Analytical and Numerical Analysis of Multidimensional Diffusion Processes in L2 Space Under Third-Kind Boundary Conditions
Year: 2026
Type: article
Abstract: This research paper investigates the solution of diffusion equations characterized by Third-Kind (Robin) boundary conditions within n-dimensional complex domains. The analysis is conducted in the L2 Hilbert space, which facilitates the substantiation of both the existence and uniqueness of solutions through variational methods. Analytical solutions are derived for multidimensional domains by employing the Fourier method and spectral analysis techniques. Complementing this theoretical framework, a high-accuracy numerical approach based on the Associated Legendre Polynomials Collocation Spectral Method (ALP-CSM) with Chebyshev–Gauss–Lobatto nodes is developed. Rigorous convergence analysis confirms spectral accuracy, with numerical examples in one, two, and three dimensions demonstrating error decay from O(10−3) to machine precision O(10−15). The mathematical impact of Third-Kind boundary conditions on the diffusion rate and the steady state of the system is demonstrated. The obtained results provide a robust tool for modeling physical processes, particularly in systems involving heat exchange on the surfaces of complex-structured domains, offering both theoretical insight and computational efficiency.
Source: Axioms
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,896
OpenAlex ID: W7161642234