International Scientific Journal


In the present paper we investigate the fractal boundary value problems for the Fredholm\Volterra integral equations, heat conduction and wave equations by using the local fractional decomposition method. The operator is described by the local fractional operators. The four illustrative examples are given to elaborate the accuracy and reliability of the obtained results. [Projekat Ministarstva nauke Republike Srbije, br. OI 174001, III41006 i br. TI 35006]
PAPER REVISED: 2013-07-14
PAPER ACCEPTED: 2013-07-16
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  1. Majumder, M., et al., Nanoscale Hydrodynamics - Enhanced Flow in Carbon Nanotubes, Nature, 438 (2005), pp. 44, doi: 10.1038/438044a, 7064
  2. Xuan, Y., Roetzel, W., Conceptions for Heat Transfer Correlation of Nanofluids, Int. J. Heat Mass Transfer, 43 (2000), 19, pp. 3701-3707
  3. Ramšak, M., Leopold, Š., Heat diffusion in fractal geometry cooling surface, Thermal Science, 16 (2012), 4, pp. 955-968
  4. Yang, X. J., Baleanu, D., Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science, 17 (2013), 2, pp. 625-628
  5. Adomian, G., A review of the decomposition method and some recent results for nonlinear equation, Mathematical and Computer Modelling, 13 (1990), 7, pp. 17-43
  6. Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Dordrecht, Netherland, 1994
  7. Wazwaz, A. M., A reliable technique for solving the weakly singular second kind Volterra-type integral equations, Applied mathematics and computation, 80 (1996), pp. 287-299
  8. Luo, X. G., Wu, Q. B., Zhang, B. Q., Revisit on partial solutions in the Adomian decomposition method: Solving heat and wave equations, Journal of Mathematical Analysis and Applications, 321 (2006), 1, pp. 353-363
  9. Nadeem, S., Akbar, N., Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: Application of Adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 11, pp. 3844-3855
  10. He, J. H., Variational iteration method-a kind of nonlinear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34 (1999), pp. 708-799
  11. He, J. H., A variational iteration approach to nonlinear problems and its applications, Mechanical Applications, 20 (1998), 1, pp. 30-31
  12. He, J. H., Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, 2012 (2012), Article ID 916793, pp. 1-130
  13. Khan, Y., Mohyud-Din, S. T., Coupling of He's polynomials and Laplace transformation for MHD viscous flow over a stretching sheet, International Journal of Nonlinear Sciences and Numerical Simulation, 11 (2010), 12, pp. 1103-1107
  14. Hristov, J., Heat-Balance integral to fractional (Half-Time) heat diffusion sub-model, Thermal Science, 14 (2010), 2, pp. 291-316
  15. Hristov, J., Approximate solutions to fractional sub-diffusion equations: The heat-balance integral method, The European Physical Journal-Special Topics, 193 (2011), pp. 229-243
  16. Li, Z. B., He, J. H., Fractional complex transform for fractional differential equations, Mathematical and Computational Applications, 15 (2010), 5, pp. 970-973
  17. He, J. H., Elagan S. K., Li, Z. B., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Physics Letters A, 376 (2012), 4, pp. 257-259
  18. Jafari H., Seifi, S., Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 5, pp. 2006-2012
  19. Zhang S., Zhang, H. Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, 375 (2011), 7, pp. 1069-1073
  20. Wu, G. C., Lee, E. W. M., Fractional variational iteration method and its application, Physics Letters A, 374 (2010), 25, pp. 2506-2509
  21. Khan, Y., Faraz, N., Yildirim, A., Wu, Q. B., Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science, Computers and Mathematics with Applications, 62 (2011), 5, pp. 2273-2278
  22. He, J. H., A short remark on fractional variational iteration method, Physics Letters A, 375 (2011), 38, pp. 3362-3364
  23. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J., Fractional calculus models and numerical methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, USA, 2012
  24. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherland, 2006
  25. Sabatier, J., Agrawal, O. P., Tenreiro Machado J.A., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, USA, 2007
  26. Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999
  27. Yang, X. J., Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher Limited, Hong Kong, China , 2011
  28. Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012.
  29. Yang, X. J., Local fractional Kernel transform in fractal space and its applications, Advances in Computational Mathematics and its Applications, 1 (2012), 2, pp. 86-93
  30. Yang, X. J., Local fractional integral equations and their applications, Advances in Computer Science and its Applications, 1 (2012), 4, pp. 234-239
  31. Yang, X. J., Zhang, F. R., Local fractional variational iteration method and its algorithms, Advances in Computational Mathematics and its Applications, 1 (2012), 3, pp. 139-145
  32. Yang, X. J., Zhang, Y. D., A new Adomian decomposition procedure scheme for solving local fractional Volterra integral equation, Advances in Information Technology and Management, 1 (2012), 4, pp. 158-161
  33. Yang, X. J., Baleanu, D, Zhong, W. P., Approximation solutions for diffusion equation on Cantor time-space, Proceeding of the Romanian Academy, Series A, 14 (2013), 2, pp. 127-133
  34. Hu, M. S., Agarwal, R. P., Yang, X. J., Local fractional Fourier series with application to wave equation in fractal vibrating string, Abstract and Applied Analysis, 2012 (2012), Article ID 567401, pp. 1-15
  35. Liao, M. K., Yang, X. J., Yan, Q., A new viewpoint to Fourier analysis in fractal space, in: Advances in Applied Mathematics and Approximation Theory (Ed. G. A. Anastassiou, O. Duman) Springer, New York, 2013, pp. 399-411
  36. Zhong, W. P., Gao, F., Shen, X. M., Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral, Advanced Materials Research, 461 (2012), pp. 306-310
  37. Guo, Y., Local fractional Z transform in fractal space, Advances in Digital Multimedia, 1 (2012), 2, pp. 96-102.
  38. Hu, M. S., Baleanu, D., Yang, X. J., One-phase problems for discontinuous heat transfer in fractal media, Mathematical Problems in Engineering, 2013 (2013), Article ID 358473, pp. 1-3
  39. Yang, A. M., Yang, X. J., Li, Z. B., Local fractional series expansion method for solving wave and diffusion equations on Cantor set, Abstract and Applied Analysis, 2013 (2013), Article ID 351057, pp. 1-5
  40. Yang, X. J., Baleanu, D., Tenreiro Machado J. A., Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis, Boundary Value Problems, 2013 (2013), 131, pp. 1-16
  41. Yang, Y. J., Baleanu, D, Yang, X. J., Analysis of wave equations by local fractional Fourier series method, Advances in Mathematical Physics, 2013 (2013), Article ID 632309, pp. 1-7

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