About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 671321, 9 pages
http://dx.doi.org/10.1155/2013/671321
Research Article

A Generalized Version of a Low Velocity Impact between a Rigid Sphere and a Transversely Isotropic Strain-Hardening Plate Supported by a Rigid Substrate Using the Concept of Noninteger Derivatives

1Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9300 Bloemfontein, South Africa
2Department of Mathematics, Texas A&M University-Kingsville, MSC 172, 700 University Boulevard, USA
3Department of Mathematics, Faculty of Art & Sciences, Celal Bayar University, Muradiye Campus, 45047 Manisa, Turkey

Received 26 January 2013; Accepted 4 March 2013

Academic Editor: Hassan Eltayeb

Copyright © 2013 Abdon Atangana 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.

Linked References

  1. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH · View at MathSciNet
  2. 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
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet
  4. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  5. M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, part II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967. View at Publisher · View at Google Scholar
  6. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  7. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet
  8. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2008. View at MathSciNet
  9. A. Yildirim, “An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 4, pp. 445–450, 2009.
  10. S. G. Samko, A. A. Kilbas, and O. I. Maritchev, “Integrals and derivatives of the fractional order and some of their applications,” Nauka i TekhnIka, Minsk, 1987 (Russian).
  11. I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. View at Zentralblatt MATH · View at MathSciNet
  12. A. Atangana, “New class of boundary value problems,” Information Sciences Letters, vol. 1, no. 2, pp. 67–76, 2012.
  13. A. Atangana, “Numerical solution of space-time fractional derivative of groundwater flow equation,” in Proceedings of the International Conference of Algebra and Applied Analysis, vol. 2, no. 1, p. 20, Istanbul, Turkey, June 2012.
  14. G. Jumarie, “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,” Applied Mathematics Letters, vol. 18, no. 7, pp. 817–826, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Davison and C. Essex, “Fractional differential equations and initial value problems,” The Mathematical Scientist, vol. 23, no. 2, pp. 108–116, 1998. View at Zentralblatt MATH · View at MathSciNet
  17. C. F. M. Coimbra, “Mechanics with variable-order differential operators,” Annalen der Physik, vol. 12, no. 11-12, pp. 692–703, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. I. Andrianov and J. Awrejcewicz, “Construction of periodic solutions to partial differential equations with non-linear boundary conditions,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 4, pp. 327–332, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  19. C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons Jr., “A new perturbative approach to nonlinear problems,” Journal of Mathematical Physics, vol. 30, no. 7, pp. 1447–1455, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Delamotte, “Nonperturbative (but approximate) method for solving differential equations and finding limit cycles,” Physical Review Letters, vol. 70, no. 22, pp. 3361–3364, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Shariyat, Automotive Body: Analysis and Design, K. N. Toosi University Press, Tehran, Iran, 2006.
  22. R. Ollson, “Impact response of orthotropic composite plates predicted form a one-parameter differential equation,” American Institute of Aeronautics and Astronautics Journal, vol. 30, no. 6, pp. 1587–1596, 1992. View at Publisher · View at Google Scholar
  23. A. S. Yigit and A. P. Christoforou, “On the impact of a spherical indenter and an elastic-plastic transversely isotropic half-space,” Composites, vol. 4, no. 11, pp. 1143–1152, 1994. View at Publisher · View at Google Scholar
  24. A. S. Yigit and A. P. Christoforou, “On the impact between a rigid sphere and a thin composite laminate supported by a rigid substrate,” Composite Structures, vol. 30, no. 2, pp. 169–177, 1995. View at Publisher · View at Google Scholar
  25. A. P. Christoforou and A. S. Yigit, “Characterization of impact in composite plates,” Composite Structures, vol. 43, pp. 5–24, 1998.
  26. A. P. Christoforou and A. S. Yigit, “Effect of flexibility on low velocity impact response,” Journal of Sound and Vibration, vol. 217, no. 3, pp. 563–578, 1998. View at Publisher · View at Google Scholar
  27. A. S. Yigit and A. P. Christoforou, “Limits of asymptotic solutions in low-velocity impact of composite plates,” Composite Structures, vol. 81, pp. 568–574, 2007. View at Publisher · View at Google Scholar
  28. A. Atangana and A. Secer, “A note on fractional order derivatives and Table of fractional derivative of some specials functions,” Abstract Applied Analysis. In press.
  29. T. H. Solomon, E. R. Weeks, and H. L. Swinney, “Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow,” Physical Review Letters, vol. 71, pp. 3975–3978, 1993. View at Publisher · View at Google Scholar
  30. R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Connecticut, UK, 2006.
  31. R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance, vol. 190, pp. 255–270, 2008. View at Publisher · View at Google Scholar
  32. A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, “Fractional diffusion in inhomogeneous media,” Journal of Physics, vol. 38, no. 42, pp. L679–L684, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. F. Santamaria, S. Wils, E. de Schutter, and G. J. Augustine, “Anomalous diffusion in purkinje cell dendrites caused by spines,” Neuron, vol. 52, no. 4, pp. 635–648, 2006. View at Publisher · View at Google Scholar · View at Scopus
  34. H. G. Sun, W. Chen, and Y. Q. Chen, “Variable order fractional differential operators in anomalous diffusion modelling,” Journal of Physics A, vol. 388, pp. 4586–4592, 2009.
  35. A. Atangana and A. Secer, “The time-fractional coupled-Korteweg-de-vries equations,” Abstract Applied Analysis, vol. 2013, Article ID 947986, 8 pages, 2013. View at Publisher · View at Google Scholar
  36. A. Atangana and J. F. Botha, “Analytical solution of groundwater flow equation via Homotopy Decomposition Method,” Journal of Earth Science & Climatic Change, vol. 3, p. 115, 2012.
  37. M. Shariyat, R. Ghajar, and M. M. Alipour, “An analytical solution for a low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate,” Journal of Engineering Mathematics, vol. 75, pp. 107–125, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. J. Awrejcewicz, V. A. Krysko, O. A. Saltykova, and Yu. B. Chebotyrevskiy, “Nonlinear vibrations of the Euler-Bernoulli beam subjected to transversal load and impact actions,” Nonlinear Studies, vol. 18, no. 3, pp. 329–364, 2011. View at Zentralblatt MATH · View at MathSciNet
  39. C. C. Poe Jr. and W. Illg, “Strength of a thick graphite/epoxy rocket motor case after impact by a blunt object.,” in Test Methods for Design Allowable for Fibrous Composites, C. C. Chamis, Ed., vol. 2, pp. 150–179, ASTM, Philadelphia, Pa, USA, 1989, ASTM STP 1003.
  40. C. C. Poe Jr., “Simulated impact damage in a thick graphite/epoxy laminate using spherical indenters,” NASA TM 100539, 1988.