About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 462535, 5 pages
http://dx.doi.org/10.1155/2013/462535
Research Article

Analytical Solutions of the One-Dimensional Heat Equations Arising in Fractal Transient Conduction with Local Fractional Derivative

1College of Science, Hebei United University, Tangshan 063009, China
2College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China
3Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy
4Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
5International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa
6Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 26 September 2013; Accepted 17 October 2013

Academic Editor: Abdon Atangana

Copyright © 2013 Ai-Ming Yang 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 one-dimensional heat equations with the heat generation arising in fractal transient conduction associated with local fractional derivative operators are investigated. Analytical solutions are obtained by using the local fractional Adomian decomposition method via local fractional calculus theory. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

1. Introduction

The Adomian decomposition method [13] was applied to process linear and nonlinear problems in the fields of science and engineering. Tatari and Dehghan [4] applied Adomian decomposition method to process the multipoint boundary value problem. Wazwaz [5] used Adomian decomposition method to deal with the Bratu-type equations. Daftardar-Gejji and Jafari [6] considered Adomian decomposition method to analyze the Bagley Torvik equation. Larsson [7] presented the solution for Helmholtz equation by using the Adomain decomposition method. Tatari and coworkers [8] investigated solution for the Fokker-Planck equation by Adomain decomposition method.

Fractional calculus [912] was applied to model the physical and engineering problems for expressions of stress-strain constitutive relations of different viscoelastic fractional order properties of materials, diffusion processes with fractional order properties, fractional order flows, analytical mechanics of fractional order discrete system vibrations [1315], and so on. Recently, the application of Adomian decomposition method for solving the linear and nonlinear fractional partial differential equations in the fields of the physics and engineering had been established in [16, 17]. Adomian decomposition method was applied to handle the time-fractional Navier-Stokes equation [18], fractional space diffusion equation [19], fractional KdV-Burgers equation [20], linear and nonlinear fractional diffusion and wave equations [21], KdV-Burgers-Kuramoto equation [22], fractional Burgers’ equation [23], and so on. For more details on some methods for solving fractional differential equations, see [2428].

Recently, local fractional calculus theory was applied to model some nondifferentiable problems for mathematical physics (see [2936] and the references therein). The Adomian decomposition method, as one of efficient tools for solving the linear and nonlinear differential equations, was extended to find the solutions for local fractional differential equations [3740] and nondifferentiable solutions were obtained.

The partial differential equationfs describing thermal process of fractal heat conduction were suggested in [30, 38] in the following form: The initial and boundary conditions are where the operator is the local fractional differential operator [29, 30, 34, 37, 38], which is applied to model the heat conduction problems in fractal media, fractal materials, fractal fracture mechanics, fractal wave behavior, Navier-Stokes equations on Cantor sets, Schrödinger equation with local fractional derivative, and diffusion equations on cantor space-time.

The one-dimensional heat equations with the heat generation arising in fractal transient conduction were considered in [30] as where is the heat generation term.

We use initial and boundary conditions as follows: The aim of this paper is to investigate the one-dimensional heat equations with the heat generation arising in fractal transient conduction by using the local fractional Adomian decomposition method.

This paper is structured as follows. In Section 2, we give the basic notations and definitions of local fractional operators. In Section 3, local fractional Adomian decomposition method for heat generation arising in fractal transient conduction is presented. Three examples are shown in Section 4. Finally, Section 5 presents conclusions.

2. Preliminaries

In this section we present some basic definitions and notations of the local fractional operators which are used further through the paper.

Let us denote local fractional continuity of as

Definition 1. Local fractional derivative operator of at the point is given by [29, 30, 3438]: where and .
Local fractional derivative of high order and local fractional partial derivative of high order are written in the form [29, 30, 38] respectively.
As inverse of local fractional differential operator, the local fractional integral operator of in the interval is defined as [29, 30, 3638] where a partition of the interval is denoted as , and , , and .
The properties are only presented as follows [29, 30, 37]:

3. Analysis of the Method

Let us rewrite the heat equations with the heat generation arising in fractal transient conduction in the form subject to the initial and boundary conditions where and symbolize and , respectively.

By defining the twofold local fractional integral operator as , we have so that Hence, we get where So, from (15) we have iterative formula as follows: where .

Finally, the exact solution can be constructed as follows:

4. Illustrative Examples

Example 1. In view of (3), we consider , , and .
We have subject to the initial value condition From (19) we have the following recursive relations: In view of (21), the first few terms of the decomposition series read From (25) we get Therefore, the exact solution of (19) can be written as The value of the fractal-dimension order of the behavior of the solution is shown in Figure 1.

462535.fig.001
Figure 1: Solution for the one-dimensional heat equations with a fixed value .

Example 2. When , , and , we get We give the initial value condition as follows: From (19) we have the following recursive relations: From (27), we have the first few terms of the decomposition series as follows: Hence, we get So, the exact solution of (19) reads The solution with fractal-dimension order is shown in Figure 2.

462535.fig.002
Figure 2: Solution for the one-dimensional heat equations with a fixed value .

Example 3. When , , and , we get The initial value condition is presented as follows: From (19) the recursive relations follow In view of (27), we get the few terms of the series; namely, Hence, we get So, the exact solution of (19) reads Figure 3 shows the exact solution when .

462535.fig.003
Figure 3: The surface shows the exact solution with a fixed value .

5. Conclusions

In this work, analytical solutions for the one-dimensional heat equations with the heat generation arising in fractal transient conduction associated with local fractional derivative operators were discussed. The obtained solutions are nondifferentiable functions, which are Cantor functions and they discontinuously depend on the local fractional derivative. It is shown that the local fractional Adomian decomposition method is an efficient and simple tool for solving local fractional differential equations.

Conflict of Interests

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

Acknowledgments

This paper was supported by the National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 11126213 and no. 61170317), and the National Natural Science Foundation of the Hebei Province (no. E2013209215).

References

  1. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, Orlando, Fla, USA, 1986. View at MathSciNet
  2. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. View at MathSciNet
  3. G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Tatari and M. Dehghan, “The use of the Adomian decomposition method for solving multipoint boundary value problems,” Physica Scripta, vol. 73, no. 6, pp. 672–676, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. A.-M. Wazwaz, “Adomian decomposition method for a reliable treatment of the Bratu-type equations,” Applied Mathematics and Computation, vol. 166, no. 3, pp. 652–663, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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
  7. E. Larsson, “A domain decomposition method for the Helmholtz equation in a multilayer domain,” SIAM Journal on Scientific Computing, vol. 20, no. 5, pp. 1713–1731, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Tatari, M. Dehghan, and M. Razzaghi, “Application of the Adomian decomposition method for the Fokker-Planck equation,” Mathematical and Computer Modelling, vol. 45, no. 5-6, pp. 639–650, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. A. T. Machado, D. Baleanu, and A. C. Luo, Nonlinear Dynamics of Complex Systems: Applications in Physical, Biological and Financial Systems, Springer, New York, NY, USA, 2011.
  10. J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, World Scientific Publishing, Singapore, 2012. View at MathSciNet
  11. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  12. 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, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. K. S. Hedrih, “Analytical mechanics of fractional order discrete system vibrations,” Advances in Nonlinear Sciences, vol. 3, pp. 101–148, 2011.
  14. K. Hedrih, “Modes of the homogeneous chain dynamics,” Signal Processing, vol. 86, no. 10, pp. 2678–2702, 2006. View at Publisher · View at Google Scholar · View at Scopus
  15. K. Hedrih, “Dynamics of coupled systems,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 2, pp. 310–334, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. 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
  17. H. Jafari and V. Daftardar-Gejji, “Solving a system of nonlinear fractional differential equations using Adomian decomposition,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 644–651, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. S. Ray and R. K. Bera, “Analytical solution of a fractional diffusion equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 329–336, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Q. Wang, “Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1048–1055, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. H. Jafari and V. Daftardar-Gejji, “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 488–497, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. Safari, D. D. Ganji, and M. Moslemi, “Application of He's variational iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2091–2097, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. El-Shahed, “Adomian decomposition method for solving Burgers equation with fractional derivative,” Journal of Fractional Calculus, vol. 24, pp. 23–28, 2003. View at Zentralblatt MATH · View at MathSciNet
  24. 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.
  25. 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
  26. H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, “Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations,” Abstract and Applied Analysis, vol. 2013, Article ID 587179, 5 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. 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
  28. A. Atangana and A. K. 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
  29. X.-J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, 2011.
  30. X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
  31. 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
  32. A. Carpinteri and A. Sapora, “Diffusion problems in fractal media defined on Cantor sets,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 90, no. 3, pp. 203–210, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. A. K. Golmankhaneh, V. Fazlollahi, and D. Baleanu, “Newtonian mechanics on fractals subset of real-line,” Romania Reports in Physics, vol. 65, pp. 84–93, 2013.
  34. X.-J. Yang, H. M. Srivastava, J.-H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,” Physics Letters A, vol. 377, no. 28–30, pp. 1696–1700, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  35. X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of Navier-Stokes equations on Cantor sets,” Mathematical Problems in Engineering, vol. 2013, Article ID 769724, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  36. X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,” Boundary Value Problems, vol. 2013, no. 1, pp. 131–146, 2013. View at Publisher · View at Google Scholar
  37. 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.
  38. X.-J. Yang, D. Baleanu, M. P. Lazarević, and M. S. Cajić, “Fractal boundary value problems for integral and differential equations with local fractional operators,” Thermal Science, 2013. View at Publisher · View at Google Scholar
  39. A. M. Yang, Y. Z. Zhang, and Y. Long, “The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar,” Thermal Science, vol. 17, no. 3, pp. 707–713, 2013. View at Publisher · View at Google Scholar
  40. 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 Publisher · View at Google Scholar