A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method

Yalçın Öztürk; Ayşe Anapalı Şenel; Gül Gözde Biçer Şarlak; Dilara Altan Koç; Mustafa Gülsu

doi:10.36753/mathenot.1836575

A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method

Abstract

This study presents efficient direct numerical solutions for the nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs). The main component of the numerical scheme is the collocation method, which uses shifted Chebyshev polynomials. The method converts the given NVFIDEs into a system of linear/nonlinear algebraic equations with unknown coefficients of the truncated shifted Chebyshev series. With the aid of Maple, unknown coefficients are obtained, and so, the desired approximation solution is achieved. Several numerical examples are presented to assess the accuracy by comparing them with other existing techniques in the literature and to evaluate the method's efficiency. From the tables and figures, the examples reveal that only a small number of shifted Chebyshev series terms are needed to achieve small errors.

Keywords

Collocation method, Fredholm-Volterra integro-differential equations, Nonlinear integro-differential equation, Shifted Chebyshev polynomials

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Details

Primary Language

English

Subjects

Mathematical Methods and Special Functions

Journal Section

Research Article

Authors

Yalçın Öztürk
0000-0002-4142-5633
Türkiye

Ayşe Anapalı Şenel ^*
0000-0002-0198-1862
Türkiye

Gül Gözde Biçer Şarlak
0000-0002-2872-5324
Türkiye

Dilara Altan Koç
0000-0001-9838-3174
Türkiye

Mustafa Gülsu
0000-0001-6139-0266
Türkiye

Early Pub Date

March 27, 2026

Publication Date

March 27, 2026

Submission Date

December 5, 2025

Acceptance Date

March 10, 2026

Published in Issue

Year 2026 Volume: 14 Number: 1

DOI

https://doi.org/10.36753/mathenot.1836575

IZ

https://izlik.org/JA87TJ27EG

APA

Öztürk, Y., Anapalı Şenel, A., Biçer Şarlak, G. G., Altan Koç, D., & Gülsu, M. (2026). A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method. Mathematical Sciences and Applications E-Notes, 14(1), 45-56. https://doi.org/10.36753/mathenot.1836575

AMA

1.Öztürk Y, Anapalı Şenel A, Biçer Şarlak GG, Altan Koç D, Gülsu M. A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method. Math. Sci. Appl. E-Notes. 2026;14(1):45-56. doi:10.36753/mathenot.1836575

Chicago

Öztürk, Yalçın, Ayşe Anapalı Şenel, Gül Gözde Biçer Şarlak, Dilara Altan Koç, and Mustafa Gülsu. 2026. “A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method”. Mathematical Sciences and Applications E-Notes 14 (1): 45-56. https://doi.org/10.36753/mathenot.1836575.

EndNote

Öztürk Y, Anapalı Şenel A, Biçer Şarlak GG, Altan Koç D, Gülsu M (March 1, 2026) A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method. Mathematical Sciences and Applications E-Notes 14 1 45–56.

IEEE

[1]Y. Öztürk, A. Anapalı Şenel, G. G. Biçer Şarlak, D. Altan Koç, and M. Gülsu, “A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method”, Math. Sci. Appl. E-Notes, vol. 14, no. 1, pp. 45–56, Mar. 2026, doi: 10.36753/mathenot.1836575.

ISNAD

Öztürk, Yalçın - Anapalı Şenel, Ayşe - Biçer Şarlak, Gül Gözde - Altan Koç, Dilara - Gülsu, Mustafa. “A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method”. Mathematical Sciences and Applications E-Notes 14/1 (March 1, 2026): 45-56. https://doi.org/10.36753/mathenot.1836575.

JAMA

1.Öztürk Y, Anapalı Şenel A, Biçer Şarlak GG, Altan Koç D, Gülsu M. A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method. Math. Sci. Appl. E-Notes. 2026;14:45–56.

MLA

Öztürk, Yalçın, et al. “A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method”. Mathematical Sciences and Applications E-Notes, vol. 14, no. 1, Mar. 2026, pp. 45-56, doi:10.36753/mathenot.1836575.

Vancouver

1.Yalçın Öztürk, Ayşe Anapalı Şenel, Gül Gözde Biçer Şarlak, Dilara Altan Koç, Mustafa Gülsu. A Direct Polynomial Solution of Nonlinear Volterra-Fredholm Integro-Differential Equations via the Shifted Chebyshev Collocation Method. Math. Sci. Appl. E-Notes. 2026 Mar. 1;14(1):45-56. doi:10.36753/mathenot.1836575