Theoretical and Computational Study of Variable-Order Nonlinear Fractional Differential Equations with Applications to Heat Transfer

Source abstract

This paper investigates a class of nonlinear variable-order (VO) fractional differential equations subject to initial value conditions, motivated by heat transfer processes in heterogeneous composite materials exhibiting spatially varying memory effects. Unlike constant-order fractional models, the proposed VO formulation allows the memory characteristics of the medium to vary across the computational domain, providing a more realistic description of heat conduction in materials with nonuniform thermal properties. A piecewise constant variable-order framework is introduced to represent abrupt changes in thermal behavior across material interfaces while preserving the nonlocal nature of the governing equations. The mathematical analysis establishes the existence of solutions via Schauder's fixed point theorem, proves uniqueness using Banach's contraction principle under suitable Lipschitz conditions, and investigates Ulam–Hyers stability to guarantee the robustness of solutions with respect to small perturbations. The novelty of the proposed approach lies in combining a piecewise variable-order formulation with rigorous well-posedness and stability analysis, thereby extending the applicability of classical fixed-point techniques to heterogeneous variable-order fractional models. The theoretical results are validated through computational examples and a heat transfer application, where the numerical approximations verify the analytical assumptions and illustrate the influence of the variable fractional order on the thermal response of layered composite materials. The proposed framework provides an effective mathematical tool for modeling memory-dependent heat transfer and other complex dynamical systems governed by variable-order nonlocal operators.