Overcoming Large-Domain Instability: A Block-Structured Newton-Nyström Algorithm for Nonlinear Integro-Differential Systems

Nonlinear Fredholm integro-differential equations of the second kind play a pivotal role in modeling complex physical systems involving nonlocal interactions, memory effects, and long-range dependencies. However, when posed on extended intervals (τ ≫ 1), these equations present severe numerical challenges due to the loss of operator compactness, spectral degradation, and instability of conventional discretization schemes. Developing robust, structure-preserving algorithms for such large-domain problems remains an open and critical challenge in computational mathematics.

This work introduces a novel Dual Transformation–Linearization–Discretization (DT-L-D) framework specifically designed for nonlinear Fredholm integro-differential equations on extended intervals. The approach first applies a dual transformation: differentiating the original equation and decomposing the global domain into quasi-uniform subintervals, yielding a block-structured system that preserves coupling topology. Nonlinearity is then handled via Newton–Kantorovich iterative linearization in Banach space, followed by Nyström discretization to obtain a finite-dimensional linear algebraic system. Rigorous convergence analysis establishes quadratic convergence under explicit Lipschitz continuity and uniform invertibility conditions.

Numerical experiments on benchmark problems confirm the method's high accuracy, with global errors reaching 10⁻⁹ and residuals down to 10⁻¹⁵. The algorithm demonstrates remarkable stability across intervals up to τ = 500, requiring only 5–13 iterations for convergence. Crucially, the discrete iterates converge to the exact solution as the iteration count increases, independently of the discretization size—a theoretical advantage over classical discretize-then-linearize approaches.

The DT-L-D framework provides a reliable, efficient, and structure-preserving paradigm for solving nonlinear integro-differential equations on large domains. Its dimension-reduction capability (O(nN) complexity) and parallelization potential make it particularly valuable for scientific computing applications requiring high-precision solutions over extended intervals.