Numerical solution of fractional volterra integral equations by Bernstein-Durrmeyer type approximation operators

Abstract

In this article define a new numerical technique for solving fractional Volterra integral equations of first and second kind by using the Bernstein-Durrmeyer operators. An extensive theoretical framework is presented including moment identities, Korovkin's theorem-based uniform convergence, and a Voronovskaya-type asymptotic equation. The corresponding numerical methods convert the fractional Volterra equations into a system of linear algebraic equations which can be computed efficiently. We prove the error and convergence results under the stated assumptions of regularity and stability. Further, numerical tests demonstrate the performance of the proposed method for fractional Volterra equations with weakly singular Riemann-Liouville kernels. Finally, the approximation results show that the Bernstein-Durrmeyer operator provides a precise approximation framework with high accuracy at moderate polynomial degrees, while the numerical performance is influenced by the conditioning of the resulting algebraic systems.