Two-Dimensional Heat Transfer Model with Spatiotemporal Relaxation: Analytical and Numerical Solution
Description
This dataset contains the Python code for implementing and validating a generalized two-dimensional heat transfer model that incorporates spatiotemporal relaxation effects simultaneously in both the energy balance equation and the thermal conductivity operator. This extension leads to a nonlocal partial differential equation with mixed derivatives, enabling the description of dispersion and finite-speed propagation of thermal disturbances — phenomena not captured by classical Fourier theory. The model preserves Onsager symmetry and thermodynamic consistency, and reduces to well-known limits such as Fourier, Cattaneo–Vernotte, and DPL under specific parameter choices. Numerical examples demonstrate that accounting for spatiotemporal relaxation is essential for modeling fast transient thermal processes, including laser heating, microscale heat transfer, biothermal waves, and heat conduction in nanofilms. The code includes both an analytical solution (Fourier series) and an implicit finite-difference numerical scheme for comparison and validation. It is intended for research and educational use in computational heat transfer and non-Fourier thermodynamics. Funding: This work was supported by the Russian Science Foundation (RSF), grant No. 23-79-10044. Related publication: This code accompanies the manuscript “Two-dimensional heat transfer model with spatiotemporal relaxation: analytical and numerical solution” submitted to Vestnik of Samara State Technical University. Series: Physics and Mathematics.
Files
Steps to reproduce
To reproduce the results: Install Python 3.8 or later. Install required packages: pip install numpy matplotlib scipy (optional, for sparse solvers). Open the provided Jupyter notebook (*.ipynb) in a Jupyter environment (e.g., Jupyter Lab or Jupyter Notebook). Run all cells sequentially. The code will automatically generate plots comparing analytical and numerical solutions at specified Fourier numbers (Fo). To explore different scenarios, modify global parameters at the top of the notebook: Fo_T_compare, Fo_q_compare — relaxation parameters. fast_mode = True (for quick test) or False (for high-resolution simulation). Mmax, Nmax, Nx, Ny, dFo — grid and time step settings. All output figures are displayed inline. Numerical profiles along x- and y-axes are computed and plotted for comparison. No additional data files are required — all calculations are performed internally.
Institutions
- Samara State Technical University