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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 462535, 5 pages
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.
- G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, Orlando, Fla, USA, 1986.
- G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, World Scientific Publishing, Singapore, 2012.
- F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010.
- 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.
- K. S. Hedrih, “Analytical mechanics of fractional order discrete system vibrations,” Advances in Nonlinear Sciences, vol. 3, pp. 101–148, 2011.
- K. Hedrih, “Modes of the homogeneous chain dynamics,” Signal Processing, vol. 86, no. 10, pp. 2678–2702, 2006.
- K. Hedrih, “Dynamics of coupled systems,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 2, pp. 310–334, 2008.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- M. El-Shahed, “Adomian decomposition method for solving Burgers equation with fractional derivative,” Journal of Fractional Calculus, vol. 24, pp. 23–28, 2003.
- 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.
- J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,” Thermal Science, vol. 14, no. 2, pp. 291–316, 2010.
- 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.
- 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.
- 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.
- 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.
- 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 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.