Bell Wavelet Method for Solving the Fredholm–Volterra Integro‐Differential Equations

This paper focuses on solving the Fredholm–Volterra integro‐differential equations (IDEs) using a numerical method based on function approximation with generalized fractional‐order Bell wavelets (GFOBWs). Previous studies have shown that the Legendre polynomials are highly effective for approximating solutions to fractional differential equations, while the Bell polynomial collocation method provides significantly better results than the Legendre collocation method for IDEs. Building on these findings, we propose a novel collocation technique using fractional‐order Bell wavelets to address the Fredholm–Volterra IDEs. An exact formula is derived for the Riemann–Liouville fractional integral operator (RLFIO) of the GFOBW. This formula is combined with the collocation method to convert the IDEs into a system of algebraic equations, enabling an efficient numerical solution method. Additionally, the absolute error of the numerical solution obtained from the proposed method is estimated. Various examples are provided to demonstrate the accuracy of the method.