Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives

Abstract

The current work aims at finding the approximate solution to solve the nonlinear fractional type Volterra integro-differential equation \begin{document}$ \begin{equation*} \sum\limits_{k = 1}^{m}F_{k}(x)D^{(k\alpha )}y(x)+\lambda \int_{0}^{x}K(x, t)D^{(\alpha )}y(t)dt = g(x)y^{2}(x)+h(x)y(x)+P(x). \end{equation*} $\end{document} In order to solve the aforementioned equation, the researchers relied on the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method. So, the equation becomes nonlinear system of equations. By solving the former nonlinear system equation, we get the approximate solution in form of Bernstein's fractional series. Besides, we will present some examples with the estimate of the error.