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Peer-reviewedCC BYAxioms

Analytical and Numerical Analysis of Multidimensional Diffusion Processes in L2 Space Under Third-Kind Boundary Conditions

  • Zafar Duman Abbasov1
  • Ghadah Albeladi2
  • Mohamed Gamal3
  • Y. H. Youssri4
4 affiliations
  1. 1Department of Mathematical Analysis, Ganja State University, Ganja AZ2000, Azerbaijan
  2. 2Department of Mathematics, Faculty of Sciences and Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
  3. 3Department of Mathematics, Faculty of Science, Qena University, Qena 83523, Egypt
  4. 4Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
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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.

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Cite

BibTeXRIS
BibTeX
@article{abbasov2026analytical,
  title = {{Analytical and Numerical Analysis of Multidimensional Diffusion Processes in L2 Space Under Third-Kind Boundary Conditions}},
  author = {Zafar Duman Abbasov and Ghadah Albeladi and Mohamed Gamal and Y. H. Youssri},
  journal = {Axioms},
  year = {2026},
  month = may,
  doi = {10.3390/axioms15050380},
  url = {https://doi.org/10.3390/axioms15050380},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Analytical and Numerical Analysis of Multidimensional Diffusion Processes in L2 Space Under Third-Kind Boundary Conditions
AU  - Zafar Duman Abbasov
AU  - Ghadah Albeladi
AU  - Mohamed Gamal
AU  - Y. H. Youssri
PY  - 2026
DA  - 2026/05/19
JO  - Axioms
DO  - 10.3390/axioms15050380
UR  - https://doi.org/10.3390/axioms15050380
AB  - 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.
SN  - 2075-1680
ER  -