Abstract
This paper provides an efficient recursive approach of the spectral Tau method, to approximate the solution of a system of generalized Abel-Volterra integral equations. In this regard, we first investigate the existence, uniqueness as well as smoothness of the solutions under various assumptions on the given data. Next, from a numerical perspective, we express approximated solution as a linear combination of suitable canonical polynomials which are constructed by an easy-to-use recursive formula. Mostly, the unknown parameters are calculated by solving low-dimensional algebraic systems independent of the degree of approximation which prevents high computational costs. Obviously, due to the singular behavior of the exact solution, using classical polynomials to construct canonical polynomials, leads to low accuracy results. In this regard, we develop new fractional-order canonical polynomials using Müntz-Legendre polynomials which have the same asymptotic behavior with the solution of the underlying problem. The convergence analysis is discussed, and the familiar spectral accuracy is achieved in \(L^{\infty }\)-norm. Finally, the reliability of the method is evaluated using various examples and applying it in solving a class of fractional differential equations systems.
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References
Al-Humedi, H.O., Abdul-hasan, A.S.: The reproducing kernel Hilbert space method for solving system of linear weakly singular Volterra integral equations. Journal of Advances in Mathematics 15, 8070–8080 (2018)
Brunner, H., Crisci, M.R., Russo, E., Vecchio, A.: Continuous and discrete time waveform relaxation methods for Volterra integral equations with weakly singular kernels. Ricerche Di Matematica 51(2), 201–222 (2002)
Brauer, F., Castillo-Chvez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2012)
Crisci, M.R., Russo, E.: An extension of Ortiz’ recursive formulation of the \(\tau \)-method to certain linear systems of ordinary differential equations. Math. Comput. 41(163), 27–42 (1983)
Capobianco, G., Cardone, A.: A parallel algorithm for large systems of Volterra integral equations of Abel type. J. Comput. Appl. Math. 220(1), 749–758 (2008)
Conte, D., Shahmorad, S., Talaei, Y.: New fractional Lanczos vector polynomials and their application to system of Abel-Volterra integral equations and fractional differential equations. J. Comput. Appl. Math. 366, Art. 112409 (2020). https://doi.org/10.1016/j.cam.2019.112409
Diethelm, K.: The Analysis of Fractional Differential Equations. Lectures Notes in Mathematics, Springer, Berlin (2010)
Eldaou, M.K., Khajah, H.G.: Iterated solutions of linear operator equations with the Tau method. Math. Comput. 66(217), 207–213 (1997)
Faghih, A., Mokhtary, P.: A new fractional collocation method for a system of multi-order fractional differential equations with variable coefficients. J. Comput. Appl. Math. 382, 113–139 (2021)
Freilich, J., Ortiz, E.: Numerical solution of systems of ordinary differential equations with the Tau method: an error analysis. Math. Comput. 39, 467–479 (1982)
Ferras, L.L., Ford, N.J., Morgado, M.L., Rebelo, M.: A hybrid numerical scheme for fractional-order systems. International Conference on Innovation, Engineering and Entrepreneurship 505, 735–742 (2018)
Faghih, A., Mokhtary, P.: An efficient formulation of Chebyshev Tau method for constant coefficients systems of multi-order FDEs. J. Sci. Comput. 82(6), Art. 6 (2020). https://doi.org/10.1007/s10915-019-01104-z
Ghanbari, F., Mokhtary, P., Ghanbari, K.: Numerical solution of a class of fractional order integro-differential algebraic equations using Müntz-Jacobi Tau method. J. Comput. Appl. Math. 362, 172–184 (2019)
Gorenflo, R., Vessella, S.: Abel Integral Equations: Analysis and Applications. Springer-Verlag, Berlin-New York (1991)
Ghanbari, F., Mokhtary, P., Ghanbari, K.: Numerical solution of a class of fractional order integro-differential algebraic equations using Müntz-Jacobi Tau method. J. Comput. Appl. Math. 362, 172–184 (2019). https://doi.org/10.1016/j.cam.2019.05.026
Hille, E.: Lectures on Ordinary Differential Equations. Addison-Wesley, United States (1969)
Katani, R., Shahmorad, S.: A block by block method for solving system of Volterra integral equations with continuous and Abel kernels. Math. Model. Anal. 20(6), 737–753 (2015)
Lanczos, C.: Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ (1956)
Lighthill, J.M.: Contributions to the theory of the heat transfer through a laminar boundary layer. Proc. R. Soc. London 202A, 359–377 (1950)
Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985)
Maleknejad, K., Salimi Shamloo, A.: Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices. Appl. Math. Comput. 195, 500–505 (2008)
Mokhtary, P., Ghoreishi, F., Srivastava, H.M.: The Müntz-Legendre Tau method for fractional differential equations. Appl. Math. Model. 40(2), 671–684 (2016)
Metzler, F., Schick, W., Kilian, H.G., Nonnenmacher, T.F.: Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995)
Ortiz, E.L.: The Tau method. SIAM J. Numer. Anal. 6, 480–492 (1969)
Rossikhin, Y., Shitikova, M.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50, 15–67 (1997)
Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms. Analysis and Applications. Springer, Heidelberg (2011)
Shen, J., Wang, Y.: Müntz-Galerkin methods and applications to mixed Dirichlet Neumann boundary value problems. SIAM J. Sci. Comput. 38, 2357–2381 (2016)
Tao, L., Yong, H.: Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind. J. Math. Anal. Appl. 324(1), 225–237 (2006)
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, Art. 50 (2019). https://doi.org/10.1007/s10092-019-0347-y
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Talaei, Y., Shahmorad, S., Mokhtary, P. et al. 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). https://doi.org/10.1007/s13540-022-00070-y
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DOI: https://doi.org/10.1007/s13540-022-00070-y
Keywords
- Recursive approach of the Tau method
- System of generalized Abel-Volterra integral equations
- Müntz-Legendre polynomials
- Fractional vector canonical polynomials
- Convergence analysis