High-precision newton-kantorovich method for nonlinear integral equations

TL;DR

The paper addresses solving nonlinear Volterra integral equations where rounding errors can destabilize standard Newton–Kantorovich iterations. It introduces a high-precision NK scheme using $mpmath$ for quadrature and stable interpolation, along with a theoretical convergence result under Lipschitz conditions on the derivative of the integral operator. Numerical experiments, including the Bratu equation, demonstrate quadratic convergence within a contraction region and improved stability in stiff regimes. The approach broadens the applicability of NK methods to physically and engineering-relevant problems requiring high numerical accuracy.

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.

Figures (1)

• Figure 1: Norm of successive differences for various values of $\lambda$.