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- W2999335523 abstract "This Dissertation is devoted to the study of boundary value problems and concerns two research areas. The first one is related to the perturbation analysis of boundary value problems in perforated domains and its application to the investigation of effective properties of composite materials. We investigate the dependence of the solutions of transmission boundary value problems upon some parameters and their behavior when the parameter corresponding to the size of the inclusions tends to zero, and the other parameters tend to some fixed values. Then we apply our results to study the effective conductivity of periodic composites. We also investigate the behavior of the solution of the Dirichlet problem for the Poisson equation in the domain in R3 which consists of a periodic array of cylinders upon perturbation of the shape of the cross-section of the cylinders and the periodic structure. Moreover, we apply our results to study the behavior of the longitudinal permeability of a periodic array of cylinders upon such perturbation. The second part of the Dissertation is related to the development of tools for solving boundary value problems for functions taking values in commutative Banach algebras. In particular, we investigate the properties of logarithmic residues of monogenic (continuous and differentiable in the sense of Gateau) functions and the behavior of the certain Cauchy type integral on the boundary of its definition. The Dissertation consists of two parts and is organized as follows. Part I consists of three chapters. In Chapter 1 we investigate the asymptotic behavior of the solutions of singularly perturbed (ideal and nonideal nonlinear) transmission problems in a periodically perforated domain. In Chapter 2 we apply the results of Chapter 1 to study the asymptotic behavior of the effective thermal conductivity of a periodic two-phase dilute composite. Chapter 3 is devoted to the study of the behavior of the longitudinal permeability of a periodic array of cylinders upon perturbation of the shape of the cross section of the cylinders and of the periodic structure. Part II consists of two chapters. In Chapter 4 we introduce a three-dimensional commutative algebra over C with a one-dimensional radical and study the logarithmic residues of monogenic functions in this algebra. Chapter 5 is devoted to the investigation of a certain analog of Cauchy type integral taking values in the mentioned algebra and its limiting values on the boundary of the domain of definition. At the end of the Dissertation, we have enclosed three appendices with some results which we have exploited in the Dissertation." @default.
- W2999335523 created "2020-01-23" @default.
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- W2999335523 date "2018-09-30" @default.
- W2999335523 modified "2023-09-27" @default.
- W2999335523 title "Periodic and hypercomplex potentials. Properties and applications" @default.
- W2999335523 hasPublicationYear "2018" @default.
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