Araştırma Makalesi
BibTex RIS Kaynak Göster

Yıl 2023, Sayı: 43, 43 - 53, 30.06.2023

Dilek Varol

https://doi.org/10.53570/jnt.1260801

Öz

Kaynakça

  • S. Abbas, M. Benchohra, G. M. N’Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
  • A. A. Hamoud, K. P. Ghadle, \emph{The Approximate Solutions of Fractional Volterra-Fredholm Integrodifferential Equations by Using Analytical Techniques}, Problemy Analiza Issues of Analyses 7 (25) (2018) 41{--}58.
  • A. Hamoud, K. Ghadle, \emph{Usage of the Homotopy Analysis Method for Solving Fractional Volterra-Fredholm Integrodifferential Equation of the Second Kind}, Tamkang Journal of Mathematics 49 (4) (2018) 301{--}315.
  • L. Zhu, Q. Fan, \emph{Solving Fractional Nonlinear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet}, Communications in Nonlinear Science and Numerical Simulation 17 (6) (2012) 2333{--}2341.
  • A. Setia, Y. Liu, A. S. Vatsala, \emph{Solution of Linear Fractional Fredholm Integro-differential Equation by Using Second Kind Chebyshev Wavelet}, in: S. Latifi (Ed.), 11th International Conference on Information Technology: New Generations, Las Vegas, 2014, pp. 465{--}469.
  • A. M. S. Mahdy, E. M. H. Mohamed, G. M. A. Marai, \emph{Numerical Solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomials of the Third Kind Method}, Theoretical Mathematics \& Applications 6 (4) (2016) 87{--}101.
  • A. M. Mahdy, E. M. Mohamed, \emph{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 (1) (2016) 24{--}32.
  • Y. Wang, L. Zhu, \emph{SCW Method for Solving the Fractional Integro-Differential Equations with a Weakly Singular Kernel}, Applied Mathematics and Computation 275 (2016) 72{--}80.
  • A. Setia, Y. Liu, A. S. Vatsala, \emph{Numerical Solution of Fredholm-Volterra Fractional Integro-Differential Equations with Nonlocal Boundary Conditions}, Journal of Fractional Calculus and Applications 5 (2) (2014) 155{--}165.
  • S. T. Mohyud-Din, H. Khan, M. Arif, M. Rafiq, \emph{Chebyshev Wavelet Method to Nonlinear Fractional Volterra–Fredholm Integro-Differential Equations with Mixed Boundary Conditions}, Advances in Mechanical Engineering 9 (3) (2017) 1{--}8.
  • F. Zhou, X. Xu, \emph{Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equations with Mixed Boundary Conditions via Chebyshev Wavelet Method}, International Journal of Computer Mathematics 96 (2) (2019) 436{--}456.
  • M. Bayram, V. F. Hatipoğlu, S. Alkan, S. E. Das, \emph{A Solution Method for Integro-Differential Equations of Conformable Fractional Derivative}, Thermal Science 22 (1) (2018) 7{--}14.
  • A. Daşcıoğlu, D. Varol, \emph{Laguerre Polynomial Solutions of Linear Fractional Integro-Differential Equations}, Mathematical Sciences 15 (1) (2021) 47{--}54.
  • D. Varol Bayram, A. Daşcıoğlu, \emph{A Method for Fractional Volterra Integro-Differential Equations by Laguerre Polynomials}, Advances in Difference Equations 2018 (2018) Article Number 466 11 pages.
  • A. Daşcıoğlu, D. Varol Bayram, \emph{Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials}, Sains Malaysiana 48 (1) (2019) 251{--}257.
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, \emph{A New Definition of Fractional Derivative}, The Journal of Computational and Applied Mathematics 264 (2014) 65{--}70.
  • T. Abdeljawad, \emph{On Conformable Fractional Calculus}, The Journal of Computational and Applied Mathematics 279 (2015) 57{--}66.
  • J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman \& Hall /CRC, New York, 2002.
  • M. Sezer, M. Kaynak, \emph{Chebyshev Polynomial Solutions of Linear Differential Equations}, The International Journal of Mathematical Education in Science and Technology 27 (4) (1996) 607{--}618.
  • A. Akyüz-Daşcıoğlu, \emph{A Chebyshev Polynomial Approach for Linear Fredholm–Volterra Integro-Differential Equations in the Most General Form}, Applied Mathematics and Computation 181 (1) (2006) 103{--}112.

Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations

Yıl 2023, Sayı: 43, 43 - 53, 30.06.2023

Dilek Varol

https://doi.org/10.53570/jnt.1260801

Öz

In this study, Chebyshev polynomials have been applied to construct an approximation method to attain the solutions of the linear fractional Fredholm integro-differential equations (IDEs). By this approximation method, the fractional IDE has been transformed into a linear algebraic equations system with the aid of the collocation points. In the method, the conformable fractional derivatives of the Chebyshev polynomials have been calculated in terms of the Chebyshev polynomials. Using the results of these calculations, the matrix relation for the conformable fractional derivatives of Chebyshev polynomials was attained for the first time in the literature. After that, the matrix forms have been replaced with the corresponding terms in the given fractional integro-differential equation, and the collocation points have been used to have a linear algebraic system. Furthermore, some numerical examples have been presented to demonstrate the preciseness of the method. It is inferable from these examples that the solutions have been obtained as the exact solutions or approximate solutions with minimum errors.

Anahtar Kelimeler

Conformable fractional derivative Chebyshev polynomials numerical solutions

Kaynakça

  • S. Abbas, M. Benchohra, G. M. N’Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
  • A. A. Hamoud, K. P. Ghadle, \emph{The Approximate Solutions of Fractional Volterra-Fredholm Integrodifferential Equations by Using Analytical Techniques}, Problemy Analiza Issues of Analyses 7 (25) (2018) 41{--}58.
  • A. Hamoud, K. Ghadle, \emph{Usage of the Homotopy Analysis Method for Solving Fractional Volterra-Fredholm Integrodifferential Equation of the Second Kind}, Tamkang Journal of Mathematics 49 (4) (2018) 301{--}315.
  • L. Zhu, Q. Fan, \emph{Solving Fractional Nonlinear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet}, Communications in Nonlinear Science and Numerical Simulation 17 (6) (2012) 2333{--}2341.
  • A. Setia, Y. Liu, A. S. Vatsala, \emph{Solution of Linear Fractional Fredholm Integro-differential Equation by Using Second Kind Chebyshev Wavelet}, in: S. Latifi (Ed.), 11th International Conference on Information Technology: New Generations, Las Vegas, 2014, pp. 465{--}469.
  • A. M. S. Mahdy, E. M. H. Mohamed, G. M. A. Marai, \emph{Numerical Solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomials of the Third Kind Method}, Theoretical Mathematics \& Applications 6 (4) (2016) 87{--}101.
  • A. M. Mahdy, E. M. Mohamed, \emph{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 (1) (2016) 24{--}32.
  • Y. Wang, L. Zhu, \emph{SCW Method for Solving the Fractional Integro-Differential Equations with a Weakly Singular Kernel}, Applied Mathematics and Computation 275 (2016) 72{--}80.
  • A. Setia, Y. Liu, A. S. Vatsala, \emph{Numerical Solution of Fredholm-Volterra Fractional Integro-Differential Equations with Nonlocal Boundary Conditions}, Journal of Fractional Calculus and Applications 5 (2) (2014) 155{--}165.
  • S. T. Mohyud-Din, H. Khan, M. Arif, M. Rafiq, \emph{Chebyshev Wavelet Method to Nonlinear Fractional Volterra–Fredholm Integro-Differential Equations with Mixed Boundary Conditions}, Advances in Mechanical Engineering 9 (3) (2017) 1{--}8.
  • F. Zhou, X. Xu, \emph{Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equations with Mixed Boundary Conditions via Chebyshev Wavelet Method}, International Journal of Computer Mathematics 96 (2) (2019) 436{--}456.
  • M. Bayram, V. F. Hatipoğlu, S. Alkan, S. E. Das, \emph{A Solution Method for Integro-Differential Equations of Conformable Fractional Derivative}, Thermal Science 22 (1) (2018) 7{--}14.
  • A. Daşcıoğlu, D. Varol, \emph{Laguerre Polynomial Solutions of Linear Fractional Integro-Differential Equations}, Mathematical Sciences 15 (1) (2021) 47{--}54.
  • D. Varol Bayram, A. Daşcıoğlu, \emph{A Method for Fractional Volterra Integro-Differential Equations by Laguerre Polynomials}, Advances in Difference Equations 2018 (2018) Article Number 466 11 pages.
  • A. Daşcıoğlu, D. Varol Bayram, \emph{Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials}, Sains Malaysiana 48 (1) (2019) 251{--}257.
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, \emph{A New Definition of Fractional Derivative}, The Journal of Computational and Applied Mathematics 264 (2014) 65{--}70.
  • T. Abdeljawad, \emph{On Conformable Fractional Calculus}, The Journal of Computational and Applied Mathematics 279 (2015) 57{--}66.
  • J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman \& Hall /CRC, New York, 2002.
  • M. Sezer, M. Kaynak, \emph{Chebyshev Polynomial Solutions of Linear Differential Equations}, The International Journal of Mathematical Education in Science and Technology 27 (4) (1996) 607{--}618.
  • A. Akyüz-Daşcıoğlu, \emph{A Chebyshev Polynomial Approach for Linear Fredholm–Volterra Integro-Differential Equations in the Most General Form}, Applied Mathematics and Computation 181 (1) (2006) 103{--}112.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik, Uygulamalı Matematik
Bölüm Araştırma Makalesi
Yazarlar

Dilek Varol 0000-0002-5158-5614

Yayımlanma Tarihi 30 Haziran 2023
Gönderilme Tarihi 6 Mart 2023
Yayımlandığı Sayı Yıl 2023 Sayı: 43

Kaynak Göster

APA Varol, D. (2023). Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations. Journal of New Theory(43), 43-53. https://doi.org/10.53570/jnt.1260801
AMA Varol D. Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations. JNT. Haziran 2023;(43):43-53. doi:10.53570/jnt.1260801
Chicago Varol, Dilek. “Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations”. Journal of New Theory, sy. 43 (Haziran 2023): 43-53. https://doi.org/10.53570/jnt.1260801.
EndNote Varol D (01 Haziran 2023) Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations. Journal of New Theory 43 43–53.
IEEE D. Varol, “Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations”, JNT, sy. 43, ss. 43–53, Haziran 2023, doi: 10.53570/jnt.1260801.
ISNAD Varol, Dilek. “Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations”. Journal of New Theory 43 (Haziran 2023), 43-53. https://doi.org/10.53570/jnt.1260801.
JAMA Varol D. Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations. JNT. 2023;:43–53.
MLA Varol, Dilek. “Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations”. Journal of New Theory, sy. 43, 2023, ss. 43-53, doi:10.53570/jnt.1260801.
Vancouver Varol D. Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations. JNT. 2023(43):43-5.
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