Finite Difference Method For Solving Research Articles

IMPLICIT METHOD FOR NONLINEAR COMPLEX DIFFUSION WITH APPLICATIONS TO IMAGE DENOISING

In this paper we focus on the development and implementation of an implicit finite difference method for solving a complex diffusion differential equation with applications to noise filtering in images. We give a complete characterization of the implementation of the fully implicit method, including the characterization of the derivatives for Newton's method at each step. In order to show the feasibility of the method, we compare the implicit method with the semi-implicit and apply it to both \textit{in vivo} images of the human retina by optical coherence tomography (OCT) and standard images (Lena and the Cameraman). We compute noise removal performance metrics to show the effect of several parameters in the performance of the method.

Finite difference methods for solving the transport equation in the problems of optical biomedical diagnostics

Настоящая работа представляет собой ретроспективный обзор развития подхода к численному моделирования распространения излучения в биологических тканях, основанного на применении сеточных методов для решения уравнения переноса излучения (УПИ), и его применения для задач биомедицинской диагностики. Преимуществом сеточных методов является возможность получения решения прямой задачи распространения излучения в мутных средах в случаях, когда получение точного аналитического решения невозможно. В свою очередь, возможность решения УПИ открывает широкие перспективы для решения обратной задачи и восстановления структурных или функциональных характеристик объектов на основании регистрации рассеянного излучения. В работе приводится обзор результатов применения сеточных методов для решения задач ангиографии и исследования крови с помощью технологий оптической биомедицинской диагностики.

ON MATHEMATICAL MODELLING OF METALS DISTRIBUTION IN PEAT LAYERS

In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in multilayered domain. We consider the metals Fe and Ca concentration in the layered peat blocks. Using experimental data the mathematical model for calculation of concentration of metals in different points in peat layers is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations (PDEs) of second order with piece-wise diffusion coefficients in the layered domain. We develop here a finite-difference method for solving of a problem of one, two and three peat blocks with periodical boundary condition in x direction. This procedure allows to reduce the 3-D problem to a system of 2-D problems by using circulant matrix.

Action of Pulse Magnetic Field on Molten Metal.

Effect of pulse magnetic field on two-phase zone of crystallizing metal has been investigated by methods of mathematical modelling in reference to vertical continuous casting machine. On the first step of the investigation electromagnetic problem for penetration of pulse magnetic field of cylindrical inductor in depth of liquid metal has been solved and distribution of electromagnetic forces in the melt has been obtained at different time moments. With a view to solve this problem combined method has been proposed. It contains boundary element method for description of field in surrounding air medium with inductor and also finite difference method for solving of the problem in molten metal. On the second step of the investigation with the help of obtained distribution of electrodynamic forces problem for acceleration of the melt has been solved. It is determined the optimal relation for pulse duration (in the case of single pulse), when melt velocity reaches its maximum value. Besides, it is shown that magnetic field may be used with the purpose of effect on crystallization zone through shell of solidified metal.