Numerical Solutions of Nonlinear Delay Mixed Integral Equation in Two Dimensions via Collocation Method Based on Chebyshev and Legendre Polynomials

This research investigates the numerical analysis of a two-dimensional (2D) delay mixed integral equation (DMIE) of the second kind with a continuous kernel. The existence, uniqueness, stability, and convergence of the solution are rigorously established in the functional space C([a,b]×[0,T]), providing a solid theoretical foundation. A collocation method based on Chebyshev and Legendre polynomials is applied to obtain accurate numerical approximations, leading to a system of algebraic equations applicable to both linear and nonlinear cases. Numerical simulations are performed using MATLAB to evaluate the method’s performance and calculate the results, including error behavior, convergence, and stability. Several illustrative applications are presented to demonstrate the method’s numerical stability and versatility in handling both linear and nonlinear problems. The results underscore the efficiency, accuracy, and robustness of the proposed numerical schemes in solving complex 2D DMIE, highlighting the novelty of the framework and its strong potential for broad applications in contemporary scientific and engineering problems.