A Regularized Urysohn Operator Framework for Nonstationary AP/HTPB Solid-Propellant Combustion with Memory Effects, Spectral Analysis and Artificial Intelligence

Abstract

This paper develops a regularized nonlinear Urysohn-operator framework for nonstationary ammonium-perchlorate/hydroxyl-terminated-polybutadiene (AP/HTPB) solid-propellant combustion with hereditary thermal feedback, pressure coupling and delayed response. The state is governed by a causal nonlinear integral equation whose kernel depends on time, history, temperature-sensitive kinetics and the current magnitude of the state. The regularized model is uε + εLuε = f + U(uε), written as where L is a positive spectral operator and ε > 0 is a smoothing parameter. Beyond standard existence and uniqueness results, the article establishes exponential stability, robustness with respect to perturbations of the kernel, the explicit estimate ∥uε − u∥ ≤ Cε, differentiability of the solution with respect to model parameters, eigenvalue localization, a spectral-radius bound and pseudospectral resolvent estimates. A numerical study compares short and long memory, strongly nonlinear saturated kernels, several regularization levels and five iterative solvers. Threedimensional response surfaces, a spectral plot, a heat map and sensitivity diagrams illustrate the theory. The framework is also connected with DeepONet, Fourier neural operators, physics-informed neural operators and transformer memory. The resulting model provides an analytically transparent and computationally stable bridge between nonlinear integral equations, combustion dynamics and modern operator learning.