Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

In this paper, we investigate nonlinear functional Volterra-Urysohn integral equations, a class of nonlinear integral equations of Volterra type. The existence and uniqueness of the solution to the equation is proved by a technique based on the Picard iterative method. For the numerical approximation of the solution, the Euler and trapezoidal discretization methods are utilized which result in a system of nonlinear algebraic equations. Using a Gronwall inequality and its discrete version, first order of convergence to the exact solution for the Euler method and quadratic convergence for the trapezoidal method are proved. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Finally, numerical examples show the functionality of the methods. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper investigates nonlinear functional Volterra-Urysohn integral equations, proving the existence and uniqueness of solutions using the Picard iterative method. Euler and trapezoidal discretization methods are utilized for numerical approximation, showing first order convergence for the Euler method and quadratic convergence for the trapezoidal method. A new Gronwall inequality is developed to prove the convergence of the trapezoidal method, with numerical examples demonstrating the functionality of the methods.