This paper aims to investigate a generalized third-order fractional heat conduction model that incorporates both thermal memory and inertial effects. The study focuses on understanding the influence of the fractional order, nonlinear heat generation and external sources on the temperature evolution in thermal systems.
The model is formulated using a Caputo fractional derivative of order 2 < α = 3, extending classical Fourier and Cattaneo-type heat conduction equations. A semi-analytical approach based on embedding Green’s function into a fixed-point iterative scheme is presented to construct approximate solutions. The method is applied to nonlinear problems, and successive estimates are obtained to analyze convergence and accuracy.
The results show that the proposed method provides accurate approximations with only a few iterations. The fractional-order model captures memory effects, leading to slower or enhanced temperature evolution depending on the value of a. The inclusion of nonlinear source terms and inertial effects significantly influences the behavior of the solution, producing results that differ from classical heat models.
The proposed model is applicable to heat transfer processes in materials with memory and nonlocal behavior, including radiative heating, chemically reactive systems and transient or periodically forced thermal processes. It provides a useful framework for modeling heat conduction in complex media where classical models are inadequate.
This paper presents a meaningful integration of a third-order fractional heat conduction model with a Green’s function based iterative scheme. The approach offers an efficient framework for solving nonlinear fractional differential equations while capturing both thermal memory and inertial heat propagation effects. To the best of the authors’ knowledge, nonlinear third-order fractional heat conduction models combined with a Green’s function based fixed-point iterative formulation have not been investigated in the present setting.
