Spectral Approximation Using Bernstein Polynomials for Third‐Kind Delayed Volterra and Integro‐Differential Equations

ABSTRACT This work presents an efficient numerical scheme for a class of third‐kind Volterra delay integral equations (VDIEs) and integro‐differential equations with proportional delays (DVIDEs), based on the Bernstein approximation method. By exploiting the structure of Bernstein polynomials, the problems are reformulated as tractable systems of algebraic equations. The proposed method is thoroughly explained, along with its application to equations and an estimation of the associated error. The developed approach demonstrates significant efficiency, stability, and minimal computational memory demands. We thoroughly confirm the method's accuracy and reliability by testing it with various numerical examples. Furthermore, the accuracy of the scheme is validated by comparing the numerical results with analytical solutions, as well as with the moving least squares and radial basis function collocation methods, demonstrating consistency with theoretical error estimates.