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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 259125, 6 pages
Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators
1College of Science, Hebei United University, Tangshan 063009, China
2College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China
3School of Civil Engineering and Architecture, Chongqing Jiaotong University, Chongqing 400074, China
4Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4
5Department of Mathematics and Mechanics, China University of Mining and Technology, Jiangsu, Xuzhou 221008, China
Received 31 July 2013; Accepted 17 October 2013
Academic Editor: Bashir Ahmad
Copyright © 2013 Ai-Min 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.
- R. Kreß and G. F. Roach, “Transmission problems for the Helmholtz equation,” Journal of Mathematical Physics, vol. 19, no. 6, pp. 1433–1437, 1978.
- A. Deraemaeker, I. Babuška, and P. Bouillard, “Dispersion and pollution of the FEM solution for the helmholtz equation in one, two and three dimensions,” International Journal for Numerical Methods in Engineering, vol. 46, no. 4, pp. 471–499, 1999.
- A. Hannukainen, M. Huber, and J. Schöberl, “A mixed hybrid finite element method for the Helmholtz equation,” Journal of Modern Optics, vol. 58, no. 5-6, pp. 424–437, 2011.
- S. Momani and S. Abuasad, “Application of He's variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119–1123, 2006.
- M. Rafei and D. D. Ganji, “Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 3, pp. 321–328, 2006.
- J.-D. Benamou and B. Desprès, “A domain decomposition method for the Helmholtz equation and related optimal control problems,” Journal of Computational Physics, vol. 136, no. 1, pp. 68–82, 1997.
- M. M. Grigoriev and G. F. Dargush, “A fast multi-level boundary element method for the Helmholtz equation,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 3–5, pp. 165–203, 2004.
- S. Tomioka, S. Nisiyama, M. Itagaki, and T. Enoto, “Internal field error reduction in boundary element analysis for Helmholtz equation,” Engineering Analysis with Boundary Elements, vol. 23, no. 3, pp. 211–222, 1999.
- O. F. Næss and K. S. Eckhoff, “A modified Fourier-Galerkin method for the Poisson and Helmholtz equations,” Journal of Scientific Computing, vol. 17, no. 1–4, pp. 529–539, 2002.
- C. M. Linton, “The Green's function for the two-dimensional Helmholtz equation in periodic domains,” Journal of Engineering Mathematics, vol. 33, no. 4, pp. 377–401, 1998.
- A. Dienstfrey, F. Hang, and J. Huang, “Lattice sums and the two-dimensional, periodic Green's function for the Helmholtz equation,” Proceedings of the Royal Society A, vol. 457, no. 2005, pp. 67–85, 2001.
- J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
- R. L. Magin, Fractional Calculus in Bioengineering, Begerll House, West Redding, Conn, USA, 2006.
- J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.
- G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2008.
- F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific Publishing, Singapore, 2010.
- R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific Publishing, Singapore, 2011.
- J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, World Scientific Publishing, Singapore, 2012.
- S. Das, Functional Fractional Calculus, Springer, Berlin, Germany, 2nd edition, 2011.
- V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, Germany, 2011.
- A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, UK, 2012.
- H. Sheng, Y. Chen, and T. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Signals and Communication Technology, Springer, New York, NY, USA, 2012.
- D. Baleanu, J. A. T. Machado, and A. C. Luo, Fractional Dynamics and Control, Springer, New York, NY, USA, 2012.
- E. Goldfain, “Fractional dynamics, Cantorian space-time and the gauge hierarchy problem,” Chaos, Solitons & Fractals, vol. 22, no. 3, pp. 513–520, 2004.
- M. S. Samuel and A. Thomas, “On fractional Helmholtz equations,” Fractional Calculus & Applied Analysis, vol. 13, no. 3, pp. 295–308, 2010.
- P. K. Gupta, A. Yildirim, and K. N. Rai, “Application of He's homotopy perturbation method for multi-dimensional fractional Helmholtz equation,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22, no. 3-4, pp. 424–435, 2012.
- K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998.
- A. Carpinteri, B. Chiaia, and P. Cornetti, “On the mechanics of quasi-brittle materials with a fractal microstructure,” Engineering Fracture Mechanics, vol. 70, no. 16, pp. 2321–2349, 2003.
- F. Ben Adda and J. Cresson, “About non-differentiable functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 721–737, 2001.
- A. Babakhani and V. Daftardar-Gejji, “On calculus of local fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 66–79, 2002.
- Y. Chen, Y. Yan, and K. Zhang, “On the local fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 362, no. 1, pp. 17–33, 2010.
- W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1754–1758, 2010.
- 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. 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.
- 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.
- 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.
- 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.
- W.-H. Su, D. Baleanu, X.-J. Yang, and H. Jafari, “Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method,” Fixed Point Theory and Applications, vol. 2013, no. 1, article 89, pp. 1–11, 2013.
- Y.-J. Yang, D. Baleanu, and X.-J. Yang, “A local fractional variational iteration method for Laplace equation within local fractional operators,” Abstract and Applied Analysis, vol. 2013, Article ID 202650, 6 pages, 2013.
- J.-H. He, “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.