Purpose This study introduces a computational framework to solve both linear and nonlinear Volterra integro-differential equations (VIDEs) of the third kind. The framework aims to address challenges associated with solving such equations by leveraging a robust and efficient numerical method. Design/methodology/approach The methodology utilizes a collocation technique based on the moving least squares approximation combined with shifted Chebyshev polynomials to approximate solutions. The VIDEs are reformulated into a specific class of integral equations, and the Gauss–Legendre integration formula is used to compute integrals within the scheme. Error analysis is conducted to ensure accuracy. Findings The proposed framework demonstrates computational efficiency, stability, and minimal memory requirements. Numerical experiments validate the effectiveness of the method, highlighting its ability to reliably solve both linear and nonlinear VIDEs of the third kind. The results affirm the accuracy and practical utility of the approach. Originality/value This study presents a novel method for solving third-kind VIDEs that eliminates the need for preliminary approximation cells. The framework combines theoretical rigor with practical implementation, offering a significant contribution to the numerical analysis of integro-differential equations.
Aourir et al. (Sun,) studied this question.