A new approach to solve weakly singular fractional-order delay integro-differential equations using operational matrices

Document Type : Research Article

Authors

1 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

2 Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, 34149-16818, Iran

Abstract

In this paper, we propose a new approach to solve weakly singular fractional delay integro-differential equations. In the proposed approach, we apply  the operational matrices of fractional integration and delay function based on the shifted Chebyshev polynomials to approximate the solution of the considered equation. By approximating the fractional derivative of the unknown function as well as the unknown function in terms of the shifted Chebyshev polynomials and substituting these approximations into the original equation, we obtain a system of nonlinear algebraic equations. We present the convergence analysis of the proposed method. Finally, to show the accuracy and validity of the proposed method, we give some numerical examples.

Keywords

References

[1] H. Belbali, M. Benbachir, Stability for coupled systems on networks with Caputo-Hadamard fractional derivative, J. Math. Model. 9 (2021) 107–118.
[2] S.P. Bhairat, Existence and continuation of solutions of Hilfer fractional differential equations, J. Math. Model. 7 (2019) 1–20.
[3] A.H. Bhrawy, A. Alofi, The operational matrix of fractional integration for shifted Chebyshev polynomials, Appl. Math. Lett. 26 (2013) 25–31.
[4] J. Biazar, K. Sadri, Solution of weakly singular fractional integro-differential equations by using a new operational approach, J. Comput. Appl. Math. 352 (2019) 453–477.
[5] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon Breach, Science Publisher Inc., New York, 1978.
[6] M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83 (2006) 123–129.
[7] M.R. Eslahchi, M. Dehghan, M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-differential equations, J. Comput. Appl. Math. 257 (2014) 105–128.
[8] S.S. Ezz-Eldien, E.H. Doha, Fast and precise spectral method for solving pantograph type Volterra integro-differential equations, Numer. Algor. 81 (2019) 57–77.
[9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[10] P.K. Kythe, P. Puri, Computational Method for Linear Integral Equations, Birkhauser, Boston, 2002.
[11] A. Lotfi, S.A. Yousefi, M. Dehghan, Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule, J. Comput. Appl. Math. 250 (2013) 143–160.
[12] R.L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl. 59 (2010) 1586–1226.
[13] F. Mohammadi, Fractional integro-differential equation with a weakly singular kernel by using block pulse functions, U.P.B. Sci. Bull. Ser. A 79 (2017) 57–66.
[14] P. Mokhtary, F. Ghoreishi, The L2-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations, Numer. Algorithms 58 (2011) 475–496.
[15] S. Nemati, S. Sedaghat, I. Mohammadi, A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels, J. Comput. Appl. Math. 308 (2016) 231–242.
[16] A. Pedas, E. Tamme, M. Vikerpuur, Spline collocation for fractional weakly singular integro-differential equations, Appl. Numer. Math. 110 (2016) 204–214.
[17] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[18] S. Rezabeyk, S. Abbasbandy, E. Shivanian, Solving fractional-order delay integro-differential equations using operational matrix based on fractional-order Euler polynomials, Math. Sci. 14 (2020) 97–107.
[19] M. Salem Abdo, S.K. Panchal, Existence and continuous dependence for fractional neutral functional differential equations, J. Math. Model. 5 (2017) 153–170.
[20] B.Q. Tang, X.F. Li, Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput. 199 (2008) 406–413.
[21] Y. Wang, L. Zhuand, Z. Wang, Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel, Adv. Difference Equ. 2018 (2018) 254.
[22] M. Yi, J. Huang, CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel, Int. J. Comput. Math. 92 (2015) 1715–1728.
[23] S. Yüzbaşı, E. Gök, M. Sezer, Laguerre matrix method with the residual error estimation for a class of delay differential equations, Math. Meth. Appl. Sci. 37 (2014) 453–463.
[24] J. Zhao, Y. Cao, Y. Xu, Sinc numerical solution for pantograph Volterra delay-integro-differential equation, Int. J. Comput. Math. 94 (2017) 853–865.
[25] J. Zhao, J. Xiao, N.J. Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algorithms 65 (2014) 723–743.
[26] 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 (1999) 763–775.
Journal of Mathematical Modeling
Volume 11, Issue 2
July 2023
Pages 257-275

Files

Share

How to cite

Statistics