle the homogeneous and non- As the inverse operator of (1), the local fractional deriva- homogeneous IVPs with local fractional derivative. tive operator is given by [22, 23, 26–33]

𝛼 𝛼 3.1. Homogeneous IVPs with Local Fractional Derivative (𝛼) 𝑑 𝑓(𝑥) 󵄨 Δ (𝑓(𝑥) −𝑓(𝑥0)) 𝑓 (𝑥0) = 󵄨𝑥=𝑥 = lim 𝛼 , 𝛼 󵄨 0 𝑥→𝑥 (2) Example 1. The homogeneous IVPs with local fractional 𝑑𝑥 0 (𝑥 −0 𝑥 ) derivative are expressed by Δ𝛼(𝑓(𝑥) − 𝑓(𝑥 )) ≅ Γ(1 + 𝛼)Δ(𝑓(𝑥) − 𝑓(𝑥 )) with 0 0 . 𝑑2𝛼𝑦 𝑑𝛼𝑦 The Yang-Laplace transform is expressed by [22, 31–33] − +2𝑦=0. (17) 𝑑2𝛼𝑥 𝑑𝛼𝑥 ∞ ̃ 𝐿,𝛼̃ 1 𝛼 𝛼 𝛼 𝐿𝛼 {𝑓 (𝑥)}=𝑓𝑠 (𝑠) = ∫ 𝐸𝛼 (−𝑠 𝑥 )𝑓(𝑥)(𝑑𝑥) , The initial boundary conditions are presented as Γ (1+𝛼) 0 (𝛼) 𝑦 (0) =1, 𝑦 (0) =0. (18) 0<𝛼≤1, (3) From (6)wehave 𝐿̃ {𝑦(𝛼) (𝑥)} =𝑠𝛼𝐿̃ {𝑦 (𝑥)} −𝑦(0) , where 𝑓(𝑥) is a local fractional continuous function. 𝛼 𝛼 (19) The inverse Yang-Laplace transform reads as[22, 31–33] ̃ (2𝛼) 2𝛼 ̃ 𝛼 (𝛼) 𝐿𝛼 {𝑦 (𝑥)}=𝑠 𝐿𝛼 {𝑦 (𝑥)}−𝑠 𝑦 (0) −𝑓 (0) . (20) 1 𝑓 (𝑥) = 𝐿̃−1 {𝑓𝐿,𝛼 (𝑠)}= 𝛼 𝑠 (2𝜋)𝛼 Hence, making use of (19)and(20), (19)canbewrittenas

𝛽+𝑖∞ (4) 𝑠2𝛼𝐿̃ {𝑦 (𝑥)}−𝑠𝛼𝑦 (0) −𝑓(𝛼)(0) −{𝑠𝛼𝐿̃ {𝑦 (𝑥)}−𝑦(0)} 𝛼 𝛼 𝐿,𝛼̃ 𝛼 𝛼 𝛼 × ∫ 𝐸𝛼 (𝑠 𝑥 )𝑓𝑠 (𝑠)(𝑑𝑠) , 𝛽−𝑖∞ ̃ +2𝐿𝛼 {𝑦 (𝑥)}=0. 𝛼 𝛼 𝛼 𝛼 𝛼 𝛼 where 𝑠 =𝛽 +𝑖 ∞ and Re(𝑠 )=𝛽 . (21) Some properties for Yang-Laplace transform are pre- Hence, we obtain sented as follows [21, 22, 22–33]: ̃ 1 1 ̃ ̃ ̃ 𝐿𝛼 {𝑦 (𝑥)}= 𝑦 (0) = . (22) 𝐿𝛼 {𝑎𝑓(𝑥) +𝑏𝑔(𝑥)}=𝑎𝐿𝛼 {𝑓(𝑥)}+𝑏𝐿𝛼 {𝑔 (𝑥)}, (5) 𝑠𝛼 +2 𝑠𝛼 +2 𝑛 So, making use of (13), we get the solution of (17): 𝐿̃ {𝑓(𝑛𝛼) (𝑥)}=𝑠𝑛𝛼𝐿̃ {𝑓(𝑥)}−∑𝑠(𝑘−1)𝛼𝑓(𝑛−𝑘)𝛼 (0) , 𝛼 𝛼 (6) 𝛼 𝑘=1 𝑦 (𝑥) =𝐸𝛼 (−2𝑥 ) . (23) 𝑓(𝑥) = 𝑠𝛼𝐹(𝑠) , 𝛼= 2/ 3 𝑥→0lim 𝑠→∞lim (7) The solution of (17)for ln ln is shown in Figure 1. 𝛼 lim 𝑓(𝑥) = lim𝑠 𝐹(𝑠) , (8) 𝑥→∞ 𝑠→0 Example 2. Let us consider the homogeneous IVPs with local

1 𝐿,𝛼 𝑠 fractional derivative in the form 𝐿̃ {𝑓(𝑎𝑥)}= 𝑓 ( ), 𝑎>0, (9) 𝛼 𝑎𝛼 𝑠 𝑎 𝑑4𝛼𝑦 −𝑦=0 4𝛼 (24) 𝑑𝑘𝛼𝑓𝐿,𝛼 (𝑠) 𝑑 𝑥 𝐿̃ {𝑥𝑘𝛼𝑓(𝑥)}=(−1)𝑘 𝑠 , (10) 𝛼 𝑑𝑠𝑘𝛼 subject to initial boundary conditions ̃ 𝐿,𝛼 𝛼 𝛼 (𝛼) 𝐿𝛼 {𝑓(𝑥−𝑐)}=𝑓𝑠 (𝑠) 𝐸𝛼 (−𝑐 𝑠 ), (11) 𝑦 (0) =0, 𝑦 (0) =0, (25) ̃ 𝛼 𝛼 𝐿,𝛼 (2𝛼) (3𝛼) 𝐿𝛼 {𝑓(𝑥) 𝐸𝛼 (𝑐 𝑥 )} = 𝑓𝑠 (𝑠−𝑐) , (12) 𝑦 (0) =0, 𝑦 (0) =1. Γ (1+𝑘𝛼) 𝐿̃ {𝑥𝑘𝛼𝐸 (𝑐𝛼𝑥𝛼)} = , From (6)wehave 𝛼 𝛼 (𝑘+1)𝛼 (13) (𝑠−𝑐) ̃ (4𝛼) 4𝛼 ̃ 3𝛼 2𝛼 (𝛼) 𝐿𝛼 {𝑦 (𝑥)}=𝑠 𝐿𝛼 {𝑦 (𝑥)}−𝑠 𝑦 (0) −𝑠 𝑦 (0) 𝛼 𝛼 𝛼 𝑐 (26) 𝐿̃ { (𝑐 𝑥 )} = , (14) 𝛼 (2𝛼) (3𝛼) 𝛼 sin𝛼 𝑠2𝛼 +𝑐2𝛼 −𝑠 𝑦 (0) −𝑓 (0) , 𝛼 ̃ 𝛼 𝛼 𝑠 so that 𝐿𝛼 {cos𝛼 (𝑐 𝑥 )} = , (15) 𝑠2𝛼 +𝑐2𝛼 4𝛼 ̃ 3𝛼 2𝛼 (𝛼) 𝛼 (2𝛼) 𝑠 𝐿𝛼 {𝑦 (𝑥)}−𝑠 𝑦 (0) −𝑠 𝑦 (0) −𝑠 𝑦 (0) ̃ 𝑘𝛼 Γ (1+𝑘𝛼) (27) 𝐿𝛼 {𝑥 } = . (16) (3𝛼) ̃ 𝑠(𝑘+1)𝛼 −𝑓 (0) − 𝐿𝛼 {𝑦 (𝑥)}=0. Abstract and Applied Analysis 3

2 0

1.9 −0.2

1.8 −0.4 1.7 −0.6 1.6 y(x) y(x) −0.8 1.5 −1 1.4

1.3 −1.2

−1.4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x

Figure 1: Graph of 𝑦(𝑥) for 𝛼=ln 2/ ln 3. Figure 2: Graph of 𝑦(𝑥) for 𝛼=ln 2/ ln 3.

Hence, (27)canbewrittenas so that 𝑠4𝛼𝐿̃ {𝑦 𝑥 }−1−𝐿̃ {𝑦 𝑥 }=0, 3 1 1 1 1 𝛼 ( ) 𝛼 ( ) (28) 𝐿̃ {𝑦 (𝑥)}= ( − )− . 𝛼 4 𝑠𝛼 −1 𝑠𝛼 +1 2 𝑠2𝛼 +1 (34) which leads to So, 1 𝐿̃ {𝑦 (𝑥)} = . 𝛼 4𝛼 (29) 3 3 1 𝑠 −1 𝑦 (𝑥) = 𝐸 (−𝑥𝛼)− 𝐸 (𝑥𝛼)− (𝑥𝛼). 4 𝛼 4 𝛼 2sin𝛼 (35) Therefore, we get The exact solution of (31)for𝛼=ln 2/ ln 3 is shown in 1 𝑦 (𝑥) = 𝐿̃−1 { } Figure 3. 𝛼 𝑠4𝛼 −1 1 1 1 1 1 1 Example 4. The non-homogeneous IVPs with local fractional = 𝐿̃−1 { ( − − )} 𝛼 2 2 𝑠𝛼 −1 2 𝑠𝛼 +1 𝑠2𝛼 +1 (30) derivative are 1 1 1 𝑑2𝛼𝑦 = 𝐸 (−𝑥𝛼)− 𝐸 (𝑥𝛼)− (𝑥𝛼). +𝑦=𝐸 (𝑥𝛼) . (36) 4 𝛼 4 𝛼 2sin𝛼 𝑑2𝛼𝑥 𝛼

The exact solution of (24)for𝛼=ln 2/ ln 3 is shown in The initial boundary conditions are Figure 2. (𝛼) 𝑦 (0) =1, 𝑦 (0) =0. (37) 3.2. Nonhomogeneous IVPs with Local Fractional Derivative In view of (6), we give Example 3. We now consider the non-homogeneous IVPs with local fractional derivative 1 𝑠𝛼 𝐿̃ {𝑦 (𝑥)}= + . 𝛼 (𝑠𝛼 +1) (𝑠2𝛼 +1) 𝑠2𝛼 +1 (38) 𝑑2𝛼𝑦 −𝑦= (𝑥𝛼) (31) 𝑑2𝛼𝑥 sin𝛼 So, we obtain subject to initial boundary conditions 𝑥 𝛼 1 𝛼 𝛼 𝛼 𝑦 (𝑥) = cos𝛼 (𝑥 )+ ∫ 𝐸𝛼(𝑥−𝑡) sin𝛼 (𝑡 ) (𝑑𝑡) (𝛼) Γ (1+𝛼) 𝑦 (0) =0, 𝑦 (0) =1. (32) 0 1 = (𝑥𝛼)+ By using (6), we have cos𝛼 Γ (1+𝛼) ̃ (2𝛼) 2𝛼 ̃ 𝛼 (𝛼) 𝑥 𝐿𝛼 {𝑦 (𝑥)}=𝑠 𝐿𝛼 {𝑦 (𝑥)}−𝑠 𝑦 (0) −𝑓 (0) , 𝛼 𝛼 𝛼 × ∫ 𝐸𝛼 (𝑡 )(sin𝛼 (𝑥 ) cos𝛼 (𝑡 ) (33) 0 ̃ 𝛼 1 𝐿𝛼 {sin𝛼 (𝑥 )} = 𝛼 𝛼 𝛼 𝑠2𝛼 +1 −cos𝛼 (𝑥 ) sin𝛼 (𝑡 )) (𝑑𝑡) 4 Abstract and Applied Analysis

0 0

−0.5 −0.5 −1

−1 −1.5 y(x)

y(x) −2 −1.5

−2.5 −2 −3

−3.5 −2.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x

Figure 3: Graph of 𝑦(𝑥) for 𝛼=ln 2/ ln 3. Figure 4: Graph of 𝑦(𝑥) for 𝛼=ln 2/ ln 3.

𝛼 = cos𝛼 (𝑥 ) Acknowledgments 𝑥 𝛼 1 𝛼 𝛼 𝛼 This work was supported by national scientific and techno- + sin𝛼 (𝑥 ){ ∫ 𝐸𝛼 (𝑡 ) cos𝛼 (𝑡 ) (𝑑𝑡) } Γ (1+𝛼) 0 logical support projects (no. 2012BAE09B00), the national 𝑥 natural Science Foundation of China (no. 61202259 and no. 𝛼 1 𝛼 𝛼 𝛼 61170317), the national natural Science Foundation of Hebei − cos𝛼 (𝑥 ){ ∫ 𝐸𝛼 (𝑡 ) sin𝛼 (𝑡 ) (𝑑𝑡) } Γ (1+𝛼) 0 Province (no. A2012209043 and no. E2013209215), and Hebei = (𝑥𝛼) province of China Scientific Research Subject in Twelfth Five- cos𝛼 Year Plan (no. 13090074). (𝑥𝛼){𝐸 (𝑥𝛼)[ (𝑥𝛼)+ (𝑥𝛼)]−1} + sin𝛼 𝛼 cos𝛼 sin 2 References (𝑥𝛼){𝐸 (𝑥𝛼)[ (𝑥𝛼)− (𝑥𝛼)] + 1} − cos𝛼 𝛼 sin𝛼 cos𝛼 [1]N.S.Koshlyakov,M.M.Smirnov,andE.B.Gliner,Differential 2 Equations of Mathematical Physics,North-Holland,NewYork, NY, USA, 1964. 1 𝛼 𝛼 𝛼 = [cos𝛼 (𝑥 )−sin𝛼 (𝑥 )+𝐸𝛼 (𝑥 )] . [2] U. Tyn Myint, Partial Differential Equations of Mathematical 2 Physics, Elsevier, New York, NY, USA, 1973. (39) [3] A. H. Stroud, “Initial value problems for ordinary differential equations,” in Numerical Quadrature and Solution of Ordinary The exact solution of (36)for𝛼=ln 2/ ln 3 is shown in Differential Equations,pp.207–303,Springer,NewYork,NY, Figure 4. USA, 1974. [4] H. Rutishauser, “Initial value problems for ordinary differential equations,” in Lectures on Numerical Mathematics,pp.208–277, 4. Conclusions Birkhauser,¨ Boston, Mass, USA, 1990. In this work we have used the Yang-Laplace transform to [5] J. A. Gatica, V. Oliker, and P. Waltman, “Singular nonlinear handle the homogeneous and non-homogeneous IVPs with boundary value problems for second-order ordinary differential looselocal fractional derivative. Some illustrative examples of equations,” Journal of Differential Equations,vol.79,no.1,pp. 62–78, 1989. approximate solutions for local fractional IVPs are discussed. The nondifferentiable solutions for fractal dimension 𝛼= [6]S.F.DavisandJ.E.Flaherty,“Anadaptivefiniteelement ln 2/ ln 3 are shown graphically. The obtained results illustrate method for initial-boundary value problems for partial differential equations,” SIAM Journal on Scientific and Statistical that the Yang-Laplace transform is an efficient mathematical Computing,vol.3,no.1,pp.6–27,1982. tool to solve the homogeneous and non-homogeneous IVPs with local fractional derivative. [7] C. Cattani and A. Kudreyko, “Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind,” Applied Mathematics and Computation,vol.215, Conflict of Interests no. 12, pp. 4164–4171, 2010. [8] C. Cattani, “Harmonic wavelet solutions of the Schrodinger¨ The authors declare that there is no conflicts of interests equation,” International Journal of Fluid Mechanics Research, regarding publication of this paper. vol.30,no.5,pp.463–472,2003. Abstract and Applied Analysis 5

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