A new approach to solve weakly singular fractional-order delay integro-differential equations using operational matrices. (English) Zbl 07814801
Summary: 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.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45J05 | Integro-ordinary differential equations |
| 34K37 | Functional-differential equations with fractional derivatives |
Keywords:
operational matrices; fractional delay integro-differential equation; weakly singular kernelReferences:
| [1] | H. Belbali, M. Benbachir, Stability for coupled systems on networks with Caputo-Hadamard frac-tional derivative, J. Math. Model. 9 (2021) 107-118. · Zbl 1488.34022 |
| [2] | S.P. Bhairat, Existence and continuation of solutions of Hilfer fractional differential equations, J. Math. Model. 7 (2019) 1-20. · Zbl 1449.34013 |
| [3] | A.H. Bhrawy, A. Alofi, The operational matrix of fractional integration for shifted Chebyshev polynomials, Appl. Math. Lett. 26 (2013) 25-31. · Zbl 1255.65147 |
| [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. · Zbl 1410.45004 |
| [5] | T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon Breach, Science Publisher Inc., New York, 1978. · Zbl 0389.33008 |
| [6] | M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83 (2006) 123-129. · Zbl 1087.65119 |
| [7] | M.R. Eslahchi, M. Dehghan, M. Parvizi, Application of the collocation method for solving nonlin-ear fractional integro-differential equations, J. Comput. Appl. Math. 257 (2014) 105-128. · Zbl 1296.65106 |
| [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. · Zbl 1447.65014 |
| [9] | A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equa-tions, Elsevier, Amsterdam, 2006. · Zbl 1092.45003 |
| [10] | P.K. Kythe, P. Puri, Computational Method for Linear Integral Equations, Birkhauser, Boston, 2002. · Zbl 1023.65134 |
| [11] | A. Lotfi, S.A. Yousefi, Mehdi 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. · Zbl 1286.49030 |
| [12] | R.L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl. 59 (2010) 1586-1226. · Zbl 1189.92007 |
| [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. · Zbl 1496.45009 |
| [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. · Zbl 1270.65078 |
| [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. · Zbl 1348.65182 |
| [16] | A. Pedas, E. Tamme, M. Vikerpuur, Spline collocation for fractional weakly singular integro-differential equations, Appl. Numer. Math. 110 (2016) 204-214. · Zbl 1351.65103 |
| [17] | I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. · Zbl 0924.34008 |
| [18] | S. Rezabeyk, S. Abbasbandy, E. Shivanian, Solving fractional-order delay integro-differential equa-tions using operational matrix based on fractional-order Euler polynomials, Math. Sci. 14 (2020) 97-107. · Zbl 1452.65143 |
| [19] | M. Salem Abdo, S.K. Panchal, Existence and continuous dependence for fractional neutral func-tional differential equations, J. Math. Model. 5 (2017) 153-170. · Zbl 1385.34053 |
| [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. · Zbl 1148.45002 |
| [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. · Zbl 1446.45009 |
| [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. · Zbl 1317.65261 |
| [23] | S ¸. Yüzbas ¸i, 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. · Zbl 1288.34067 |
| [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. · Zbl 1369.65177 |
| [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. · Zbl 1298.65197 |
| [26] | V.V. Zozulya, P.I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-d elas-ticity and fracture mechanics, J. Chin. Inst. Eng. 22 (1999) 763-775. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
