@article{Zaky2025, 
author = {Mahmoud A. Zaky and Weam G. Alharbi and Marwa M. Alzubaidi and R.T. Matoog},
title = {A Legendre tau approach for high-order pantograph Volterra-Fredholm integro-differential equations},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {3},
pages = {7067-7085},
keywords = {pantograph Volterra-Fredholm, High-order integro-differential equations, spectral method, Legendre tau},
url = {https://www.sciopen.com/article/10.3934/math.2025322},
doi = {10.3934/math.2025322},
abstract = {High-order pantograph Volterra-Fredholm integro-differential equations (P-VF-IDEs) arise in various scientific and engineering applications, where systems exhibit delay or scaling in their dynamics. This paper investigates a class of high-order P-VF-IDEs characterized by the presence of both Volterra and Fredholm integral terms as well as pantograph delay elements. We propose a spectral tau approach to approximate the solution P-VF-IDEs in both one- and two-dimensions. In particular, we employ the operational differentiation and integration matrices to approximate the integro-differential operator, transforming the continuous problem into a system of algebraic equations, and providing high accuracy with fewer computational modes. Numerical experiments illustrate the accuracy and convergence properties of the spectral Legendre tau method in solving high-order P-VF-IDEs and demonstrate its efficacy compared to other spectral approaches.}
}