Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations
Year 2024,
Volume: 7 Issue: 1, 38 - 45, 18.03.2024
Dilek Varol
,
Ayşegül Daşcıoğlu
Abstract
This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.
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Year 2024,
Volume: 7 Issue: 1, 38 - 45, 18.03.2024
Dilek Varol
,
Ayşegül Daşcıoğlu
References
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- [2] 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), 406413 .
- [3] P. K. Kythe, P. Puri, Computational Method for Linear Integral Equations, Birkhauser, Boston, 2002.
- [4] 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 (2002), 763775 .
- [5] C. Li, Y. Wang, Numerical algorithm based on Adomian decomposition for fractional differential equations Comput. Math. with Appl., 57(10) (2009), 1672-1681 .
- [6] B. Ghanbari, S. Djilali, Mathematical and numerical analysis of a three-species predator-prey model with herd behavior and time fractional-order derivative Math. Method Appl. Sci., 43(4) (2020), 1736-1752 .
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- [23] Z. Meng, L. Wang, H. Li, W. Zhang, Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math., 92(6) (2015), 1275-1291.
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- [30] F. Mohammadi, Fractional integro-differential equation with a weakly singular kernel by using block pulse functions, U.P.B. Sci. Bull. Series A., 79(1) (2017).
- [31] P. Rahimkhani, Y. Ordokhani, E. Babolian, Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations, Appl. Numer. Math., 122 (2017), 66–81.
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- [41] B. Gürbüz, M. Sezer, Laguerre polynomial solutions of a class of delay partial functional differential equations, Acta Phys. Polon. A., 132(3) (2017c), 558-560.
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- [43] B. Gürbüz, M. Sezer, A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, Int. J. Appl. Phys. Math., 7(1) (2017a), 49-58.
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- [48] D. Varol Bayram, A. Das¸cıo˘glu, A method for fractional Volterra integro-differential equations by Laguerre polynomials, Adv. Differ. Equ., 2018 (2018), 466.
- [49] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
- [50] W. W. Bell, Special Functions for Scientists and Engineers, D. Van Nostrand Company, London, 1968.