A Numerical Method for Solving First-Order Nonlinear Volterra-Fredholm Integro-differential Equations Using Power Series Polynomials

Abstract

This is a numerical approach for solving first-order Nonlinear Volterra-Fredholm Integro-differential equations with power series polynomials. The approach approximated Volterra-Fredholm Integro-differential equations using power series polynomials. The modelled problem is converted to a system of algebraic equations, which is then solved using the standard collocation approach. After determining the approach's uniqueness and convergence, numerical examples were used to assess its effectiveness. The results showed that the method is competitive with other methods. The proposed method transforms the given Integro-differential equation into an equivalent system of algebraic equations by approximating the unknown function with a finite power series expansion. The coefficients of the series are determined by substituting the approximation into the original equation and enforcing the equality at selected collocation points within the domain of interest. This technique ensures high accuracy and rapid convergence with minimal computational effort. Several illustrative examples are provided to demonstrate the efficiency and reliability of the method. The obtained results show excellent agreement with exact solutions, confirming the suitability of the power series polynomial approach for solving first-order Volterra–Fredholm Integro-differential equations arising in applied mathematics, physics and engineering models.