Variable-Order Fractional Differential Equations: Existence, Stability, and Application to 3D Noise Evolution - Tharmalingam Gunasekar; Jaya Priya Dhanasekar; Anakira, Nidal; Osama Ogilat; Sheimat, Amani (Statistics, Optimization & Information Computing, 2026)
This paper studies a boundary value (BV) problem characterized by variable order(VO) Caputo fractional derivatives to study the existence, uniqueness, and stability of solutions under well-defined boundary conditions. Fixed point theory is used, where the Banach contraction principle ensures the uniqueness of solutions, while the Krasnoselskii theorem confirms their existence. Furthermore, the notion of Ulam-Hyers stability is used to investigate the response of the solutions to small perturbations. Numerical examples are presented to illustrate the theoretical results and to validate the approach under practical conditions. Additionally, an application concerning the evolution of features in three - dimensional noise fields is included. The highlights its engineering relevance, particularly in image processing and signal analysis, where modeling noise behavior and memory effects is important for tasks such as denoising and feature extraction