A numerical approach to fractional Volterra–Fredholm integro-differential problems using shifted Chebyshev spectral collocation

This study presents an innovative numerical framework for addressing initial value problems (IVPs) in linear fractional Volterra–Fredholm integro-differential equations (FVFIDEs). The approach utilizes a spectral collocation method grounded in shifted Chebyshev polynomials of the second kind to construct an approximate solution. By integrating this approximation into the governing equation and applying collocation constraints at predefined nodes, the IVP is converted into a system of linear algebraic equations. This system is subsequently resolved using the Newton–Raphson iteration, ensuring computational precision and rapid convergence. To validate the method's efficacy, a series of benchmark examples are analyzed, highlighting its stability, efficiency, and adaptability. The findings underscore the scheme's high-order accuracy, positioning it as a robust computational tool for fractional Volterra–Fredholm integro-differential problems in applied mathematics and engineering.