An asymptotic homogenization formula for complex permittivity and its application

Yıl 2023, Cilt: 7 Sayı: 1, 243 - 252, 31.03.2023

Vladimir Mityushev , Tatjana Gric Zh. Kh. Zhunussova , Karlygash Dosmagulova

https://doi.org/10.31197/atnaa.1223064

Öz

The $\mathbb R$-linear boundary value problem in a multiply connected domain on a flat torus is considered. This problem is closely related to the Riemann-Hilbert problem on analytic functions. The considered problem arises in the homogenization procedure of random media with complex constants which express the permittivity of components. A new asymptotic formula for the effective permittivity tensor is derived. The formula contains location of inclusions in symbolic form. The application of the derived formula to investigation of the morphology of the tumor cells in disordered biological media is discussed.

Anahtar Kelimeler

Composites, Asymptotic formulas, homogenization

Destekleyen Kurum

Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan

Proje Numarası

AP08856381

Kaynakça

• [1] I. Andrianov, V. Mityushev, "Exact and exact formulae in the theory of composites," in Modern Problems in Applied Analysis. Trends in Mathematics, P. DrygaÅŻ, S. Rogosin, Ed, BirkhÃďuser, Cham. (2018).
• [2] I. Andrianov, S. Gluzman, V. Mityushev, (eds.), Mechanics and Physics of Structured Media: Asymptotic and Integral Methods of Leonid Filshtinsky, Academic Press, London (2022).
• [3] N. S. Bakhvalov, G. Panasenko, Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems" in The Mechanics of Composite Materials, Vol. 36, Springer Science and Business Media (2012).
• [4] Bliev, N. K. (1997). Generalized analytic functions in fractional spaces, Boca Raton, CRC Press.
• [5] F. D. Gakhov, Boundary Value Problems, Elsevier (2014).
• [6] S. Gluzman, V. Mityushev, W. Nawalaniec, Computational Analysis of Structured Media, Academic Press, Elsevier, Amsterdam (2018).
• [7] T. Gric, S. G. Sokolovski, N. Navolokin, O. Semyachkina-Glushkovskaya, and E. U. Rafailov, "Metamaterial formalism approach for advancing the recognition of glioma areas in brain tissue biopsies," Opt. Mater. Express 10, 1607-1615 (2020).
• [8] Kalmenov, T. S., Sadybekov, M. A. (2017). On a Frankl-type problem for a mixed parabolic-hyperbolic equation. Siberian Mathematical Journal, 58(2), 227-231.
• [9] Mikhlin, S. G. (2014). Integral equations: and their applications to certain problems in mechanics, mathematical physics, and technology. Elsevier.
• [10] V. Mityushev, S. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions: Theory and Applications, Chapman & Hall / CRC, Monographs, and Surveys in Pure and Applied Mathematics, Boca Raton, etc. (2000).
• [11] V. Mityushev, "R-linear problem on torus and its application to composites," Complex Variables 50(7-10), 621-630 (2005).
• [12] Mityushev, V., Rylko, N. (2022). Effective properties of two-dimensional dispersed composites. Part I. Schwarz’s alternating method. Computers and Mathematics with Applications, 111, 50-60.
• [13] V. Mityushev, T. Gric, E. Rafailov, "On the study of the THz metamaterials to deal ith the dielectric response of the cancerous biological tissues", Proc. SPIE 12230, Terahertz Emitters, Receivers, and Applications XIII, 122300E (30 September 2022); http://dx.doi.org/10.1117/12.2632733
• [14] N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 2008. Translated from the 2nd Russian edition by 1953.
• [15] Z. Zhang, H. Ding, X. Yan, L. Liang, D. Wei, M. Wang, Q. Yang, and J. Yao, Sensitive detection of cancer cell apoptosis based on the non-bianisotropic metamaterials biosensors in terahertz frequency, Opt. Mater. Express 8(3), 659667 (2018).
• [16] Mityushev, V., Zhunussova, Z. (2021). Optimal Random Packing of Spheres and Extremal Effective Conductivity. Symmetry, 13(6), 1063 https://doi.org/10.3390/sym13061063
• [17] Zari, I., Gul, T., Dosmagulova, K., Saee, T., Haq, S. (2023). Heat transfer analysis of Radiative-Marangoni Convective flow in nanofluid comprising Lorentz forces and porosity effects. Advances in the Theory of Nonlinear Analysis and its Applications, 7(1), 61-81.
• [18] Zhunussova, Zh.Kh., Ashimov, Ye.K., Dosmagulova, K.A., Zhunussova, L.Kh. (2022). Optimal Packing of Two Disks on Torus. Applied Mathematics and Information Sciences, 16(4), 549-554.