Second kind Chebyshev collocation technique for Volterra-Fredholm fractional order integro-differential equations
DOI:
https://doi.org/10.21580/jnsmr.2022.8.2.13021Keywords:
Approximate solutions, Bernstein collocation technique, Caputo derivative, fractional order integro-differential equations, Volterra-FredholmAbstract
In this work, we present the numerical solution of fractional order Volterra–Fredholm integro-differential equations using the second kind of Chebyshev collocation technique. First, we transformed the problem into a system of linear algebraic equations, which are then solved using matrix inversion to obtain the unknown constants. Furthermore, numerical examples are used to outline the method’s accuracy and efficiency using tables and figures. The results show that the method performed better in terms of improving accuracy and requiring less rigorous work.
©2022 JNSMR UIN Walisongo. All rights reserved.
Downloads
References
L Adam, “Fractional Calculus: History, Definition and Application for the engineer,’’ Department of Aerospace and Mechanical Engineering University of Notre Dame IN 46556, U.S.A. 2004.
M. Caputo, “Linear models of dissipation whose Q is almost frequency Independent,’’ Geophysical Journal International, 13(5):529 – 539,1967,
doi.org/10.1111/j.1365-246X.1967.tb02303.x.
S. Momani and A. Qaralleh, “An efficient
Method for solving systems of
Fractional Integro-Differential equations,’’ Computer and Mathematical with Applications, 52(3):459-570, 2006. doi.org/10.1016.
S.G. Samko, A.A. Kilbas, and O.I. Marichev, “Fractional Integrals and Derivatives. Theory and Applications,’’ Gordon and Breach, Yverdon. 1993.
R.C. Mittal, and R. Nigam, “Solution of Fractional Integro-differential Equations by Adomian Decomposition Method,’’ International Journal of Applied Mathematics and Mechanics, 4(2): 87-94, 2008.
H.M. Osama, and A.A. Sarmad, “Approximate solution of Fractional Integro-Differential Equations by using Bernstein Polynomials,’’ Engineering and Technology Journal, 30(8):1362-1373,2012.
D. Sh. Mohammed, ‘’Numerical solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomials,’’ Mathematical Problems in Engineering Article ID 431965, 1, 5, 2014.
A.M.S Mahdy, and E.M.H. Mohamed, E.M.H., “Numerical studies for solving system of linear Fractional Integro-Differential Equations by using Least Squares Method and Shifted Chebyshev Polynomials,’’ Journal of Abstract and Computational Mathematics, 1(24):24-32,2016.
V.B. Dilek, and D. Aysegül, “Applied Collocation Method using Laguerre Polynomials as the basis functions’’. Advances in Difference Equations a Springer Open Journal 1-11, 2018, doi.org/10.1186/s13662-018-1924-0.
T. Oyedepo, C.Y. Ishola, T.F., Aminu, and A.A. Victor, “Bernstein Modified Bernstein Collocation Method for the Solution of Fractional Integro-Differential Equations,’’ Journal of Science Technology and Education ,8(1): 65 – 72,2020.
D. Aysegul, and V.B. Dilek, “Solving
Fractional Fredholm integro-differential
Equations by Laguerre polynomials,’’
Sains Malaysiana, 48(1):251-
,2019,dx.dio
.org/10.17576/jsm-2019-4801-29.
S. Alkan, and V.F. Hatipoglu,
“Approximate solutions of Volterra-
Fredholm Integro-Differential Equations of Fractional order,’’ Tbilisi Mathematical Journal, 10(2):1-13,2017, doi: 10.1515/tmj-2017-0021.
S.T. Mohyud-Din, H. Khan, H., M. Arif, and M. Rafiq, “Chebyshev Wavelet Method to nonlinear Fractional Volterra–Fredholm Integro-Differential Equations with mixed Boundary Conditions’’ Advances in Mechanical Engineering 9(3):1-8,2017, doi.org/10.1177/1687814017694802.
F. Zhou, and X. Xu, “Numerical solution of Fractional Volterra-Fredholm Integro- Differential Equations with mixed Boundary Conditions via Chebyshev Wavelet Method,” International Journal of Computer Mathematics 96:1–18,2018 doi:10.1080/00207160.2018.1521517.
H. Dehestani, Y. Ordokhani, and M. Razzaghi, “Combination of Lucas wavelets with Legendre–Gauss Quadrature for Fractional Fredholm-Volterra Integro-Differential Equations,’’ Journal of Computational and Applied Mathematics, 382:1–35,2021,doi: 10.1016/j.cam.2020.113070.
N.K. Salman, and M.M. Mustafa, “Numerical solution of fractional Volterra-Fredholm Integro-Differential Equation using Lagrange Polynomials,’’ Baghdad Science Journal 17(4): 234-1240,2020 doi.org/10.21123/bsj.2020.17.4.1234.
N. Rajagopal, S. Balaji,
R. Seethalakshmi,and V.S. Balaji''A New Numerical Method for Fractional order Volterra Integro-Differential Equations," Ain Shams Engineering Journal,11(1):171-177,2020, doi.org/10.1016/j.asej.2019.08.004.
M. Lotfi, and A. Alipanah,'' Legendre
Spectral Element Method for Solving
Volterra-Integro Differential equations
,7:1-11,2020,
doi.org/10.1016/j.rinam.2020.100116.
Z. Meng, L. Wang, H. Li, and W. Zhang, “Legendre wavelets method for solving fractionalintegro-differential equations” International Journal of Computer Mathematics, 92(6): 1275–1291,2014 doi.org/10.1080/00207160.2014.932909.
J. R. Loh, C. Phang, and A. Isah, “New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations” Advances in Mathematical Physics 2017, Article ID 3821870, doi. org/10.1155/2017/3821870.
E. Keshavarz, Y. Ordokhani, and M. Razzaghi, “Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets” Computer Methods Differential Equation , 7(2) :163–176, 2019.
Y. Ordokhani, and H. Dehestani, “Numerical solution of linear Fredholm-Volterra integro-differential equations of fractional order” World Journal Modelling Simulation 12(3): 204–216, 2016.
Y. Ordokhani, N. Rahimi, “Numerical solution of fractional Volterra integro-differential equations via the rationalized Haar functions” Journal of Science Kharazmi University ,14(3): 211–224, 2014.
T. Oyedepo, C.Y. Ishola, O.A. Uwaheren, M.L. Olaosebikan, M.O. Ajisope, and A.A., Victor “Least-squares Bernstein Method for Solving Fractional Integro-differential Equations” Journal of Science Technology and Education, 10(1): 104 – 144, 2022.
T. Oyedepo, O.A. Taiwo, A.J. Adewale, A.A. Ishaq, and A.M. Ayinde, “Numerical Solution of System of Linear Fractional Integro-Differential Equations by Least Squares Collocation Chebyshev Technique” Mathematics and Computational Science 3(2): 10 - 21, 2022.
T. Oyedepo, O.A. Taiwo, A.J. Adewale, A.A. Ishaq, and A.M. Ayinde, “Numerical Solution of System of Linear Fractional Integro-Differential Equations by Least Squares Collocation Chebyshev Technique” Mathematics and Computational Science 3(2): 10 - 21, 2022.
A.H. Bhrawya, E. Tohidic, F. Soleymanic, “ A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals” Applied Mathematics and Computation 219(2): 482–497, 2012.
C. Edwards,” Math 312 Fractional calculus final presentation,’’ [video] Retrieved fromhttps://www.youtube.com/watch?v=CsJa3XiOmfs [Accessed 20 Sept. 2018].
Downloads
Published
Issue
Section
License
Copyright
The copyright of the received article shall be assigned to the publisher of the journal. The intended copyright includes the right to publish the article in various forms (including reprints). The journal maintains the publishing rights to published articles. Authors are allowed to use their articles for any legal purposes deemed necessary without written permission from the journal, but with an acknowledgment to this journal of initial publication.
Licensing
