Variable-Order Fractional Differential Equations: Existence, Stability, and Application to 3D Noise Evolution

This paper studies a boundary value (BV) problem characterized by variable order(VO) Caputo fractionalderivatives to study the existence, uniqueness, and stability of solutions under well-defined boundary conditions. Fixedpoint theory is used, where the Banach contraction principle ensures the uniqueness of solutions, while the Krasnoselskiitheorem confirms their existence. Furthermore, the notion of Ulam-Hyers stability is used to investigate the response of thesolutions to small perturbations. Numerical examples are presented to illustrate the theoretical results and to validate theapproach under practical conditions. Additionally, an application concerning the evolution of features in three - dimensionalnoise fields is included. The highlights its engineering relevance, particularly in image processing and signal analysis, wheremodeling noise behavior and memory effects is important for tasks such as denoising and feature extraction