Skip to main navigation menu Skip to main content Skip to site footer
cjpl

Articles

CJPLS: VOL. 12, NO. 2, DECEMBER 2024 (Ongoing)

Computational Analysis of Fractional Integro-Differential Equations using Chebyshev series solution of second kind.

Submitted
December 10, 2024
Published
2024-12-21

Abstract

A new numerical scheme via a Chebyshev series method is used to solve a family of linear fractional integro-differential equations, especially the Fredholm and Volterra equations. The linear fractional integro-differential equation can be transformed into a system of equations for the unknown function itself and its m derivatives by taking into account the mth order Chebyshev series of the unknown function at any point. Using MATLAB 2009 software, this approach provides a straightforward and closed-form solution to a linear fractional integro-differential equation. Finally, examples are provided to illustrate comparisons between the proposed Chebyshev series solution and some existing methods in this direction. The results obtained performed better in terms of quick convergence and stability over the existing Taylor series expansion methodologies.

References

  1. M. A. Ezza,”Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer”, Physics B vol. 406, 2011, pp. 30-36.
  2. S. G. Samko, A. A. Kilbas abd., O. I. Marichev, “Fractional Integrals and Derivatives”, Gordon and Breach Science Publishers, Yverdon, 1993
  3. I.Podlubny, Fractional Differential Equations, Academic Press, New York, 1999
  4. K., B. Miller Ross, “An Introduction to the Fractional Calculus and Fractional Differential”, Wiley, New York, 1993.
  5. A. Arikoglu, and I. Ozkol,”Solution of fractional differential equations by using differential transform method,” Chaos Solitons Fractals, vol. 14, 2007, 1473- 1481.
  6. X. Zhang, B. Tang and Y. He, “Homotopy analysis method for higher-order fractional Integro-Differential Equations”, Comput. Math. Appl. Vol. 62, 2011, 3194-3203.
  7. S. Momani, and M. A. Nour, “Numerical methods for fourth-order fractional integro-differential equations”. Appl. Math. Comput. Vol. 182, 2006, 754-760..
  8. Y. Nawaz, “Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equation”, Comput. Math. Appl. Vol. 61, 2011, 2330-2341.
  9. E. A. Rawashdeh, “Numerical solution of fractional integro-differential equations by collocation method”, Appl. Math. Comput. Vol. 176, 2006, 1-6,
  10. S. Yaabasi, N. Sahin and A. Yildirim, “A collocation approach for solving high-order linear Fredholm-Volterra integro-differential equations”, Math. Comput. Model. Vol. 55, 2012, 547-563.
  11. M. T. Raji, O Fagbemiro and B. I. Olajuwon “Computational Analysis of Fractional Volterra Fredholm Integro-Differential Equations using Eulerian Polynomial basis function”. Journal of Nigerian Association of Mathematical Physics, 68 (1), 2024, 141 – 150.
  12. W. Volk. “The iterated Galerkin methods for linear integro-differential equations”, J. Comput. Appl. Math. 21, 1988, 63–74.
  13. H. Brunner, A. Makroglou, K. Miller “Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution”, Appl. Numer. Math. 23, 1997, 381–402.
  14. Q. Hu, “Interpolation correction for collocation solutions of Fredholm integro-differential equation”, Math. Comput. 223, 1998, 987–999.
  15. M. T. Rashed “Numerical solution of a special type of integro-differential equations”, Appl. Math. Comput. 143, 2003, 73–88.
  16. A. Setia, Y. Liu & A. S. Vatsala “Numerical solution of Fredholm-Volterra Fractional Integro-Differential Equations with Nonlocal Boundary conditions”. Journal of Fractional Calculus and Applications. 5(2), 2013, 155-165.
  17. R.P. Kanwall & K.C. Liu “A Taylor expansion approach for solving integral equations”, Int. J. Math. Ed. Sci. Technol. 20, 1989, 411–414.
  18. M. Sezer, “A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials”, Int. J. Math. Ed. Sci. Technol. 27, 1996, 821–834.
  19. M. Sezer “Taylor polynomial solution of Volterra integral equations”, Int. J. Math. Ed. Sci. Technol. 25, 1994, 625–633.
  20. S. Yalcinbas, “Taylor polynomial solutions of nonlinear Volterra C Fredholm integral equations”, Appl. Math. Comput. 127, 2002, 195–206.
  21. S. Yalcinbas & M. Sezer “The approximate solution of high-order linear Volterra Fredholm integro-differential equations in terms of Taylor polynomials”, Appl. Math. Comput. 112, 2000, 291–308.
  22. L. Huang, X-F, Li, Y. Zhan, X-Y. Duan,”Approximate solution of fractional integro-differential equations by Taylor expansion method”, Computers and Mathematics with Applications, 62, 2011, 1127 – 1134.
  23. S.S. Ray, R. K Bera “Analytical solution of the Bagley Torvik equation by Adomian decomposition method”, Appl. Math. Comput. 168, 2005, 398–410.