This study presents efficient direct numerical solutions for the nonlinear Volterra-Fredholm integro-differential equations (NVFIDEs). The main component of the numerical scheme is the collocation method, which uses shifted Chebyshev polynomials. The method converts the given NVFIDEs into a system of linear/nonlinear algebraic equations with unknown coefficients of the truncated shifted Chebyshev series. With the aid of Maple, unknown coefficients are obtained, and so, the desired approximation solution is achieved. Several numerical examples are presented to assess the accuracy by comparing them with other existing techniques in the literature and to evaluate the method's efficiency. From the tables and figures, the examples reveal that only a small number of shifted Chebyshev series terms are needed to achieve small errors.
Öztürk et al. (Fri,) studied this question.