Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

  • Yang, Yin (Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University) ;
  • Chen, Yanping (School of Mathematical Sciences South China Normal University) ;
  • Huang, Yunqing (Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University)
  • Received : 2013.05.20
  • Published : 2014.01.01
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Abstract

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Fredholm-Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in $L^{\infty}$ norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.

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References

  1. P. Agrawal and P. Kumar, Comparison of five numerical schemes for fractional differential equations, Advances in fractional calculus, 43-60, Springer, Dordrecht, 2007.
  2. W. M. Ahmad and R. EL-Khazali, Fractional-order dynamical models of love, Chaos Solitons Fractals 33 (2007), no. 4, 1367-1375. https://doi.org/10.1016/j.chaos.2006.01.098
  3. P. Baratella and A. Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004), no. 2, 401-418. https://doi.org/10.1016/j.cam.2003.08.047
  4. A. H. Bhrawy and M. A. Alghamdi, A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals, Bound. Value Probl. 1 (2012), no. 62, 1-13.
  5. A. H. Bhrawy and A. S. Alofi, The operational matrix of fractional integration for shifted Chebyshev polynomials, Appl. Math. Lett. 26 (2013), no. 1, 25-31. https://doi.org/10.1016/j.aml.2012.01.027
  6. A. H. Bhrawy and M. Alshomrani, A shifted Legendre spectral method for fractional-order multi-point boundary value problems, Advan Differ Eqs. 2012 (2012), 1-8. https://doi.org/10.1186/1687-1847-2012-1
  7. A. H. Bhrawy, A. S. Alofi, and S. S. Ezz-Eldien, A quadrature tau method for fractional differential equations with variable coefficients, Appl. Math. Lett. 24 (2011), no. 12, 2146-2152. https://doi.org/10.1016/j.aml.2011.06.016
  8. A. H. Bhrawy, M. M. Tharwat, and A. Yildirim, A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations, Appl. Math. Model. 37 (2013), no. 6, 4245-4252. https://doi.org/10.1016/j.apm.2012.08.022
  9. C. Canuto, M. Y. Hussaini, and A. Quarteroni, Spectral Methods, Fundamentals in single domains, Springer-Verlag, Berlin, 2006.
  10. Y. Chen and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel, Math. Comp. 79 (2010), no. 269, 147-167. https://doi.org/10.1090/S0025-5718-09-02269-8
  11. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Heidelberg, 2nd Edition, 1998.
  12. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model. 35 (2011), no. 12, 5662-5672. https://doi.org/10.1016/j.apm.2011.05.011
  13. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldie, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl. 62 (2011), no. 5, 2364-2373. https://doi.org/10.1016/j.camwa.2011.07.024
  14. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldie, A new Jacobi operational matrix: an application for solving fractional differential equations, Appl. Math. Model. 36 (2012), no. 10, 4931-4943. https://doi.org/10.1016/j.apm.2011.12.031
  15. J. H. He, Nonlinear oscillation with fractional derivative and its applications, In: International Conference on Vibrating Engineering, Dalian, China, 1998, 288-291.
  16. J. H. He, Some applications of nonlinear fractional differential equations and therir approximations, Bull.Sci. Technol. 15 (1999), 86-90.
  17. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1989.
  18. F. Huang and F. Liu, The time fractional diffusion equation and the advection-dispersion equation, ANZIAM J. 46 (2005), 317-330. https://doi.org/10.1017/S1446181100008282
  19. H. Jafari and S. A. Yousefi, Application of Legendre wavelets for solving fractional differential equations, Comput. Math. Appl. 62 (2011), no. 3, 1038-1045. https://doi.org/10.1016/j.camwa.2011.04.024
  20. M. M. Khader and A. S. Hendy, An efficient numerical scheme for solving fractional optimal control problems, Int. J. Nonlinear Sci. 14 (2012), no. 3, 287-297.
  21. M. M. Khader, N. H. Sweilam, and A. M. S. Mahdy, An efficient numerical method for solving the fractional diffusion equation, J. Appl. Math. Bioinf. 1 (2011), no. 2, 1-12.
  22. A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.
  23. Y. L. Li, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput. 216 (2010), no. 8, 2276-2285. https://doi.org/10.1016/j.amc.2010.03.063
  24. Y. Luchko and R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivatives, Preprint series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998.
  25. F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics (Udine, 1996), 291-348, CISM Courses and Lectures, 378, Springer, Vienna, 1997.
  26. G. Mastroianni and D. Occorsto, Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey, J. Comput. Appl. Math. 134 (2001), no. 1-2, 325-341. https://doi.org/10.1016/S0377-0427(00)00557-4
  27. P. Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282 (1984), no. 2, 669-698. https://doi.org/10.1090/S0002-9947-1984-0732113-4
  28. A. Pedas and E. Tamme, Piecewise polynomial collocation for linear boundary value problems of fractional differential equations, J. Comput. Appl. Math. 236 (2012), no. 13, 3349-3359. https://doi.org/10.1016/j.cam.2012.03.002
  29. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  30. D. L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc. 150 (1970), 41-53. https://doi.org/10.1090/S0002-9947-1970-0410210-0
  31. D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc. 162 (1971), 157-170.
  32. E. A. Rawashdeh, Legendre wavelets method for fractional integro-differential equations, Appl. Math. Sci. 5 (2011), no. 49-52, 2467-2474.
  33. E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput. 176 (2006), no. 1, 1-6. https://doi.org/10.1016/j.amc.2005.09.059
  34. M. Rehman and R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 11, 4163-4173. https://doi.org/10.1016/j.cnsns.2011.01.014
  35. A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. 59 (2010), no. 3, 1326-1336. https://doi.org/10.1016/j.camwa.2009.07.006
  36. A. Saadatmandi and M. Dehghan, A Legendre collocation method for fractional integro-differential equations, J. Vib. Control 17 (2011), no. 13, 2050-2058. https://doi.org/10.1177/1077546310395977
  37. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon & Breach, Yverdon, 1993.
  38. N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput. 131 (2002), no. 2-3, 517-529. https://doi.org/10.1016/S0096-3003(01)00167-9
  39. N. H. Sweilam and M. M. Khader, A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations, ANZIAM J. 51 (2010), no. 4, 464-475. https://doi.org/10.1017/S1446181110000830
  40. N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, Homotopy perturbation method for linear and nonlinear system of fractional integro-differential equations, Int. J. Comput. Math. Numer. Simul. 1 (2008), no. 1, 73-87.
  41. Y. Wei and Y. Chen, Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv. Appl. Math. Mech. 4 (2012), no. 1, 1-20. https://doi.org/10.4208/aamm.10-m1055

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