Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2016 (2016), Article ID 2913539, 5 pages
http://dx.doi.org/10.1155/2016/2913539
Research Article

The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator

Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq

Received 5 June 2016; Accepted 24 August 2016

Academic Editor: Zhenhua Guo

Copyright © 2016 Hassan Kamil Jassim. 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

We used the local fractional variational iteration transform method (LFVITM) coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.

1. Introduction

The diffusion equation is a partial differential equation that portrays density dynamics in a material that undertakes diffusion. It is also used to describe progression demonstrating diffusive-like performance, for example, the transmission of alleles in a population genetics [13]. The three-dimensional diffusion equation in fractal heat transfer involving local fractional derivatives was presented assubject to the initial conditionwhere the local fractional Laplace operator is defined as follows (see [48]): is a nondifferentiable diffusion coefficient, and is satisfied with the nondifferentiable temperature distribution, while the three-dimensional wave equation involving local fractional derivatives was presented assubject to the initial conditionsMany physical problems are governed by partial differential equations (PDEs), and the solution of these equations has been a subject of many investigators in recent years. The diffusion and wave equations have been successfully modeled for many physical and engineering phenomena such as seismic analysis, rheology, fluid flow, viscous damping, viscoelastic materials, and polymer physics [911].

Recently, the diffusion and wave problems were studied by several authors by using local fractional decomposition method [1215], local fractional variational iteration [1517], local fractional series expansion [18], local fractional functional decomposition method [19, 20], local fractional Laplace decomposition method [21], local fractional homotopy perturbation method [22], local fractional similarity solution [23], and local fractional differential transform method [24, 25]. In this paper, our aims are to present the coupling method of local fractional Laplace transform and variational iteration method, which is called the local fractional variational iteration transform method, and to use it to solve three-dimensional diffusion and wave equations with local fractional derivative.

2. Mathematical Fundamentals

In this section, we present the basic theory of local fractional calculus and concepts of local fractional Laplace transform (see [1215]).

Definition 1. One says that a function is local fractional continuous at ; if it holds,with , for and . For , it is called local fractional continuous on , denoted by .

Definition 2. Setting , the local fractional derivative of at is defined aswhere .

Definition 3. Let one denote a partition of the interval as , , and with and The local fractional integral of in the interval is given by

Definition 4. Let . The Yang-Laplace transform of is given bywhere the latter integral converges and .

Definition 5. The inverse formula of the Yang-Laplace transforms of is given bywhere ; fractal imaginary unit is , and .

The properties for local fractional Laplace transform used in the paper are given as

3. LFVITM for Three-Dimensional Diffusion Problems

We first rewrite problem (1) in the local fractional operator formwhere the local fractional differential operators , , , and are defined byAdopting the local fractional Laplace transform (denoted in this paper by ) to both sides of (12) and using the initial condition leads toOperating with the inverse of local fractional Laplace transform on both sides of (14) givesDeriving both sides of (15) with respect to , we haveBy the correction function of the irrational methodfinally, the solution is given byWe now consider the initial conditions of (2); namely, we haveConsequently, we obtainand so on.

The solution in a nondifferentiable series formis readily obtained.

Therefore, the exact solution can be written as

4. LFVITM for Three-Dimensional Wave Problems

We first rewrite the problem (4) in the local fractional operator formApplying the local fractional Laplace transform to both sides of (24) and using the initial condition leads toOperating with the inverse of local fractional Laplace transform on both sides of (25) givesDeriving both sides of (26) with respect to , we obtainBy the correction function of the irrational method,Finally, the solution is given byWe now consider the initial conditions of (5); namely,Starting with the zeroth approximation,Substituting (31) in (28) we obtain the following successive approximations:and so on.

The solution in a nondifferentiable series formis readily obtained.

Therefore, the exact solution can be written as

5. Conclusion

In this work, we studied the local fractional variational iteration transform method to solve three-dimensional diffusion and wave equations involving local fractional derivative operator and their nondifferentiable solutions were obtained. This method can also be applied to a large class of system of partial differential equations with approximations that converges rapidly to accurate solutions.

Competing Interests

The author declares that there are no competing interests regarding this paper.

Acknowledgments

Hassan Kamil Jassim acknowledges Ministry of Higher Education and Scientific Research in Iraq for its support of this work.

References

  1. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
  2. B. Q. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Computational Fluid and Solid Mechanics, Springer, London, UK, 2006. View at MathSciNet
  3. H. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics, Prentice Hall, Upper Saddle River, NJ, USA, 2007.
  4. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, 2011.
  5. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Scientific, New York, NY, USA, 2012.
  6. V. Christianto and B. Rahul, “A derivation of proca equations on cantor sets: a local fractional approach,” Bulletin of Mathematical Sciences & Applications, vol. 10, pp. 48–56, 2014. View at Publisher · View at Google Scholar
  7. H.-Y. Liu, J.-H. He, and Z.-B. Li, “Fractional calculus for nanoscale flow and heat transfer,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24, no. 6, pp. 1227–1250, 2014. View at Publisher · View at Google Scholar · View at Scopus
  8. Y.-J. Hao, H. M. Srivastava, H. Jafari, and X.-J. Yang, “Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates,” Advances in Mathematical Physics, vol. 2013, Article ID 754248, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997.
  10. A. M. Spasic and M. P. Lazarevic, “Electroviscoelasticity of liquid/liquid interfaces: fractional-order model,” Journal of Colloid and Interface Science, vol. 282, no. 1, pp. 223–230, 2005. View at Publisher · View at Google Scholar · View at Scopus
  11. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  12. X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximation solutions for diffusion equation on Cantor time-space,” Proceeding of the Romanian Academy A, vol. 14, no. 2, pp. 127–133, 2013. View at Google Scholar
  13. Z.-P. Fan, H. K. Jassim, R. K. Raina, and X.-J. Yang, “Adomian decomposition method for three-dimensional diffusion model in fractal heat transfer involving local fractional derivatives,” Thermal Science, vol. 19, supplement 1, pp. S137–S141, 2015. View at Publisher · View at Google Scholar · View at Scopus
  14. H. Jafari and H. K. Jassim, “Local fractional adomian decomposition method for solving two dimensional heat conduction equations within local fractional operators,” Journal of Advance in Mathematics, vol. 9, no. 4, pp. 2574–2582, 2014. View at Google Scholar
  15. D. Baleanu, J. A. T. Machado, C. Cattani, M. C. Baleanu, and X.-J. Yang, “Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators,” Abstract and Applied Analysis, vol. 2014, Article ID 535048, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. X.-J. Yang, D. Baleanu, Y. Khan, and S. T. Mohyud-Din, “Local fractional variational iteration method for diffusion and wave equations on Cantor sets,” Romanian Journal of Physics, vol. 59, no. 1-2, pp. 36–48, 2014. View at Google Scholar · View at MathSciNet · View at Scopus
  17. S. Xu, X. Ling, Y. Zhao, and H. K. Jassim, “A novel schedule for solving the two-dimensional diffusion problem in fractal heat transfer,” Thermal Science, vol. 19, supplement 1, pp. S99–S103, 2015. View at Publisher · View at Google Scholar · View at Scopus
  18. A.-M. Yang, X.-J. Yang, and Z.-B. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,” Abstract and Applied Analysis, vol. 2013, Article ID 351057, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Y. Cao, W. G. Ma, and L. C. Ma, “Local fractional functional method for solving diffusion equations on Cantor sets,” Abstract and Applied Analysis, vol. 2014, Article ID 803693, 6 pages, 2014. View at Publisher · View at Google Scholar
  20. S.-Q. Wang, Y.-J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,” Abstract and Applied Analysis, vol. 2014, Article ID 176395, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. H. K. Jassim, “Local fractional Laplace decomposition method for nonhomogeneous heat equations arising in fractal heat flow with local fractional derivative,” International Journal of Advances in Applied Mathematics and Mechanics, vol. 2, no. 4, pp. 1–7, 2015. View at Google Scholar
  22. Y. Zhang, C. Cattani, and X.-J. Yang, “Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains,” Entropy, vol. 17, no. 10, pp. 6753–6764, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. X.-J. Yang, D. Baleanu, and H. M. Srivastava, “Local fractional similarity solution for the diffusion equation defined on Cantor sets,” Applied Mathematics Letters, vol. 47, pp. 54–60, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. X.-J. Yang, J. A. T. Machado, and H. M. Srivastava, “A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach,” Applied Mathematics and Computation, vol. 274, pp. 143–151, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. H. Jafari, H. K. Jassim, F. Tchier, and D. Baleanu, “On the approximate solutions of local fractional differential equations with local fractional operator,” Entropy, vol. 18, no. 150, pp. 1–12, 2016. View at Publisher · View at Google Scholar