Modeling the Spread and Mitigation of Cybersecurity Threats using Non-Homogeneous Heat Equations with Boundary Conditions

Authors

How to Cite

Chinni, R. (2025). Modeling the Spread and Mitigation of Cybersecurity Threats using Non-Homogeneous Heat Equations with Boundary Conditions. International Journal of Applied Mathematical Research, 14(2), 40-56. https://doi.org/10.14419/h702mv77

Received date: June 12, 2025

Accepted date: August 9, 2025

Published date: August 15, 2025

DOI:

https://doi.org/10.14419/h702mv77

Keywords:

Cybersecurity, Mathematical Modeling, Non-Homogeneous Heat Equation, Dynamic Boundary Conditions, Stability Analysis, Real-Time Adaptation, AI-Driven Defense, Cyber-Attack Simulation, Vulnerability Propagation, Network Topology.

Abstract

This study presents a mathematical model using non-homogeneous heat equations with dynamic boundary conditions to simulate the spread and mitigation of cybersecurity threats. Our results demonstrate that incorporating adaptive boundary conditions, which represent real-time adjustments to defense mechanisms, improves the robustness of cybersecurity systems against evolving threats. Stability analysis using the Lyapunov criterion shows that the system remains stable over time, with the vulnerability (represented by the Lyapunov function) decreasing as time progresses. Numerical simulations indicate that varying the intensity of cyber-attacks and adjusting boundary conditions dynamically leads to more efficient and effective defense strategies, optimizing both computational load and system security. The mathematical model, validated through several test cases, reveals that dynamic adaptation of defense mechanisms significantly outperforms traditional static models, offering scalable solutions for real-time cyber defense. Furthermore, our results demonstrate the model's ability to simulate the impact of network topology changes and varying attack intensities, providing insights into how network configurations influence the propagation of vulnerabilities.

References

  1. U. Mohammed, R. B. Adeniyi, M. E. Semenov, ”A family of hybrid linear multi-step methods type for special third order ordinary differential equations,” Journal of the Nigerian Mathematical Society, 2018. https://jnms.ictp.it/jnms/index.php/jnms/article/download/286/53
  2. A. Pereira, J. G. Mesquita, M. Choquehuanca, ”Ordinary-Functional Differential Equations,” Summer 19 Report. http://summer.icmc.usp.br/summers/summer19/download/Summer19.pdf#page=82
  3. R. A. de Ciencias Exactas, F y Naturales, ”The 10th AIMS Conference on Dynamical Systems Differential Equations and Applications,” 2014. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=c13b82c2d8bc95cd480d1b7471000c6fdf044a4d
  4. R. Company, V. N. Egorova, L. J ´odar, ”Quadrature Integration Techniques for Random Hyperbolic PDE Problems,” MDPI Mathematics, 2021. https://www.mdpi.com/2227-7390/9/2/160
  5. M. Sherman, G. Kerr, G. Gonz ´alez-Parra, ”Comparison of symbolic computations for solving linear delay differential equations using the Laplace transform method,” Mathematical and Computational Methods, 2022. https://www.mdpi.com/2297-8747/27/5/81
  6. Fifelola, R., Linus, O. U., Femi, A. K., & Egbeja, J. S. (2024). Advanced transform techniques for the one-dimensional non-homogeneous heat equation with non-homogeneous BCs and IC. International Journal of Applied Mathematical Research, 13(2), 96-102. https://doi.org/10.14419/kyyb9f56 International Journal of Applied Mathematical Research 17
  7. S. Pinelas, J. Rossa, C. Caravela, ”International Conference on Differential and Difference Equations and Applications,” Conference Paper. https: //comum.rcaap.pt/handle/10400.26/11296
  8. V. J. Bevia, J. C. Cort ´es, C. L. P ´erez, ”A mathematical model with uncertainty quantification for allelopathy with applications to real-world data,” Springer, 2024. https://link.springer.com/article/10.1007/s10651-024-00612-y
  9. G. W. Bluman, R. de la Rosa, ”Variational and optimal control representations of conditioned and driven processes,” Royal Society Publishing, 2021. https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2020.0908
  10. J. C. Cort ´es, S. E. Delgadillo-Aleman, ”Probabilistic analysis of a class of impulsive linear random differential equations via density functions,” Elsevier Applied Mathematics, 2021. https://www.sciencedirect.com/science/article/pii/S0893965921002755
  11. M. K. Bourbatache, T. D. Le, O. Millet, ”Limits of classical homogenization procedure for coupled diffusion-heterogeneous reaction processes in porous media,” Springer, 2021. https://link.springer.com/article/10.1007/s11242-021-01683-2
  12. A. Uribe-Chavez, ”A numerical model and semi-analytic equations for determining water table elevations and discharges in non-homogeneous subsurface drainage systems,” University of Arizona, 2001. https://repository.arizona.edu/bitstream/handle/10150/289956/azu td 3010252 sip1 w.pdf?sequence=4
  13. H. Ugail, ”Partial differential equations for geometric design,” Books.google.com, 2011. https://books.google.com/books?hl=en&lr=&id= HLqZDwAAQBAJ&oi=fnd&pg=PR11&dq=Advanced+transformation+methods+for+non-homogeneous+differential+equations+Raphael+et+al&ots=hRA4IlhGAS&sig=tYACt0t3vOBihp V lw5YJwwNGo
  14. Y. N. Raffoul, ”Advanced differential equations,” Books.google.com, 2022. https://books.google.com/books?hl=en&lr=&id=fGVkEAAAQBAJ&oi= fnd&pg=PP1&dq=Advanced+transformation+methods+for+non-homogeneous+differential+equations+Raphael+et+al&ots=QgNNdACy6C&sig=xsLZ-zgx3l-KJdSb6QOrNJBa2ec
  15. T. H. Otway, ”Elliptic–Hyperbolic Partial Differential Equations: A Mini-Course in Geometric and Quasilinear Methods,” Books.google.com,2015. https://books.google.com/books?hl=en&lr=&id=N9ojCgAAQBAJ&oi=fnd&pg=PR5&dq=Advanced+transformation+methods+for+ non-homogeneous+differential+equations+Raphael+et+al&ots=uqRsNiBl8t&sig=7GGqQl1RRr-mjpdhV5O8TTDMTZA
  16. M. K. Bourbatache, ”Probabilistic analysis of a class of impulsive linear random differential equations,” ScienceDirect, 2021. https://www.sciencedirect.com/science/article/pii/S0893965921002755
  17. J. C. Cort´es, S. E. Delgadillo-Aleman, ”Probabilistic analysis of impulsive differential equations,” Springer, 2021. https://link.springer.com/article/10.1007/s11242-021-01683-2
  18. J. Awrejcewicz, ”Numerical simulations of physical and engineering processes,” Google Books, 2011. https://books.google.com/books?hl=en&lr=&id= HLqZDwAAQBAJ&oi=fnd&pg=PR11&dq=Advanced+transformation+methods+for+non-homogeneous+differential+equations+Raphael+et+al
  19. S. Pinelas, J. Rossa, ”Non-homogeneous Navier-Stokes equations,” Differential Equations and Applications, 2015. https://comum.rcaap.pt/handle/10400.26/11296
  20. R. Chetrite, H. Touchette, ”Variational and optimal control representations of conditioned and driven processes,” IOPscience, 2015. https://iopscienceiop.org/article/10.1088/1742-5468/2015/12/P12001/meta
  21. S. Schaaf, ”Cooling of non-homogeneous media with cylindrical symmetry,” University of California, 1944. https://scholar.google.com/citations?user=NHYa9hsAAAAJ&hl=en&num=20&oi=sra
  22. K. Chaganti and P. Paidy, “Strengthening Cryptographic Systems with AI-Enhanced Analytical Techniques,” International Journal of Applied Mathematical Research, vol. 14, no. 1, pp. 13–24, 2025. [Online]. Available: https://doi.org/10.14419/fh79gr07
  23. K. C. Chaganti, “A Scalable, Lightweight AI-Driven Security Framework for IoT Ecosystems: Optimization and Game Theory Approaches,” IEEE Access, vol. 99, pp. 1–1, 2025. [Online]. Available: https://doi.org/10.1109/ACCESS.2025.3558623

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How to Cite

Chinni, R. (2025). Modeling the Spread and Mitigation of Cybersecurity Threats using Non-Homogeneous Heat Equations with Boundary Conditions. International Journal of Applied Mathematical Research, 14(2), 40-56. https://doi.org/10.14419/h702mv77

Received date: June 12, 2025

Accepted date: August 9, 2025

Published date: August 15, 2025