J. Math. Computer Sci., 40 (2026), 84–100 Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs A numerical study of nonlinear mixed Volterra-Fredholm in- tegral equations using Toeplitz matrix method M. E. Nasr a , Sahar M. Abusalim a , M. A. Abdou b , K. M. Khalil a , M. A. Abdel-Aty c,∗ a Department of Mathematics, College of Science, Jouf University, Sakaka, Saudi Arabia. b Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt. c Department of Mathematics and Computer Science, Faculty of Science, Benha University, Benha 13518, Egypt. Abstract It is known that integral equations, whether linear or nonlinear, and whether the kernel is continuous or discontinuous, play a major role in explaining physical and engineering phenomena. The importance of these equations appears when calculating the effect of time and its impact on the solution. This importance increases if the study is based on nonlinear integral equations with a singular kernel. In this study, a nonlinear equation was assumed, and the effect of time during a certain period was studied, with the assumption of the singular kernel of the integral equation in a general form. All the previous singular kernels can be derived from it as special cases. Many methods, whether semi-analytical or numerical, can find solutions to integral equations. However, these methods fail to find the solution when the kernel is singular. If we deal with the orthogonal polynomial method, it treats each type of singular kernel as an independent case. Therefore, the authors in this research used the Toeplitz matrix method (TMM), considering the kernel in a general form and deriving special cases as applications of the method. Here, the existence and uniqueness of the solutions of the second-kind nonlinear mixed Volterra-Fredholm integral equation (NMV-FIE) are discussed. The integral operator is shown to be normal and continuous. We then derive a numerically solvable nonlinear algebraic system (NLAS) using the TMM. The Banach fixed point theorem is used to prove that this NLAS is solvable. When the kernel takes a logarithmic and Hilbert kernels, numerical examples are discussed and the estimation error, in each case, is calculated. Some numerical experiments are performed to show the efficiency of the presented approach, and all results are performed by using the program Wolfram Mathematica 11. Keywords: Banach fixed point theorem, nonlinear mixed Volterra-Fredholm integral equation, integral operator, Toeplitz matrix method. 2020 MSC: 45G10, 45P05, 45H10, 65R20. ©2026 All rights reserved. 1. Introduction This paper uses a new numerical approach to solve the nonlinear mixed Volterra-Fredholm integral equation (NMV-FIE) as follows: Θ(x, t)= f(x, t)+ λ Z t 0 Z Ω F(t, τ)k(|u − v|)ϖ(τ, y, Θ(y, τ))dydτ, (1.1) ∗ Corresponding author Email addresses: menasr@ju.edu.sa (M. E. Nasr), sabosalem@ju.edu.sa (Sahar M. Abusalim), abdella_777@yahoo.com (M. A. Abdou), kmshalaby@ju.edu.sa (K. M. Khalil), mohammed.abdallah@fsc.bu.edu.eg (M. A. Abdel-Aty) doi: 10.22436/jmcs.040.01.06 Received: 2024-10-16 Revised: 2025-01-06 Accepted: 2025-04-07