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An easy-to-implement recursive fractional spectral-Galerkin method for multi-term weakly singular Volterra integral equations with non-smooth solutions

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Abstract

Multi-term weakly singular Volterra integral equations arise in numerous scientific and engineering applications modeled by fractional differential equations. Numerical analysis of these types of integral equations with non-smooth solutions is one of the open problems due to singularities at the initial time leading to the destruction of the accuracy of the approximate solutions. The spectral approximations based on classical orthogonal polynomials have a low convergence rate compared with the exact non-smooth solutions. We theoretically investigate the regularity of the solution of multi-term weakly singular Volterra integral equations, which ensures the non-smooth property of the solutions. This non-smooth representation is significant in constructing highly accurate numerical schemes based on fractional-order approximate solutions. The advantage of the proposed method is that there is no need to solve an algebraic system of equations, as the unknown coefficients can be specified by a simple recursive algorithm. This makes the numerical method for solving the problem efficient and of low computational cost. The \(L_{\infty }\)-convergence analysis of the proposed method is discussed. Some numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed method.

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References

  1. Itkin, A.: Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach. Birkhäuser New York, NY (2017)

    Book  Google Scholar 

  2. El-Mesiry, A., El-Sayed, A., El-Saka, H.: Numerical methods for multiterm fractional (arbitrary) orders differential equations. Appl. Math. Comput. 160, 683–699 (2005)

    MathSciNet  Google Scholar 

  3. Faghih, A., Rebelo, M.: A spectral approach to non-linear weakly singular fractional integro-differential equations. Fract. Calc. Appl. Anal. 26, 370–398 (2023). https://doi.org/10.1007/s13540-022-00113-4

    Article  MathSciNet  Google Scholar 

  4. Basset, A.B.: On the descent of a sphere in a vicous liquid. Quart. J. 41, 369–381 (1910)

    Google Scholar 

  5. Amin, A.Z., Zaky, M.A., Hendy, A.S., Hashim, I., Aldraiweesh, A.: High-order multivariate spectral algorithms for high-dimensional nonlinear weakly singular integral equations with delay. Mathematics 10, 3065 (2022). https://doi.org/10.3390/math10173065

    Article  Google Scholar 

  6. Bhrawy, A.H., Zaky, M.A.: Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl. Math. Model 40(2), 832–845 (2016)

    Article  MathSciNet  Google Scholar 

  7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods: fundamentals in single domains. Springer-Verlag, New York (2006)

    Book  Google Scholar 

  8. Huang, C., Stynes, M.: Spectral Galerkin methods for a weakly singular Volterra integral equation of the second kind. IMA J. Numer. Anal. 37(3), 1411–1436 (2017)

    MathSciNet  Google Scholar 

  9. Hou, D., Lin, Y., Azaiez, M., et al.: A Müntz-collocation spectral method for weakly singular Volterra integral equations. J. Sci. Comput. 81, 2162–2187 (2019)

    Article  MathSciNet  Google Scholar 

  10. Conte, D., Shahmorad, S., Talaei, Y.: New fractional Lanczos vector polynomials and their application to a system of Abel-Volterra integral equations and fractional differential equations. J. Comput. Appl. Math. 366 (2020). https://doi.org/10.1016/j.cam.2019.112409

  11. Ghoreishi, F., Hadizadeh, M.: Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations. Numer. Algorithms 52, 541–55 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  12. Azizipour, G., Shahmorad, S.: A new Tau-collocation method with fractional basis for solving weakly singular delay Volterra integro-differential equations. J. Appl. Math. Comput. 68, 2435–2469 (2022). https://doi.org/10.1007/s12190-021-01626-6

    Article  MathSciNet  Google Scholar 

  13. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Bessel functions with various applications. Appl. Math. 64, 637–662 (2019). https://doi.org/10.21136/AM.2019.0279-18

  14. H. Brunner, Collocation methods for Volterra integral and related functional equations methods. Cambridge University Press (2004)

  15. Ortiz, E.L., Samara, L.: An operational approach to the Tau method for the numerical solution of nonlinear differential equations. Computing 27(1), 15–25 (1981)

    Article  MathSciNet  Google Scholar 

  16. Khosravian-Arab, H., Dehghan, M., Eslahchi, M.R.: Fractional Sturm-Liouville boundary value problems in unbounded domains: theory and applications. J. Comput. Phys. 299, 526–560 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  17. Avc, I., Mahmudov, N.I.: Numerical solutions for multi-term fractional order differential equations with fractional Taylor operational matrix of fractional integration. Mathematics (2020)

  18. Ardabili, J.S., Talaei, T.: Chelyshkov collocation method for solving the two-dimensional Fredholm-Volterra integral equations. Int. J. Appl. Comput. Math. 4, 25 (2018)

    Article  MathSciNet  Google Scholar 

  19. Shen, J., Sheng, Ch., Wang, Zh.: Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels. J. Math. Stud. 48(4), 315–329 (2019)

    Article  MathSciNet  Google Scholar 

  20. Ould Sidi, H., Zaky, M.A., Elwaled, K., Akgul, A., Hendy, A.S.: Numerical reconstruction of a space-dependent source term for multidimensional space-time fractional diffusion equations. Rom. Rep. Phys. 75, 120 (2023)

    Article  Google Scholar 

  21. Zaky, M.A., Hendy, A.S., Aldraiweesh, A.A.: Numerical algorithm for the coupled system of nonlinear variable-order time fractional Schrödinger equations. Rom. Rep. Phys. 75, 106 (2023)

    Google Scholar 

  22. Ould Sidi, H., Babatin, M., Alosaimi, M., Hendy, A.S., Zaky, M.A.: Simultaneous numerical inversion of space-dependent initial condition and source term in multi-order time-fractional diffusion models. Rom. Rep. Phys. (2023). https://rrp.nipne.ro/IP/AP722.pdf

  23. Zaky, M.A.: An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl. Numer. Math. 154, 205–222 (2020)

    Article  MathSciNet  Google Scholar 

  24. Zaky, M.A.: Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J. Comput. Appl. Math. 357, 103–122 (2019)

    Article  MathSciNet  Google Scholar 

  25. Shen, J., Wang, Y.: Müntz-Galerkin methods and applications to mixed Dirichlet Neumann boundary value problems. SIAM J. Sci. Comput. 38, 2357–2381 (2016)

    Article  MathSciNet  Google Scholar 

  26. Kant, K., Nelakanti, G.: Galerkin and multi-Galerkin methods for weakly singular Volterra-Hammerstein integral equations and their convergence analysis. (2020)

  27. Fermo, L., Mezzanotte, D., Occorsio, D.: On the numerical solution of Volterra integral equations on equispaced nodes. ETNA 59, 9–23 (2023)

    Article  MathSciNet  Google Scholar 

  28. Zaky, M.A., Doha, E.H., Tenreiro Machado, J.A.: A spectral framework for fractional variational problems based on fractional Jacobi functions. Appl. Numer. Math. 132, 51–72 (2018)

    Article  MathSciNet  Google Scholar 

  29. Zaky, M.A., Ameen, I.G.: A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with non-smooth solutions. Eng. Comput. 37, 2623–2631 (2021)

    Article  Google Scholar 

  30. Zayernouri, M., Karniadakis, G.E.: Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 257, 460–480 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  31. Alshbool, M.H.T., Bataineh, A.S., Hashim, I., Isik, O.R.: Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions. J. King. Saud. Univ. Sci. 29(1), 1–18 (2017)

    Article  Google Scholar 

  32. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of natural materials. J. Appl. Mech. Trans. ASME 51(2), 294–298 (1984)

    Article  ADS  Google Scholar 

  33. MckeeK, S.: Volterra integral and integro-differential equations arising from problems in engineering and science. Bull. Inst. Math. Appl. 24, 135–138 (1988)

    MathSciNet  Google Scholar 

  34. Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37(7), 5498–5510 (2013)

    Article  MathSciNet  Google Scholar 

  35. Sabermahani, S., Ordokhani, Y., Yousefi, S.A.: Fractional-order general Lagrange scaling functions and their applications. Bit. Numer. Math. 60, 101–128 (2020). https://doi.org/10.1007/s10543-019-00769-0

    Article  MathSciNet  Google Scholar 

  36. Micula, S.: A numerical method for weakly singular nonlinear Volterra integral equations of the second kind. Symmetry 2020, 12 (1862). https://doi.org/10.3390/sym12111862

    Article  Google Scholar 

  37. Karimi Vanani, S., Soleymani, F.: Tau approximate solution of weakly singular Volterra integral equations. Math. Comput. Model. 57, 494–502 (2013)

    Article  MathSciNet  Google Scholar 

  38. Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra type integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)

    MathSciNet  Google Scholar 

  39. Allahviranloo, T., Gouyandeh, Z., Armand, A.: Numerical solutions for fractional differential equations by Tau-Collocation method. Appl. Math. Comput. 271, 979–990 (2015)

    MathSciNet  Google Scholar 

  40. Tao, X., Xie, Z., Zhou, X.: Spectral Petrov-Galerkin methods for the second kind Volterra type integro-differential equations. Numer. Math. Theory Methods Appl. 4(2), 216–236 (2011)

    Article  MathSciNet  Google Scholar 

  41. Li, X.: Numerical solution of fractional differential equations using cubic-spline wavelet collocation method. Commun. Nonlinear Sci. Numer. Simul. 17, 3934–3946 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  42. Talaei, Y., Shahmorad, S., Mokhtary, P.: A new recursive formulation of the Tau method for solving linear Abel-Volterra integral equations and its application to fractional differential equations. Calcolo 56(50) (2019)

  43. Talaei, Y., Micula, S., Hosseinzadeh, H., Noeiaghdam, S.: A novel algorithm to solve nonlinear fractional quadratic integral equations. AIMS 7(7), 13237–13257 (2022)

    MathSciNet  Google Scholar 

  44. Talaei, Y., Shahmorad, S., Mokhtary, P., Faghih, A.: A fractional version of the recursive Tau method for solving a general class of Abel-Volterra integral equations systems. Fract. Calc. Appl. Anal. 25, 1553–1584 (2022)

    Article  MathSciNet  Google Scholar 

  45. Talaei, Y., Lima, P.M.: An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions. Comp. Appl. Math. 42, 190 (2023). https://doi.org/10.1007/s40314-023-02333-7

    Article  MathSciNet  Google Scholar 

  46. Diethelm, K.: The analysis of fractional differential equations. Lectures Notes in Mathematics. Springer, Berlin (2010)

    Book  Google Scholar 

  47. Yang, Y.: Jacobi spectral Galerkin methods for Volterra integral equations with the weakly singular kernel. Bull. Korean Math. Soc. 53(1), 247–262 (2016)

    Article  MathSciNet  Google Scholar 

  48. Li, Y., Sun, N.: Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Comput. Math. Appl. 62(301), 1046–1054 (2011)

    Article  MathSciNet  Google Scholar 

  49. Talaei, Y.: Chelyshkov collocation approach for solving linear weakly singular Volterra integral equations. J. Appl. Math. Comput. 60 (2019)

  50. Gu, Z.: Spectral collocation method for system of weakly singular Volterra integral equations. Adv. Comput. Math. 45, 2677–2699 (2019). https://doi.org/10.1007/s10444-019-09703-y

  51. Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53(2), 414–434 (2012)

    Article  MathSciNet  Google Scholar 

  52. Ma, Z., Alikhanov, A.A., Huang, C., Zhang, G.: A multi-domain spectral collocation method for Volterra integral equations with a weakly singular kernel. Appl. Numer. Math. 167, 218–236 (2021)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the anonymous reviewers for their constructive comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran

    Younes Talaei

  2. Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

    Mahmoud A. Zaky

  3. Department of Applied Mathematics, National Research Centre, Dokki, Cairo, 12622, Egypt

    Mahmoud A. Zaky

  4. Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St, Yekaterinburg, 620002, Russia

    Ahmed S. Hendy

  5. Department of Mathematics, Faculty of Science Benha University, Benha, 13511, Egypt

    Ahmed S. Hendy

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  1. Younes Talaei
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  3. Ahmed S. Hendy
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Correspondence to Younes Talaei, Mahmoud A. Zaky or Ahmed S. Hendy.

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Talaei, Y., Zaky, M.A. & Hendy, A.S. An easy-to-implement recursive fractional spectral-Galerkin method for multi-term weakly singular Volterra integral equations with non-smooth solutions. Numer Algor (2024). https://doi.org/10.1007/s11075-023-01742-3

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