High-precision newton-kantorovich method for nonlinear integral equations

TL;DR

This paper introduces a high-precision Newton-Kantorovich method utilizing the mpmath library to solve nonlinear integral equations more accurately and stably, especially for complex, nonlinear, and stiff problems.

Contribution

It presents a novel high-precision implementation of the Newton-Kantorovich method that improves stability and accuracy for challenging nonlinear integral equations.

Findings

Enhanced stability and accuracy in solving nonlinear integral equations.

Superior performance over traditional low-precision methods in stiff regimes.

Broader applicability in scientific and engineering problems.

Abstract

The paper considers the numerical solution of nonlinear integral equations using the Newton-Kantorovich method with the mpmath library. High-precision quadrature of the kernel K(t, s, u) with respect to the variable s for fixed t increases stability and accuracy in problems sensitive to rounding and dispersion. The presented implementation surpasses traditional low-precision methods, especially for strongly nonlinear kernels and stiff regimes, thereby expanding the applicability of the method in scientific and engineering computations.