Bernstein Polynomial Approximation for Fredholm Integro-Differential Equations of Fractional Order with Numerous Constant Delays (JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 2026)
Background:This study presents a useful new framework that uses Bernstein polynomials to improve a spectral collocation technique for numerically solving Fredholm integro-differential equations of fractional orders with variable coefficients and multi-time constant delay (FIFDEs-Delays) under boundary conditions.Materials and Methods:The approximate solutions are assumed to be in the form of the truncated Bernstein polynomial series. This novel approach is based on the use of a matrix technique to convert the display equation with conditions into an algebraic linear system of equations with unknown Bernstein coefficients.  Results:This approach improves the accuracy of the solutions found while simultaneously simplifying the problem. The solution of this system determines the coefficients of the assumed solution. In addition, the integral operators employed in this technique were quantitatively evaluated using the Clenshaw-Curtis formula.Conclusion:we provide specific examples to showcase the accuracy of the method, and we employ the least-squares error methodology to minimize error terms within the given domain. Ultimately, the most common application suggested for the numerical approaches is implemented in a Python program.