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On the Effects of Boundary Conditions in One-Dimensional Models of Hemodynamics
by Gerasim V. KrivovichevORCID
Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, 7/9 Universitetskaya Nab., 199034 Saint Petersburg, Russia
Mathematics 2022, 10(21), 4058; https://doi.org/10.3390/math10214058
Received: 9 October 2022 / Revised: 26 October 2022 / Accepted: 28 October 2022 / Published: 1 November 2022
(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)
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Abstract
The paper is devoted to the theoretical analysis of the effects of boundary conditions on the solutions of the system of one-dimensional (1D) hemodynamics. The integral inequalities, which realize the energy inequalities for the solutions of initial-boundary-value problems, are obtained. It is demonstrated that the unphysical unbounded solutions can take place for the case of bounded functions from boundary conditions. For the periodic boundary conditions, the integral estimation illustrates the correct behavior of the solution. For this case of boundary conditions, the effective Fourier method for the analytical solution is proposed. The analytical solutions, obtained by this approach, can be used for the comparison of different 1D blood-flow models. The results obtained in the paper allow for an the alternatively view of the stated boundary conditions and can explain some problems, which can arise in numerical simulations. They expand the possibilities of the application of analytical methods in the field of blood-flow simulation. The results can be useful for the specialists on blood-flow modeling.