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GEGENBAUER WAVELETS OPERATIONAL MATRIX METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS

  • UR REHMAN, MUJEEB (School of Natural Sciences National University of Sciences and Technology) ;
  • SAEED, UMER (School of Natural Sciences National University of Sciences and Technology)
  • Received : 2014.12.23
  • Published : 2015.09.01

Abstract

In this article we introduce a numerical method, named Gegenbauer wavelets method, which is derived from conventional Gegenbauer polynomials, for solving fractional initial and boundary value problems. The operational matrices are derived and utilized to reduce the linear fractional differential equation to a system of algebraic equations. We perform the convergence analysis for the Gegenbauer wavelets method. We also combine Gegenbauer wavelets operational matrix method with quasilinearization technique for solving fractional nonlinear differential equation. Quasilinearization technique is used to discretize the nonlinear fractional ordinary differential equation and then the Gegenbauer wavelet method is applied to discretized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Gegenbauer wavelet method. Numerical examples are provided to illustrate the efficiency and accuracy of the methods.

Keywords

References

  1. A. Ali, M. A. Iqbal, and S. T. Mohyud-Din, Hermite wavelets method for boundary value problems, Int. J. Mod. Appl. Phys. 3 (2013), no. 1, 38-47.
  2. A. Arikoglu and I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos Solitons Fractals 34 (2007), no. 5, 1473-1481. https://doi.org/10.1016/j.chaos.2006.09.004
  3. E. Babolian and F. Fattahzdeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput. 188 (2007), no. 1, 417-426. https://doi.org/10.1016/j.amc.2006.10.008
  4. R. E. Bellman, Functional equations in the theory of dynamic programming. II. Nonlinear differential equations, Proc. Natl. Acad. Sci. 41 (1955), 482-485. https://doi.org/10.1073/pnas.41.7.482
  5. R. E. Bellman, Functional equations in the theory of dynamic programming. V. Positivity and quasilinearity, Proc. Natl. Acad. Sci. 41 (1955), 743-746. https://doi.org/10.1073/pnas.41.10.743
  6. R. E. Bellman and R. E. Kalaba, Quasilinearization and nonlinear boundary-value problems, American Elsevier Publishing Company, 1965.
  7. C. Chen and C. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE P.-Contr. Theor. Appl. 144 (1997), 87-94. https://doi.org/10.1049/ip-cta:19970702
  8. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909-996. https://doi.org/10.1002/cpa.3160410705
  9. M. Dehghan and M. Lakestani, Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions, Int. J. Comput. Math. 85 (2008), no. 9, 1455-1461. https://doi.org/10.1080/00207160701534763
  10. J. V. Devi, F. A. McRae, and Z. Drici, Generalized quasilinearization for fractional differential equations, Comput. Math. Appl. 59 (2010), no. 3, 1057-1062. https://doi.org/10.1016/j.camwa.2009.05.017
  11. J. V. Devi and C. Suseela, Quasilinearization for fractional differential equations, Commun. Appl. Anal. 12 (2008), no. 4, 407-418.
  12. K. T. Elgindy and K. A. Smith-Miles, Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices, J. Comput. Appl. Math. 237 (2013), no. 1, 307-325. https://doi.org/10.1016/j.cam.2012.05.024
  13. 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, Appl. Math. Comp. 160 (2005), no. 3, 683-699. https://doi.org/10.1016/j.amc.2003.11.026
  14. S. A. El-Wakil, A. Elhanbaly, and M. A. Abdou, Adomian decomposition method for solving fractional nonlinear differential equations, Appl. Math. Comput. 182 (2006), no. 1, 313-324. https://doi.org/10.1016/j.amc.2006.02.055
  15. N. Engheta, On fractional calculus and fractional multipoles in electromagnetism, IEEE T. Antenn. Propag. 44 (1996), no. 4, 554-566. https://doi.org/10.1109/8.489308
  16. I. Hashim, O. Abdulaziz, and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 3, 674-684. https://doi.org/10.1016/j.cnsns.2007.09.014
  17. E. Hesameddini, S. Shekarpaz, and H. Latifizadeh, The Chebyshev wavelet method for numerical solutions of a fractional oscillator, Int. J. Appl. Math. Research 1 (2012), no. 4, 493-509.
  18. A. Kilicman and Z. A. A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comp. 187 (2007), no. 1, 250-265. https://doi.org/10.1016/j.amc.2006.08.122
  19. V. V. Kulish and J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng. 124 (2002), 803-806. https://doi.org/10.1115/1.1478062
  20. C. Lederman, J.-M. Roquejoffre, and N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames, Ann. Mat. Pura Appl. 183 (2004), no. 2, 173-239. https://doi.org/10.1007/s10231-003-0085-1
  21. U. Lepik, Solving PDEs with the aid of two-dimensional Haar wavelets, Comput. Math. Appl. 61 (2011), no. 7, 1873-1879. https://doi.org/10.1016/j.camwa.2011.02.016
  22. Y. Li, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), no. 9, 2284-2292. https://doi.org/10.1016/j.cnsns.2009.09.020
  23. R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl. 59 (2010), no. 5, 1586-1593. https://doi.org/10.1016/j.camwa.2009.08.039
  24. F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, pp. 291-348, Springer-Verlag, New York, 1997.
  25. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), no. 1, 80-90. https://doi.org/10.1016/j.apnum.2005.02.008
  26. R. N. Mohapatra, K. Vajravelu, and Y. Yin, An improved quasilinearization method for second order nonlinear boundary value problems, J. Math. Anal. Appl. 214 (1997), no. 1, 55-62. https://doi.org/10.1006/jmaa.1997.5583
  27. C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-order systems and controls, Advances in Industrial Control, Springer, 2010.
  28. M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, Internat. J. Systems Sci. 32 (2001), no. 4, 495-502. https://doi.org/10.1080/00207720120227
  29. M. Razzaghi and S. Yousefi, Legendre wavelets method for constrained optimal control problems, Math. Methods Appl. Sci. 25 (2002), no. 7, 529-539. https://doi.org/10.1002/mma.299
  30. M. Rehman and R. A. Khan, A numerical method for solving boundary value problems for fractional differential equations, Appl. Math. Model. 36 (2012), no. 3, 894-907. https://doi.org/10.1016/j.apm.2011.07.045
  31. U. Saeed and M. Rehman, Haar wavelet-quasilinearization technique for fractional nonlinear differential equations, Appl. Math. Comput. 220 (2013), 630-648. https://doi.org/10.1016/j.amc.2013.07.018
  32. U. Saeed and M. Rehman, Wavelet-Galerkin quasilinearization method for nonlinear boundary value problems, Abstr. Appl. Anal. 2014 (2014), Article ID 868934, 10 pages.
  33. U. Saeed and M. Rehman, Hermite wavelet method for fractional delay differential equations, J. Difference Equations 2014 (2014), Article ID 359093, 8 pages. https://doi.org/10.1186/1687-1847-2014-8
  34. L. R. Soares, H. M. de Oliveira, and R. J. D. Sobral Cintra, New compactly supported scaling and wavelet functions derived from Gegenbauer polynomials, Electrical and Computer Engineering, Canadian Conference on 2-5 May 2004 (vol.4) (2004), 2347-2350; DOI: 10.1109/CCECE.2004.1347717.
  35. S. G. Venkatesh, S. K. Ayyaswamy, and S. R. Balachandar, The Legendre wavelet method for solving initial value problems of Bratu-type, Comput. Math. Appl. 63 (2012), no. 8, 1287-1295. https://doi.org/10.1016/j.camwa.2011.12.069
  36. Y. Wang and Q. Fan, The second kind Chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comp. 218 (2012), no. 17, 8592-8601. https://doi.org/10.1016/j.amc.2012.02.022
  37. Y.Wang, H. Song, and D. Li, Solving two-point boundary value problems using combined homotopy perturbation method and Greens function method, Appl. Math. Comput. 212 (2009), no. 2, 366-376. https://doi.org/10.1016/j.amc.2009.02.036

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