In this paper, a numerical method based on shifted Legendre polynomials combined with a least-squares formulation is developed for solving a class of phase-lagged Volterra–Fredholm integral equations of the second kind. Phase-lag effects arise naturally in many physical and engineering models involving delayed responses, leading to increased analytical and computational complexity. By employing shifted Legendre polynomials as basis functions, the proposed approach transforms the original integral equation into a system of algebraic equations that can be solved efficiently. Sufficient conditions for the existence and uniqueness of the solution are established using the Banach fixed-point theorem, and a convergence analysis of the proposed method is provided. Several numerical examples are presented to demonstrate the accuracy, stability, and efficiency of the method, including comparisons with existing approaches. The results confirm that the proposed scheme offers an effective and reliable tool for the numerical solution of phase-lagged Volterra–Fredholm integral equations.
Nasr et al. (Thu,) studied this question.