Legendre Spectral Tau Method for Solving the Fractional Integro-Differential Equations with A Weakly Singular Kernel

Document Type : Research Paper


1 Associate Professor, Mathematics and Computer Science Department, Adib mazandaran institute of higher education, Sari, Iran

2 Associate Professor,Mathematica and Computer Science Department, Adib mazandaran institute of higher education, Sari, Iran

3 School of Mathematics and Computer Sciences, Damghan University, P.O.Box 36715-364, Damghan, Iran.


‎In this paper‎, ‎the Tau method based on shifted Legendre polynomials has been introduced to approximate the numerical solutions of a class of fractional integro-differential equations with a weakly singular kernel‎ . ‎By using operational matrices we reduce the problem to a set of algebraic equations‎ . ‎Also the upper bound of the error of the shifted Legendre expansion is investigated‎. ‎Finally‎, ‎several numerical examples are given to illustrate the high accuracy of the method‎.


  1. V. V. Anh, N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys., 104, 1349–1387 (2001).
  2. A. Arikoglu, I. Ozkol, Solution of fractional integro-differential equations by using frac[1]tional differential transform method, Chaos Solitons Fractals, 40, 521–529 (2009).
  3. F. Awawdeh, E.A. Rawashdeh, H.M. Jaradat, Analytic solution of fractional integro[1]differential equations, An. Univ. Craiova Ser. Mat. Inform., 38, 1–1
  4. T. Blaszczyk, M. Ciesielski, M. Klimek, J. Leszczynski, Numerical solution of fractional oscillator equation, Appl. Math. Comput., 218, 2480–2488 (2011).
  5. R. N. Bracewell, A. C. Riddle, Inversion of fan-beam scans in radio astronomy, Astro[1]physical Jornal, 150, 427–434 (1967).
  6. U. Buck, Inversion of molecular scattering data. Reviews of Modern physics, 46, 369–389 (1974).
  7. A.K. Gupta, S. Saha Ray, Travelling wave solution of fractional KdV–Burger–Kuramoto equation describing nonlinear physical phenomena, AIP Adv., 4, 097120;1–11 (2014).
  8. S. B. Healy, J. Haase, O. Lesne, Abel transform inversion of radio occultation measure[1]ments made with a receiver inside the Earth’s atmosphere, Annals of Geophysics, 20, 1253–1256 (2002).
  9. L. Huang, X. F. Li, Y. L. Zhao, X. Y. Duan, Approximate solution of fractional integro[1]differential equations by Taylor expansion method, Comput. Math. Appl. 62, 1127–1134 (2011).
  10. A. J. Jakeman, R. S. Anderssen, Abel type integral equations in stereology ,General discussion. J. Microscopy, 105, 121–133 (1975).
  11. N. A. Khan, A. Ara, S. A. Ali, A. Mahmood, Analytical study of Navier–Stokes equa[1]tion with fractional orders using He’s homotopy perturbation and variational iteration method, Int. J. Nonlinear Sci. Numer. Simul., 10, 1127–1134 (2009).
  12. P. K. Kythe, P. Puri, Computational Method for Linear Integral Equations, Birkhauser, Boston, (2002).
  13. X. H. Ma, C. M. Huang, Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl. Math. Comput., 219, 6750–6760 (2013).
  14. J. B. Macelwane, Evidence on the interior of the earth derived from seismic sources, in Internal Constitution of the Earth, ,Gutenberg B (ed). Dover, New York, 227–304 (1951).
  15. K. Maleknejad, M. N. Sahlan, A. Ostadi, Numerical solution of fractional integro[1]differential equation by using cubic B-spline wavelets, in: Proceedings of the World Congress on Engineering 2013 Vol. I, WCE 2013, July 3–5, London, UK, (2013).
  16. R. C. Mittal, Ruchi Nigam, Solution of fractional integro-differential equations by Ado[1]mian decomposition method, Int. J. Appl. Math. Mech., 4, 87–94 (2008).
  17. S. Momani, R. Qaralleh, An efficient method for solving systems of fractional integro[1]differential equations, Comput. Math. Appl., 52, 459–470 (2006).
  18. E. L. Ortiz, L. Samara, An opperational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing, 27, 15–25 (1981).
  19. I. Podlubny, The Laplace Transform Method for Linear Differential Equations of the Fractional Order, UEF-02-94, Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovakia, (1994).
  20. S. Saha Ray, Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method, Commun. Nonlinear Sci. Numer. Simul., 14, 1295–1306 (2009).
  21. S. Saha Ray, R.K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Appl. Math. Comput., 168, 398–410 (2005).
  22. B.M. Schulz and M. Schulz, Numerical investigations of anomalous diffusion effects in glasses, J. Non-Cryst. Solids., 352, 4884–4887 (2006).
  23. D. N. Susahab, M. Jahanshahi, Numerical solution of nonlinear fractional Volterra– Fredholm integro-differential equations with mixed boundary conditions, Int. J. Ind. Math., 7, 63–69 (2015).
  24. Y.wang, Li . Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Int. J. Comput. Math., 275, 72–80 (2015).
  25. M. X. Yi, J. Huang, CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel, Int. J. Comput. Math., 92, 1715–1728 (2015).
  26. L. Zhu, Q. Fan, Numerical solution of nonlinear fractional-order Volterra integro[1]differential equations by SCW, Commun. Nonlinear Sci. Numer. Simul., 18, 1203–1213 (2013).
  27. V. V. Zozulya, P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, J. Chin. Inst. Eng., 22, 763–775 (1999).