The numerical analysis of nonlinear mathematical models on graphs

Physical sciences, mathematics (01P).

The numerical algorithms for non-stationary mathematical models in nonstandard domains are investigated in the dissertation. The problem definition domain is represented by branching structures with conjugation equations considered at the branching points. The numerical analysis of the conjugation equations and non-classical boundary conditions distinguish considered problems among the classical problems of mathematical physics presented in the literature.

The scope of the dissertation covers the investigation of stability and convergence of the numerical algorithms on branching structures with different conjugation equations, the construction and implementation of parallel algorithms, the investigation of the numerical schemes for the problems with nonlocal integral conditions. The modeling of the excitation of neuron and photoexcited carrier decay in a semiconductor, also the problem of the identification of nonlinear model are considered in the dissertation.

The dissertation consists of an introduction, five chapters, main conclusions, bibliography and the list of the author’s publications. Introductory chapter covers the problem formulation and the object of research, the topicality of the thesis, the aims and objectives of the dissertation, the methodology of research, scientific novelty and the practical value of the achieved results. The defended thesis and structure of the dissertation are given in this chapter. The first chapter presents the overview of mathematical models in the non-standard domains or with non-classical boundary conditions. In second chapter the θ- implicit algorithm and two predictor-corrector algorithms for the linear parabolic problems on branching structures are given. The stability and convergence analysis of these algorithms is performed. The third chapter presents the parallel versions of the algorithms, the efficiency and scalability analysis of parallel algorithms, and the numerical simulation of neuron excitation. The fourth chapter deals with the one-dimensional parabolic problem with nonlocal integral boundary condition. The analysis of the implicit scheme is performed with different values of coefficients of the nonlocal boundary condition. The nonlinear model of excited carrier decay in a semiconductor is presented in chapter five. The analysis of the linearized numerical scheme and the results of the model fitting experiment are presented.

The research results have been published in 7 scientific articles, 6 of them in the reviewed scientific journals. The results have been presented at 13 conferences.

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DOI: https://doi.org/10.20334/2006-M