Numerical technique based on Bernstein polynomials approach for solving auto-convolution VIEs and the initial value problem of auto-convolution VIDEs

This study introduces a computational technique aimed at solving the auto-convolution Volterra integral equation (AVIE) and the auto-convolution Volterra integro-differential equation (AVIDE). In this approach, we use the Bernstein approximation method to estimate solutions for these equations. By leveraging the characteristics of Bernstein polynomials, we simplify the problem, transforming these equations into a manageable system of algebraic equations. We provide a detailed description of the approach, and then its practicality for the suggested equations is presented. The suggested algorithm is computationally efficient, has greater stability, is straightforward to implement on computers, and demands less computer memory. This approach first converts these equations into a class of integral equations and then uses the proposed approach to estimate the solution. Some theorems have been proposed to demonstrate the existence and uniqueness of the suggested approach. In addition, an estimate of the error bound for this approach is provided. A comparison of this technique with previously known methods is examined. Finally, representative numerical tests are reported to demonstrate the precision and efficiency of the proposed solving method.