- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2014 (2014), Article ID 386459, 5 pages

http://dx.doi.org/10.1155/2014/386459

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

^{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

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.

#### Abstract

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

#### 1. Introduction

In recent years, the ordinary and partial differential equations have found applications in many problems in mathematical physics [1, 2]. Initial value problems (IVPs) for ordinary and partial differential equations have been developed by some authors in [3–6]. There are analytical methods and numerical methods for solving the differential equations, such as the finite element method [6], the harmonic wavelet method [7–9], the Adomian decomposition method [10–12], the homotopy analysis method [13, 14], the homotopy decomposition method [15, 16], the heat balance integral method [17, 18], the homotopy perturbation method [19], the variational iteration method [20], and other methods [21].

Recently, owing to limit of classical and fractional differential equations, the local fractional differential equations have been applied to describe nondifferentiable problems for the heat and wave in fractal media [22, 23], the structure relation in fractal elasticity [24], and Fokker-Planck equation in fractal media [25]. Some methods were utilized to solve the local fractional differential equations. For example, the local fractional variation iteration method was used to solve the heat conduction in fractal media [26, 27]. The local fractional decomposition method for solving the local fractional diffusion and heat-conduction equations was considered in [28, 29]. The local fractional series expansion method for solving the Schrödinger equation with the local fractional derivative was presented [30]. The Yang-Laplace transform structured in 2011 [22] was suggested to deal with local fractional differential equations [31, 32]. The coupling method for variational iteration method within Yang-Laplace transform for solving the heat conduction in fractal media was proposed in [33].

In this paper, our aim is to use the Yang-Laplace transform to solve IVPs with local fractional derivative. The structure of the paper is as follows. In Section 2, some definitions and properties for the Yang-Laplace transform are given. Section 3 is devoted to the solutions for the homogeneous and nonhomogeneous IVPs with local fractional derivative. Finally, conclusions are presented in Section 4.

#### 2. Yang-Laplace Transform

In this section we show some definitions and properties for the Yang-Laplace transform.

The local fractional integral operator is defined as [22, 23, 26–33] where , , , , , is a partition of the interval .

As the inverse operator of (1), the local fractional derivative operator is given by [22, 23, 26–33] with .

The Yang-Laplace transform is expressed by [22, 31–33] where is a local fractional continuous function.

The inverse Yang-Laplace transform reads as [22, 31–33]
where and *. *

Some properties for Yang-Laplace transform are presented as follows [21, 22, 22–33]:

#### 3. IVPs with Local Fractional Derivatives

In this section we handle the homogeneous and non-homogeneous IVPs with local fractional derivative.

##### 3.1. Homogeneous IVPs with Local Fractional Derivative

*Example 1. *The homogeneous IVPs with local fractional derivative are expressed by

The initial boundary conditions are presented as

From (6) we have

Hence, making use of (19) and (20), (19) can be written as

Hence, we obtain

So, making use of (13), we get the solution of (17):

The solution of (17) for is shown in Figure 1.

*Example 2. *Let us consider the homogeneous IVPs with local fractional derivative in the form

subject to initial boundary conditions

From (6) we have

so that

Hence, (27) can be written as

which leads to

Therefore, we get

The exact solution of (24) for is shown in Figure 2.

##### 3.2. Nonhomogeneous IVPs with Local Fractional Derivative

*Example 3. *We now consider the non-homogeneous IVPs with local fractional derivative
subject to initial boundary conditions

By using (6), we have
so that

So,

The exact solution of (31) for is shown in Figure 3.

*Example 4. *The non-homogeneous IVPs with local fractional derivative are

The initial boundary conditions are

In view of (6), we give

So, we obtain

The exact solution of (36) for is shown in Figure 4.

#### 4. Conclusions

In this work we have used the Yang-Laplace transform to handle the homogeneous and non-homogeneous IVPs with looselocal fractional derivative. Some illustrative examples of approximate solutions for local fractional IVPs are discussed. The nondifferentiable solutions for fractal dimension are shown graphically. The obtained results illustrate that the Yang-Laplace transform is an efficient mathematical tool to solve the homogeneous and non-homogeneous IVPs with local fractional derivative.

#### Conflict of Interests

The authors declare that there is no conflicts of interests regarding publication of this paper.

#### Acknowledgments

This work was supported by national scientific and technological support projects (no. 2012BAE09B00), the national natural Science Foundation of China (no. 61202259 and no. 61170317), the national natural Science Foundation of Hebei Province (no. A2012209043 and no. E2013209215), and Hebei province of China Scientific Research Subject in Twelfth Five-Year Plan (no. 13090074).

#### References

- N. S. Koshlyakov, M. M. Smirnov, and E. B. Gliner,
*Differential Equations of Mathematical Physics*, North-Holland, New York, NY, USA, 1964. View at MathSciNet - U. Tyn Myint,
*Partial Differential Equations of Mathematical Physics*, Elsevier, New York, NY, USA, 1973. - A. H. Stroud, “Initial value problems for ordinary differential equations,” in
*Numerical Quadrature and Solution of Ordinary Differential Equations*, pp. 207–303, Springer, New York, NY, USA, 1974. View at Google Scholar - H. Rutishauser, “Initial value problems for ordinary differential equations,” in
*Lectures on Numerical Mathematics*, pp. 208–277, Birkhäuser, Boston, Mass, USA, 1990. View at Google Scholar - J. A. Gatica, V. Oliker, and P. Waltman, “Singular nonlinear boundary value problems for second-order ordinary differential equations,”
*Journal of Differential Equations*, vol. 79, no. 1, pp. 62–78, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. F. Davis and J. E. Flaherty, “An adaptive finite element method for initial-boundary value problems for partial differential equations,”
*SIAM Journal on Scientific and Statistical Computing*, vol. 3, no. 1, pp. 6–27, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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, no. 12, pp. 4164–4171, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Cattani, “Harmonic wavelet solutions of the Schrödinger equation,”
*International Journal of Fluid Mechanics Research*, vol. 30, no. 5, pp. 463–472, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - C. Cattani, “Harmonic wavelets towards the solution of nonlinear PDE,”
*Computers & Mathematics with Applications*, vol. 50, no. 8-9, pp. 1191–1210, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. S. Duan, R. Rach, and A. M. Wazwaz, “Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems,”
*International Journal of Non-Linear Mechanics*, vol. 49, pp. 159–169, 2013. View at Google Scholar - C. Li and Y. Wang, “Numerical algorithm based on Adomian decomposition for fractional differential equations,”
*Computers & Mathematics with Applications*, vol. 57, no. 10, pp. 1672–1681, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2006–2012, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 3, pp. 674–684, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Atangana and A. Secer, “The time-fractional coupled-Korteweg-de-Vries equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 947986, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Atangana and A. Kılıçman, “Analytical solutions of boundary values problem of 2D and 3D poisson and biharmonic equations by homotopy decomposition method,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 380484, 9 pages, 2013. View at Publisher · View at Google Scholar - J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,”
*Thermal Science*, vol. 14, no. 2, pp. 291–316, 2010. View at Publisher · View at Google Scholar · View at Scopus - J. Hristov, “Transient flow of a generalized second grade fluid due to a constant surface shear stress: an approximate integral-balance solution,”
*International Review of Chemical Engineering*, vol. 3, no. 6, pp. 802–809, 2011. View at Google Scholar - S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary value problems,”
*Mathematical Problems in Engineering*, vol. 2007, Article ID 98602, 15 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 9, no. 2, pp. 141–156, 2008. View at Google Scholar · View at Scopus - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific Publishing, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - X.-J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic Publisher, Hong Kong, 2011. - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science Publisher, New York, NY, USA, 2012. - A. Carpinteri, B. Chiaia, and P. Cornetti, “The elastic problem for fractal media: basic theory and finite element formulation,”
*Computers and Structures*, vol. 82, no. 6, pp. 499–508, 2004. View at Publisher · View at Google Scholar · View at Scopus - K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,”
*Physical Review Letters*, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,”
*Thermal Science*, vol. 17, no. 2, pp. 625–628, 2013. View at Google Scholar - J. H. He and F. J. Liu, “Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy,”
*Nonlinear Science Letters A*, vol. 4, no. 1, pp. 15–20, 2013. View at Google Scholar - X.-J. Yang, D. Baleanu, and W. P. Zhong, “Approximate solutions for diffusion equations on cantor space-time,”
*Proceedings of the Romanian Academy A*, vol. 14, no. 2, pp. 127–133, 2013. View at Google Scholar - X. J. Yang, D. Baleanu, M. P. Lazarevic, and M. S. Cajic, “Fractal boundary value problems for integral and differential equations with local fractional operators,”
*Thermal Science*, pp. 103–103, 2013. View at Publisher · View at Google Scholar - Y. Zhao, D.-F. Cheng, and X.-J. Yang, “Approximation solutions for local fractional Schrödinger equation in the one-dimensional Cantorian system,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 291386, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He, “Asymptotic methods for solitary solutions and compactons,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. P. Zhong and F. Gao, “Application of the Yang-Laplace transforms to solution to nonlinear fractional wave equation with fractional derivative,” in
*Proceeding of the 3rd International Conference on Computer Technology and Development*, pp. 209–213, 2011. - C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at Google Scholar