Numerical Methods For Solving Nonlinear Volterra Integral Equations

Abstract

Nonlinear Volterra integral equations constitute an important class of integral equations with significant applications in mathematical physics, population dynamics, viscoelasticity, engineering, biological systems, control theory, and models involving hereditary and memory effects. Due to the nonlinear structure of these equations, exact analytical solutions are generally difficult to obtain, thereby making the development of reliable numerical techniques essential. This study investigates numerical methods for solving nonlinear Volterra integral equations of the second kind. Particular attention is given to the rectangle quadrature method, the trapezoidal Nyström method combined with Newton iteration, and the Chebyshev spectral collocation method. The mathematical formulations of the methods are developed, while their consistency, stability, convergence behaviour, computational requirements, and accuracy are examined. Numerical test problems with known exact solutions are employed to evaluate the performance of the methods through maximum absolute errors and experimental convergence orders. The analysis indicates that the rectangle quadrature approach provides a simple computational procedure but exhibits relatively low-order convergence. The trapezoidal Nyström method considerably improves numerical accuracy and generally demonstrates second-order convergence for sufficiently smooth problems. The Chebyshev spectral collocation technique offers substantially faster convergence when both the kernel and solution possess sufficient regularity. The study demonstrates that the appropriate numerical method for nonlinear Volterra integral equations depends on the smoothness of the solution, kernel properties, computational cost, and required degree of accuracy.