UNVEILING THE POWER OF SOBOLEV SPACES FOR ENGINEERING APPLICATIONS IN LIPSCHITZ-BOUNDED DOMAINS
Keywords:
Universal extension, Sobolev spaces, Lipschitz domains, generalized Regular decompositionsAbstract
A family of universal extension operators for Sobolev spaces of differential forms on Lipschitz-bounded domains is the major goal of this paper. We want to show that Sobolev spaces of differential forms are essential to the analysis of bounded Lipschitz domains. The goal is to connect mathematics theory to engineering and science applications in complicated geometric contexts. Sobolev spaces, a staple of mathematical analysis, are widely used to explore partial differential equations and provide attractive Euclidean solutions. These conventional spaces struggle with Lipschitz-limited domains due to their complicated geometric properties. Universal extension operators provide a systematic and robust way to apply Sobolev spaces to more complex scenarios. Within the scope of this research, we investigate the theory of Sobolev spaces for differential forms as well as their application to Lipschitz-constrained domains. The construction approach maintains essential aspects of mathematics by building atop smooth forms, L2-spaces, and Sobolev spaces. Sobolev spaces have wide-ranging applications across engineering and science. In structural mechanics, they model stress distribution in irregular components, optimizing material use in constructions like reinforced concrete beams. For image processing, Sobolev spaces enable precise denoising, vital in medical imaging and satellite analysis. Electrical engineers employ Sobolev-based methods for electromagnetic field analysis in designing antennas and circuits. In aerospace, Sobolev spaces simulate airflow around complex surfaces, improving aerodynamics. In material science, they aid in understanding nanoscale behaviors, advancing quantum properties and nanomaterial applications. These case studies highlight Sobolev spaces' adaptability, providing a mathematical foundation to tackle intricate challenges across scientific and engineering disciplines.
References
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28. F. L. Hasan and M. A. Abdoon, "The generalized (2+1) and (3+1)-dimensional with advanced analytical wave solutions via computational applications," International Journal of Nonlinear Analysis and Applications, vol. 12, no. 2, pp. 1213-1241, 2021.
29. A. Qazza and R. Saadeh, "On the Analytical Solution of Fractional SIR Epidemic Model," Applied Computational Intelligence and Soft Computing, 2023, pp. 1–16.
30. R. Saadeh, "Numerical algorithm to solve a coupled system of fractional order using a novel reproducing kernel method," Alexandria Engineering Journal, vol. 60, no. 5, pp. 4583-4591, 2021.
31. A. Qazza, R. Saadeh, and E. Salah, "Solving fractional partial differential equations via a new scheme," AIMS Mathematics, vol. 8, no. 3, pp. 5318–5337, 2022.
32. R. Edwan, R. Saadeh, S. Hadid, M. Al-Smadi, and S. Momani, "Solving time-space-fractional Cauchy problem with constant coefficients by finite-difference method," Computational Mathematics and Applications, pp. 25-46, 2020.
33. A. Qazza, A. Burqan, R. Saadeh, and R. Khalil, "Applications on Double ARA–Sumudu Transform in Solving Fractional Partial Differential Equations," Symmetry, vol. 14, no. 9, pp. 1817, 2022.
34. M. Fornasier, G. Savaré, and G. E. Sodini, "Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces," Journal of Functional Analysis, vol. 285, no. 11, pp. 110153, 2023.
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37. C. Amrouche and M. Moussaoui, "The Dirichlet problem for the Laplacian in Lipschitz domain," Abstract, arXiv preprint arXiv:2204.02831, 2022.
38. S. Talukder and R. Kumar, "Robust Stability of Neural-Network-Controlled Nonlinear Systems With Parametric Variability," IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2023.
39. M. Revay, R. Wang, and I. R. Manchester, "Recurrent equilibrium networks: Flexible dynamic models with guaranteed stability and robustness," IEEE Transactions on Automatic Control, 2023.
40. M. Kanatov and L. Atymtayeva, "Deep convolutional neural network based person detection and people counting system," Advanced Engineering Technology and Application, vol. 7, no. 3, pp. 5-9, 2018.
41. A. A. Alwan, H. A. Naser, and A. E. Hashoosh, "The solutions of Mixed Hemiequilibrium Problem with application in Sobolev space," in IOP Conference Series: Materials Science and Engineering, vol. 571, no. 1, p. 012019, IOP Publishing, July 2019.
42. R. O. Ogbumba, M. S. Shagari, M. Alansari, T. A. Khalid, E. A. E. Mohamed, and A. A. Bakery, "Advancements in Hybrid Fixed Point Results and F-Contractive Operators," Symmetry, vol. 15, no. 6, p. 1253, Jun. 2023.
43. L. Bourgeois, S. Fliss, J. F. Fritsch, C. Hazard, and A. Recoquillay, "Scattering in a partially open waveguide: the forward problem," IMA Journal of Applied Mathematics, vol. 88, no. 1, pp. 102-151, 2023.
44. A. Araujo, A. Havens, B. Delattre, A. Allauzen, and B. Hu, "A unified algebraic perspective on lipschitz neural networks," arXiv preprint arXiv:2303.03169, 2023.
45. B. Delattre, A. Araujo, Q. Barthélemy, and A. Allauzen, "The Lipschitz-Variance-Margin Tradeoff for Enhanced Randomized Smoothing," arXiv preprint arXiv:2309.16883, 2023.
46. A. Qazza and R. Hatamleh, "The existence of a solution for semi-linear abstract differential equations with infinite B-chains of the characteristic sheaf," International Journal of Applied Mathematics, vol. 31, no. 5, 2018.
47. R. Hiptmair, J. Li, and J. Zou, "Universal extension for Sobolev spaces of differential forms and applications," Journal of Functional Analysis, vol. 263, no. 2, pp. 364-382, 2012.
48. E. M. Stein, "Singular integrals and differentiability properties of functions," Princeton University Press, 1970.
49. O. Karadsheh, A. Ahmad, A. Hosny, E. Omar, H. Bushra, and A. Qusef, "The Quality of Civil Engineering Graduates: Case of Jordan," Journal of Statistics Applications & Probability, vol. 12, no. 1, pp. 267-276, 2023.
50. S. Bukenov and A. Akshabayev, "Using Neural Networks to Improve Emotional State of Person," Advanced Engineering Technology and Application, vol. 5, no. 3, pp. 19-22, 2016..
2. F. Della Pietra, C. Nitsch, F. Oliva, and C. Trombetti, "On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1," Advances in Calculus of Variations, vol. 16, no. 4, pp. 1123-1135, 2023.
3. P. Pauli, R. Wang, I. R. Manchester, and F. Allgöwer, "Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian," arXiv preprint arXiv:2303.11835, 2023.
4. R. Wang and I. Manchester, "Direct parameterization of lipschitz-bounded deep networks," in International Conference on Machine Learning, pp. 36093-36110, July 2023.
5. E. Di Nezza, G. Palatucci, and E. Valdinoci, "Hitchhiker's guide to the fractional Sobolev spaces," Bulletin des sciences mathématiques, vol. 136, no. 5, pp. 521-573, 2012.
6. R. J. DiPerna and P. L. Lions, "Ordinary differential equations, transport theory and Sobolev spaces," Inventiones mathematicae, vol. 98, no. 3, pp. 511-547, 1989.
7. L. Tartar, An introduction to Sobolev spaces and interpolation spaces, vol. 3, Springer Science & Business Media, 2007.
8. B. Azzaoui, B. Tellab, and K. Zennir, "Positive solutions for integral nonlinear boundary value problem in fractional Sobolev spaces," Mathematical Methods in the Applied Sciences, vol. 46, no. 3, pp. 3115-3131, 2023.
9. G. E. Comi and G. Stefani, "A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I," Revista Matemática Complutense, vol. 36, no. 2, pp. 491-569, 2023.
10. P. Pucci and L. Temperini, "On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces," Math. Eng., vol. 5, no. 21, 2023..
11. D. K. Almutairi, M. A. Abdoon, S. Y. M. Salih, S. A. Elsamani, F. E. Guma, and M. Berir, "Modeling and Analysis of a Fractional Visceral Leishmaniosis with Caputo and Caputo–Fabrizio derivatives," Journal of the Nigerian Society of Physical Sciences, pp. 1453-1453, 2023.
12. M. Elbadri, M. A. Abdoon, M. Berir, and D. K. Almutairi, "A Numerical Solution and Comparative Study of the Symmetric Rossler Attractor with the Generalized Caputo Fractional Derivative via Two Different Methods," Mathematics, vol. 11, no. 13, pp. 2997, 2023.
13. M. Elbadri, M. A. Abdoon, M. Berir, and D. K. Almutairi, "A Symmetry Chaotic Model with Fractional Derivative Order via Two Different Methods," Symmetry, vol. 15, no. 6, 2023.
14. F. E. Guma, O. M. Badawy, M. Berir, and M. A. Abdoon, "Numerical Analysis of Fractional-Order Dynamic Dengue Disease Epidemic in Sudan," Journal of the Nigerian Society of Physical Sciences, pp. 1464-1464, 2023.
15. F. E. Guma, O. M. Badawy, A. G. Musa, B. O. Mohammed, M. A. Abdoon, M. Berir, and S. Y. M. Salih, "Risk factors for death among COVID-19 Patients admitted to isolation Units in Gedaref state, Eastern Sudan: a retrospective cohort study," Journal of Survey in Fisheries Sciences, vol. 10, no. 3S, pp. 712-722, 2023.
16. R. Saadeh, M. A. Abdoon, A. Qazza, and M. Berir, "A Numerical Solution of Generalized Caputo Fractional Initial Value Problems," Fractal And Fractional, vol. 7, no. 4, pp. 332, 2023.
17. A. Qazza, M. Abdoon, R. Saadeh, and M. Berir, "A New Scheme for Solving a Fractional Differential Equation and a Chaotic System," European Journal of Pure and Applied Mathematics, vol. 16, no. 2, pp. 1128-1139, 2023.
18. M. A. Abdoon and F. L. Hasan, "Advantages of the Differential Equations for Solving Problems in Mathematical Physics with Symbolic Computation," Mathematical Modelling of Engineering Problems, vol. 9, no. 1, 2022.
19. M. A. Abdoon, R. Saadeh, M. Berir, and F. E. Guma, "Analysis, modeling and simulation of a fractional-order influenza model," Alexandria Engineering Journal, pp. 231-240, vol. 74, 2023.
20. T. Hamadneh, A. Hioual, R. Saadeh, M. A. Abdoon, D. K. Almutairi, T. A. Khalid, and A. Ouannas, "General Methods to Synchronize Fractional Discrete Reaction–Diffusion Systems Applied to the Glycolysis Model," Fractal and Fractional, vol. 7, no. 11, pp. 828, 2023.
21. A. Qazza, A. Burqan, and R. Saadeh, "Application of ARA-Residual Power Series Method in Solving Systems of Fractional Differential Equations," Mathematical Problems in Engineering, 2022, pp. 1–17.
22. E. Salah, A. Qazza, R. Saadeh, and A. El-Ajou, "A hybrid analytical technique for solving multi-dimensional time-fractional Navier-Stokes system," AIMS Mathematics, vol. 8, no. 1, pp. 1713–1736, 2023.
23. A. B. M. Alzahrani, M. A. Abdoon, M. Elbadri, M. Berir, and D. E. Elgezouli, "A Comparative Numerical Study of the Symmetry Chaotic Jerk System with a Hyperbolic Sine Function via Two Different Methods," Symmetry, 2023.
24. D. K. Almutairi, M. A. Abdoon, M. Berir, R. Saadeh, and A. Qazza, "A Numerical Confirmation of a Fractional SEITR for Influenza Model Efficiency," Appl. Math, vol. 17, no. 5, pp. 741-749, 2023.
25. E. Elshoubary, M. A. Abdoon, A. Bahatheg, and M. M. Albeladi, "Stability analysis and numerical simulation of fractional model of Leishmaniasis," Results in Nonlinear Analysis, vol. 6, no. 2, pp. 9-17, 2023.
26. T. Hamadneh, A. Hioual, R. Saadeh, M. A. Abdoon, D. K. Almutairi, T. A. Khalid, and A. Ouannas, "General Methods to Synchronize Fractional Discrete Reaction–Diffusion Systems Applied to the Glycolysis Model," Fractal and Fractional, 2023, vol. 7, pp. 828.
27. M. A. Abdoon, F. L. Hasan, and N. E. Taha, "Computational technique to study analytical solutions to the fractional modified kdv-zakharov-kuznetsov equation," in Abstract and Applied Analysis, vol. 2022, Hindawi, June 2022.
28. F. L. Hasan and M. A. Abdoon, "The generalized (2+1) and (3+1)-dimensional with advanced analytical wave solutions via computational applications," International Journal of Nonlinear Analysis and Applications, vol. 12, no. 2, pp. 1213-1241, 2021.
29. A. Qazza and R. Saadeh, "On the Analytical Solution of Fractional SIR Epidemic Model," Applied Computational Intelligence and Soft Computing, 2023, pp. 1–16.
30. R. Saadeh, "Numerical algorithm to solve a coupled system of fractional order using a novel reproducing kernel method," Alexandria Engineering Journal, vol. 60, no. 5, pp. 4583-4591, 2021.
31. A. Qazza, R. Saadeh, and E. Salah, "Solving fractional partial differential equations via a new scheme," AIMS Mathematics, vol. 8, no. 3, pp. 5318–5337, 2022.
32. R. Edwan, R. Saadeh, S. Hadid, M. Al-Smadi, and S. Momani, "Solving time-space-fractional Cauchy problem with constant coefficients by finite-difference method," Computational Mathematics and Applications, pp. 25-46, 2020.
33. A. Qazza, A. Burqan, R. Saadeh, and R. Khalil, "Applications on Double ARA–Sumudu Transform in Solving Fractional Partial Differential Equations," Symmetry, vol. 14, no. 9, pp. 1817, 2022.
34. M. Fornasier, G. Savaré, and G. E. Sodini, "Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces," Journal of Functional Analysis, vol. 285, no. 11, pp. 110153, 2023.
35. M. U. Awan, M. A. Noor, M. V. Mihai, and K. I. Noor, "Generalized Coordinated Nonconvex Functions and Integral Inequalities," Appl. Math, vol. 12, no. 2, pp. 337-344, 2008.
36. L. Béthune, T. Masséna, T. Boissin, Y. Prudent, C. Friedrich, F. Mamalet, and D. Vigouroux, "DP-SGD Without Clipping: The Lipschitz Neural Network Way," arXiv preprint arXiv:2305.16202, 2023.
37. C. Amrouche and M. Moussaoui, "The Dirichlet problem for the Laplacian in Lipschitz domain," Abstract, arXiv preprint arXiv:2204.02831, 2022.
38. S. Talukder and R. Kumar, "Robust Stability of Neural-Network-Controlled Nonlinear Systems With Parametric Variability," IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2023.
39. M. Revay, R. Wang, and I. R. Manchester, "Recurrent equilibrium networks: Flexible dynamic models with guaranteed stability and robustness," IEEE Transactions on Automatic Control, 2023.
40. M. Kanatov and L. Atymtayeva, "Deep convolutional neural network based person detection and people counting system," Advanced Engineering Technology and Application, vol. 7, no. 3, pp. 5-9, 2018.
41. A. A. Alwan, H. A. Naser, and A. E. Hashoosh, "The solutions of Mixed Hemiequilibrium Problem with application in Sobolev space," in IOP Conference Series: Materials Science and Engineering, vol. 571, no. 1, p. 012019, IOP Publishing, July 2019.
42. R. O. Ogbumba, M. S. Shagari, M. Alansari, T. A. Khalid, E. A. E. Mohamed, and A. A. Bakery, "Advancements in Hybrid Fixed Point Results and F-Contractive Operators," Symmetry, vol. 15, no. 6, p. 1253, Jun. 2023.
43. L. Bourgeois, S. Fliss, J. F. Fritsch, C. Hazard, and A. Recoquillay, "Scattering in a partially open waveguide: the forward problem," IMA Journal of Applied Mathematics, vol. 88, no. 1, pp. 102-151, 2023.
44. A. Araujo, A. Havens, B. Delattre, A. Allauzen, and B. Hu, "A unified algebraic perspective on lipschitz neural networks," arXiv preprint arXiv:2303.03169, 2023.
45. B. Delattre, A. Araujo, Q. Barthélemy, and A. Allauzen, "The Lipschitz-Variance-Margin Tradeoff for Enhanced Randomized Smoothing," arXiv preprint arXiv:2309.16883, 2023.
46. A. Qazza and R. Hatamleh, "The existence of a solution for semi-linear abstract differential equations with infinite B-chains of the characteristic sheaf," International Journal of Applied Mathematics, vol. 31, no. 5, 2018.
47. R. Hiptmair, J. Li, and J. Zou, "Universal extension for Sobolev spaces of differential forms and applications," Journal of Functional Analysis, vol. 263, no. 2, pp. 364-382, 2012.
48. E. M. Stein, "Singular integrals and differentiability properties of functions," Princeton University Press, 1970.
49. O. Karadsheh, A. Ahmad, A. Hosny, E. Omar, H. Bushra, and A. Qusef, "The Quality of Civil Engineering Graduates: Case of Jordan," Journal of Statistics Applications & Probability, vol. 12, no. 1, pp. 267-276, 2023.
50. S. Bukenov and A. Akshabayev, "Using Neural Networks to Improve Emotional State of Person," Advanced Engineering Technology and Application, vol. 5, no. 3, pp. 19-22, 2016..
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Published
2023-12-31
How to Cite
Elmhadi Afif. (2023). UNVEILING THE POWER OF SOBOLEV SPACES FOR ENGINEERING APPLICATIONS IN LIPSCHITZ-BOUNDED DOMAINS. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 4(12), 146–161. Retrieved from https://cajmtcs.casjournal.org/index.php/CAJMTCS/article/view/589
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