In this thesis, a numerical method for solving nonlinear Volterra integro-differential equations based on barycentric rational interpolation is presented. In the proposed method, derivatives are approximated using the barycentric rational differential matrix, while integrals are evaluated via an indirect quadrature rule derived from the same interpolation framework. The existence and uniqueness of the obtained numerical solution are rigorously established under suitable conditions. Furthermore, convergence analysis demonstrates that the method achieves a high order of accuracy. To assess numerical stability, the linear stability properties of the method are investigated using the convolution test equation. For validation, numerical results obtained by the proposed method are compared with those produced by the spectral collocation method. Numerical experiments conducted for exampels of nonlinear Volterra integro-differential equations confirm the high accuracy, computational efficiency, and excellent agreement with theoretical predictions.
