Department of Applied Mathematics

Properties of singular solutions of differential equations, spectral analysis of difference systems and modeling of nonlinear processes

**The composition of the scientific group:** Volodymyr Derkach, Doctor of Physiсs and Mathematics, Professor, Head of the Department (head), Ihor Skrypnik, Doctor of Physiсs and Mathematics, Corresponding Member of the NASU (co-head), Kateryna Buryachenko, Ph.D. in Physiсs and Mathematics, Assistant Professor, senior researcher, Mariya Shan, Ph.D. in Physiсs and Mathematics, junior researcher, teacher, Olha Trofymenko, Ph.D. in Physiсs and Mathematics, Assistant Professor of the Department, senior researcher, Yuliya Orban’, Ph.D. in Physiсs and Mathematics, Assistant Professor of the Department, senior researcher

Nonnegative weak solutions of quasilinear parabolic equations were studied. The model cases of these equations are represented by a double nonlinear anisotropic parabolic equation with an absorption term and an anisotropic porous medium equation, including one with gradient absorption. For them point-by-point upper estimates were obtained, which were written down as a distance to area boundary. Point-by-point estimates were obtained through the nonlinear Ries potential in the right-hand side of the equation. This helped to study the regularity of solutions —prove the local boundedness in continuity of weak solutions of the anisotropic porous medium equation. New classes of generalized boundary triplets for symmetric operators in Hilbert space are also introduced – B-generalized, S-generalized, and ES-generalized boundary triplets, and the corresponding Weyl functions are investigated. It is shown that the Weyl function class of B-generalized boundary triplets coincides with the class of strict non-Vanlinian operator-functions, the Weyl function class of S-generalized boundary triplets coincides with the class of non-Vanlinian operator-functions with invariant domain definition, the Weyl function class of ES-generalized boundary triplets coincides with the class of non-Vanlinian operator-functions, such that the quadratic form of an imaginary part has an invariant domain. For each of these classes, the problem of implementing a function operator in the form of Weyl functions is solved. The obtained results are applied to the classification of boundary value problems for the Laplace operator in domains with smooth boundary and Lipschitz boundary, as well as mixed boundary value problems.

Two-dimensional eigenfunctions of magnetic and electric vector potentials in the form of their decomposition into series by orthogonal Chebyshev polynomials of the 1st and 2nd kind are constructed to describe the current density in a transmission line with stepwise inhomogeneity of finite length in it.

A method of calculation was developed and the resonant properties of a two-layer distributed inhomogeneity in the form of a comb slit resonator in the grounding layer of a microstrip transmission line combined with a stepped (capacitive) inhomogeneity in it were studied. It is proved that such a structure in general has a complex scattering characteristic with wide pass bands and stop bands, and therefore can be used to design multifunctional devices containing resonant reflection and resonant signal transmission frequencies.

The results obtained within the project will improve the current knowledge in this scientific field and will be used by scientists from Ukraine (Taras Shevchenko National University of Kyiv, M. Drahomanov National Pedagogical University, Sikorsky National Polytechnic University, Kharkiv National University V. Karazin, Institute of Applied Mathematics and Mechanics of NASU, Institute of Mathematics of the NASU, Tymoshenko Institute of Mechanics of the NASU), Italy (La Sapienza University, Universities of Parma, Naples, Florence), France (The Pierre and Marie Curie University, University of Tours), Austria (University of Graz), Germany (University of Lübeck), Finland (University of Vaasa).

In the field of science – mathematical physics, theory of functions, functional analysis, applied electrodynamics. In the industry –high frequency and microwave radio equipment; design bureaus of companies producing transmitters for household and defense purposes; means of communication (mobile market, etc.).

The *practical value* of the project results is that quasilinear equations of diffusion-strong nonlinear absorption structure, anisotropic elliptic and parabolic equations can be used in modeling of nonlinear physical processes. Such processes include processes with irregular and singular data, as well as those that occur in highly inhomogeneous media, such as anisotropy, diffusion (including multiphase), which can be used in metal physics (in the study of diffusion processes of alloys of different materials).

Developed techniques and algorithms can be used for the analysis of multilayer inhomogeneities in planar type transmission lines to design compact filtering devices with advanced functionality. These are low pass filters, bandpass filters, harmonic filters and output links of high-efficiency active devices with control of higher harmonics.

__Skrypnik I. I.__Harnack’s Inequality for Quasilinear Elliptic Equations with Singular Absorption Term / Potential Analysis. — 2019. — Vol. 50. — P. 521—539. https://link.springer.com/article/10.1007/s11118-018-9691-9__Shan’ M.__A priori estimates of Keller-Osserman type for twice nonlinear anisotropic, parabolic equations with absorption / Proceedings of the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine. – 2018. – Vol. 32. – P. 149-159 (in Ukrainian). https://drive.google.com/file/d/1S2sOT8I9RiX5Dw43qkH8Jc8CNtImdILk/view__Shan M. A., Skrypnik I. I.__Keller-Osserman estimates and removability result for the anisotropic porous medium equation with gradient absorption term / Mathematische Nachrichten. — 2019. — Vol. 292. — P. 436—453. https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.201700177__Buryachenko K.O.__Riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation /__Buryachenko K.O., Skrypnik I.I.__//*Nonlinear Analysis, Theory, Methods and Applications (2019) N 178, p. 56-85.*https://www.mendeley.com/catalogue/riesz-potentials-pointwise-estimates-solutions-anisotropic-porous-medium-equation/__Buryachenko K.O.__Local subestimates of solutions to double-phase parabolic equations via nonlinear parabolic potentials /__K.O.Buryachenko__*//Journal of Mathematical Sciences (United States) (2019) 242(6), p. 772-786.*https://www.mendeley.com/catalogue/local-subestimates-solutions-doublephase-parabolic-equations-via-nonlinear-parabolic-potentials/__Derkach V.A__. Rigged de Branges-Pontryagin spaces and their applications to extentions and embedding/__V. A. Derkach__, H. Dym//*Journal of Functional Analysis- (2019), 277(1), p. 31-110.*https://www.mendeley.com/catalogue/rigged-brangespontryagin-spaces-application-extensions-embedding/__D. Strelnikov__. Boundary triples for integral systems on the half-line. Methods of Functional Analysis and Topology, V.25, (2019), no. 1, pp. 84-96. (http://mfat.imath.kiev.ua/article/?id=1143)__L. Pestov, D. Strelnikov__. Approximate controllability of the wave equation with mixed boundary conditions. Journal of Mathematical Sciences, V.239, (2019), no. 2, pp. 75-85. (https://doi.org/10.1007/s10958-019-04289-8)__Olga Trofimenko__, Anastasiya Minenkova. Integral identities for polyanalytic functions // Topics in Classical and Modern Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, 2019. – p.279-291. https://doi.org/10.1007/978-3-030-12277-5_17

Department of Applied Mathematics