Bernstein computational algorithm for integro-differential equations

In this study, we introduce a computational algorithm for solving Integro-Differential Equations (IDEs) using Bernstein polynomials as basis functions. The algorithm approximates the solution by expressing it in terms of Bernstein polynomials and substituting this assumed solution into the IDE. Collocating the resulting equation at evenly spaced points yields a system of linear algebraic equations, which is solved via matrix inversion to find the Bernstein coefficients. These coefficients are then used to construct the approximate solution. Numerical examples demonstrate the method's accuracy and efficiency, highlighting its advantages in reducing computational effort.