Adaptive Decomposition for Fractional Volterra Integral Equations in Diverse Systems

Abstract

We develop a comprehensive, mathematically rigorous framework for fractional-order Volterra integral equations (F-VIEs), a class of equations in which the kernel encodes power-law or Mittag--Leffler memory rather than the classical exponential decay. Such equations arise naturally when a system's memory decays algebraically --- a feature observed in anomalous diffusion, viscoelastic soft matter, epidemic spreading on heterogeneous contact networks, rough stochastic volatility in financial markets, and non-Markovian quantum open systems. The paper makes five original contributions: (i) a clarified Fractional Adomian Decomposition Method (F-ADM) including an explicit nonlinear Adomian-polynomial construction for genuinely nonlinear F-VIEs with a complete proof of uniform convergence under H\"older-continuous kernel conditions; (ii) adaptive Gauss--Jacobi quadrature rules tuned to the weakly singular structure of fractional kernels, with explicit error constants and a corrected analyticity hypothesis governing spectral accuracy; (iii) a hybrid F-ADM/Jacobi-GQT switching algorithm, with a formal convergence guarantee, that reduces function evaluations by up to 61 percent against pure quadrature; (iv) a suite of eight application modules spanning mathematical biology, mechanical engineering, quantitative finance, non-Markovian quantum dynamics, biomedical pharmacokinetics, agricultural hydrology, climate science, and neural population dynamics, each with fully worked numerical examples validated against published benchmark data or field measurements; and (v) eight high-resolution figures providing graphical verification of all major results. Relative to standalone quadrature, the hybrid algorithm cuts function evaluations by 38--61\% while matching or improving accuracy across thirty benchmark problems. Theorems on existence, uniqueness, H\"older stability, ADM convergence, and hybrid-method convergence are proved in full. An open-problem list identifies five directions where the framework can be extended, including kernel learning via Gaussian processes and tensor-train methods for high-dimensional coupled systems.