The approximation of the solution of wave problems by spectral expansions connected with elliptic differential operators
DOI:
https://doi.org/10.37934/arfmts.86.2.101106Keywords:
Spectral expansion, wave problem, elliptic differential operatorAbstract
In this research, we investigate the spectral expansions connected with elliptic differential operators in the space of singular distributions, which describes the vibration process made of thin elastic membrane stretched tightly over a circular frame. The sufficient conditions for summability of the spectral expansions connected with wave problems on the disk are obtained by taking into account that the deflection of the membrane during the motion remains small compared to the size of the membrane and for wave propagation problems, the disk is made of some thermally conductive material.
References
Alimov Sh, A. "On the spectral decompositions of distributions." In Doklady Mathematics, vol. 331, pp. 661-662. 1993.
Alimov, S. A., and A. A. Rakhimov. "Localization of spectral expansions of distributions." Differential Equations 32, no. 6 (1996): 798-802.
Rakhimov, Abdumalik A. "On The Approximation of The Spectral Expansions of Distributions in The Weak Topology." Australian Journal of Basic and Applied Sciences 6, no. 2 (2012): 20-24.
Fargana, A., A. A. Rakhimov, A. A. Khan, and T. B. H. Hassan. "Equiconvergence in Summation Associated with Elliptic Polynomial." In Journal of Physics: Conference Series, vol. 949, no. 1, p. 012001. IOP Publishing, 2017. https://doi.org/10.1088/1742-6596/949/1/012001
Rakhimov, A. A. "On the equiconvergence of the Fourier series and the Fourier integral of distributions." In AIP Conference Proceedings, vol. 1739, no. 1, p. 020060. AIP Publishing LLC, 2016. https://doi.org/10.1063/1.4952540
Rakhimov, Abdumalik A., Torla Bin Hj Hassan, and Ahmad Fadly Nurullah bin Rasedee. "On equiconvergence of Fourier series and Fourier integral." In Journal of Physics: Conference Series, vol. 819, no. 1, p. 012025. IOP Publishing, 2017. https://doi.org/10.1088/1742-6596/819/1/012025
Fargana, A., A. A. Rakhimov, A. A. Khan, and T. B. H. Hassan. "Optimization of the Regularization of the Solution of the Plate Vibration Problem." International Journal of Applied Engineering Research 13, no. 8 (2018): 6364-6368.
Rakhimov, A. "On the summability of the spectral expansions associated with the elliptic differential operators." In Journal of Physics: Conference Series, vol. 1132, no. 1, p. 012013. IOP Publishing, 2018. https://doi.org/10.1088/1742-6596/1132/1/012013
Rakhimov, Abdumalik. "Localization of the Spectral Expansions Associated with the Partial Differential Operators." In Mathematical Methods in Engineering, pp. 217-233. Springer, Cham, 2019. https://doi.org/10.1007/978-3-319-91065-9_11
Ahmedov, Anvarjon A., Nur Amalina Binti Jamaludin, and Abdumalik Rakhimov. "Uniformly Convergence of The Spectral Expansions of The Schrödinger Operator on A Closed Domain." In Journal of Physics: Conference Series, vol. 435, no. 1, p. 012014. IOP Publishing, 2013. https://doi.org/10.1088/1742-6596/435/1/012014
Ahmedov, Anvarjon A., Ahmad Fadly Nurullah bin Rasedee, and Abdumalik Rakhimov. "On the sufficient conditions of the localization of the Fourier-Laplace series of distributions from liouville classes." In Journal of Physics: Conference Series, vol. 435, no. 1, p. 012016. IOP Publishing, 2013. https://doi.org/10.1088/1742-6596/435/1/012016
Alimov, Sh A. "Generalized localization of Riesz means of spectral expansions of distributions." In Doklady Mathematics, vol. 86, no. 2, pp. 597-599. SP MAIK Nauka/Interperiodica, 2012. https://doi.org/10.1134/S1064562412050018
Ashurov, Ravshan, Almaz Butaev, and Biswajeet Pradhan. "On generalized localization of Fourier inversion associated with an elliptic operator for distributions." In Abstract and Applied Analysis, vol. 2012. Hindawi, 2012. https://doi.org/10.1155/2012/649848
Mustafa, Wan Azani, Haniza Yazid, and Noratikah Mazlan. "An enhancement method on illumination images: A survey." Journal of Advanced Review on Scientific Research 28, no. 1 (2016): 33-41.
