View in EDS

Legendre spectral collocation method for one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations.

Saved in:
Bibliographic Details
Title: Legendre spectral collocation method for one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations.
Authors: Ezz-Eldien, S. S.1 (AUTHOR) s_sezeldien@yahoo.com, Alalyani, A.2 (AUTHOR)
Source: International Journal of Modern Physics C: Computational Physics & Physical Computation. Feb2026, Vol. 37 Issue 2, p1-11. 11p.
Subjects: Nonlinear equations, Pantograph, Legendre's polynomials, Volterra equations, Integro-differential equations, Numerical analysis, Approximation algorithms
Abstract: The integration of functional, integral and delay components of pantograph Volterra–Fredholm integro-differential equations provides a powerful framework for modeling complex systems with interdependent dynamics, particularly where nonlinearity influences and proportional feedback are essential. Numerical approximations to multi-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations present significant challenges due to the integration of nonlinearity, proportional delays and mixed integral terms, necessitating adaptive methods to achieve highly accurate approximations. In this work, we extend the Legendre spectral approximation to the one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations. In this method, the Legendre differentiation matrix and the pantograph operational matrix are used to manage the proportional delay terms inherent in these equations. To demonstrate the superiority of the proposed scheme, we present comparisons with other established spectral methods, highlighting the advantages of the Legendre spectral approach. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Modern Physics C: Computational Physics & Physical Computation is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
Description
Abstract:The integration of functional, integral and delay components of pantograph Volterra–Fredholm integro-differential equations provides a powerful framework for modeling complex systems with interdependent dynamics, particularly where nonlinearity influences and proportional feedback are essential. Numerical approximations to multi-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations present significant challenges due to the integration of nonlinearity, proportional delays and mixed integral terms, necessitating adaptive methods to achieve highly accurate approximations. In this work, we extend the Legendre spectral approximation to the one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations. In this method, the Legendre differentiation matrix and the pantograph operational matrix are used to manage the proportional delay terms inherent in these equations. To demonstrate the superiority of the proposed scheme, we present comparisons with other established spectral methods, highlighting the advantages of the Legendre spectral approach. [ABSTRACT FROM AUTHOR]
ISSN:01291831
DOI:10.1142/S0129183125500615