Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2009, Article ID 730465, 11 pages
http://dx.doi.org/10.1155/2009/730465
Research Article

On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations

1Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey
2Department of Mathematics, ITTU, 744012 Ashgabat, Turkmenistan
3Department of Mathematics, Ege University, 35100 Izmir, Turkey

Received 26 February 2009; Accepted 17 July 2009

Academic Editor: Paulo Batista Gonçalves

Copyright © 2009 Allaberen Ashyralyev 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. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  2. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993. View at Zentralblatt MATH · View at MathSciNet
  3. J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240–268, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. V. E. Tarasov, “Fractional derivative as fractional power of derivative,” International Journal of Mathematics, vol. 18, no. 3, pp. 281–299, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. E. M. El-Mesiry, A. M. A. El-Sayed, and H. A. A. El-Saka, “Numerical methods for multi-term fractional (arbitrary) orders differential equations,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 683–699, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. M. A. El-Sayed and F. M. Gaafar, “Fractional-order differential equations with memory and fractional-order relaxation-oscillation model,” Pure Mathematics and Applications, vol. 12, no. 3, pp. 296–310, 2001. View at Google Scholar · View at MathSciNet
  7. A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “Numerical solution for multi-term fractional (arbitrary) orders differential equations,” Computational & Applied Mathematics, vol. 23, no. 1, pp. 33–54, 2004. View at Google Scholar · View at MathSciNet
  8. R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, Vienna, Austria, 1997. View at Google Scholar · View at MathSciNet
  9. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in System Application, vol. 2, Lille, France, 1996. View at Google Scholar
  10. A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  11. I. Podlubny and A. M. A. El-Sayed, On Two Definitions of Fractional Calculus, Slovak Academy of Science-Institute of Experimental Physics, 1996.
  12. S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow, Russia, 1967. View at MathSciNet
  13. P. E. Sobolevskii and L. M. Chebotaryeva, “Approximate solution by method of lines of the Cauchy problem for an abstract hyperbolic equations,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 5, pp. 103–116, 1977 (Russian). View at Google Scholar
  14. A. Ashyralyev, M. Martinez, J. Paster, and S. Piskarev, “Weak maximal regularity for abstract hyperbolic problems in function spaces, further progress in analysis,” in Proceedings of the 6th International ISAAC Congress, pp. 679–689, World Scientific, Ankara, Turkey, August 2007.
  15. A. Ashyralyev and N. Aggez, “A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations,” Numerical Functional Analysis and Optimization, vol. 25, no. 5-6, pp. 439–462, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Ashyralyev and I. Muradov, “On difference schemes a second order of accuracy for hyperbolic equations,” in Modelling Processes of Explotation of Gas Places and Applied Problems of Theoretical Gasohydrodynamics, pp. 127–138, Ilim, Ashgabat, Turkmenistan, 1998.
  17. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004. View at Zentralblatt MATH · View at MathSciNet
  18. A. Ashyralyev and Y. Ozdemir, “On nonlocal boundary value problems for hyperbolic-parabolic equations,” Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1075–1089, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Ashyralyev and O. Yildirim, “On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations,” Taiwanese Journal of Mathematics, vol. 13, 22 pages, 2009. View at Google Scholar
  20. A. A. Samarskii, I. P. Gavrilyuk, and V. L. Makarov, “Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces,” SIAM Journal on Numerical Analysis, vol. 39, no. 2, pp. 708–723, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. Ashyralyev and P. E. Sobolevskii, “Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 183–213, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Ashyralyev and M. E. Koksal, “On the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space,” Numerical Functional Analysis and Optimization, vol. 26, no. 7-8, pp. 739–772, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. Ashyralyev and M. E. Koksal, “On the stability of the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 57491, 26 pages, 2007. View at Google Scholar
  24. M. Ashyraliyev, “A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space,” Numerical Functional Analysis and Optimization, vol. 29, no. 7-8, pp. 750–769, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. Ashyralyev and P. E. Sobolevskii, “A note on the difference schemes for hyperbolic equations,” Abstract and Applied Analysis, vol. 6, no. 2, pp. 63–70, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, Russia, 1975.
  27. A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2 of Iterative Methods, Birkhäuser, Basel, Switzerland, 1989. View at MathSciNet