A new generalized Bell wavelet and its applications for solving linear and nonlinear integral equations

In this study, a new fractional function based on the Bell wavelet is defined to solve the Fredholm and Volterra integral equations. The aim of this study is to approximate the unknown function of the linear (or nonlinear) integral equations by truncating the Bell wavelet series. Firstly, by using generalized Bell polynomials, the fractional Bell wavelet functions are defined. Secondly, the operational matrices are derived and transformed into matrix form. Then, a numerical scheme is developed to apply both linear and nonlinear test problems from the literature, including equations with exact solutions. By applying the generalized Bell wavelet in combination with the collocation method, the original problems are converted into a system of linear or nonlinear algebraic equations. These equations are then solved using classical techniques to determine the unknown coefficients. To evaluate the effectiveness of the proposed approach, test problems are compared with results from several established methods, and the outcomes are visually represented. This method demonstrates significantly improved accuracy compared to those found in the existing literature.