Home – Home – Jornals – Numerical Analysis and Applications 2019 number 1

In this paper, we obtain the existence and uniqueness of periodic solutions for a nonlinear fourth-order differential equation utilizing an explicit Green's function and fixed point index theorem combining with an operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish a theorem on the existence of positive solutions for fourth-order boundary value problem with variable parameter. Finally, we give an example to illustrate our results.

The existence and uniqueness of the generalized solution of the boundary value problem for the system of magnetoporosity equations in the dissipative approximation have been proved. The results of the numerical solution obtained by the finite element method of the test boundary value problem of magnetoporosity in the frequency domain are presented.

The Kalman filter algorithm is currently one of the most popular approaches to solving the data assimilation problem. The major line of the application of the Kalman filter to the data assimilation is the ensemble approach. In this paper, we propose a version of the Kalman stochastic ensemble filter. In the algorithm presented the ensemble perturbations analysis is attained by means of transforming an ensemble of forecast perturbations. The analysis step is made only for a mean value. Thus, the ensemble π-algorithm is based on the advantages of the stochastic filter and the efficiency and locality of the square root filters. The numeral method of the ensemble π-algorithm realization is proposed, the applicability of this method has been proved. This algorithm is implemented for the problem in the three-dimensional domain. The results of the numeral experiments with the model data for estimating the efficiency of the offered numeral algorithm are presented. The comparative analysis of the root-mean-square error behavior of the ensemble π-algorithm and the Kalman ensemble filter by means of the numeral experiments with a one-dimensional Lorentz model is made.

This paper realizes a symbiosis of two approaches to the numerical solution of second order ODEs with a small parameter having singularities such as interior and boundary layers, namely, the application of both compact schemes of high orders and layer-resolving grids. The generation of layer-resolving grids, based on estimates of solution derivatives and formulations of coordinate transformations eliminating solution singularities, is a generalization of the methodology early developed for the first order scheme. This paper presents the formulas of the coordinate transformations and numerical experiments for the schemes of the first, second, and third orders on uniform and layer-resolving grids for the equations with boundary, interior, exponential and power layers of the first and second scales. The experiments conducted confirm the uniform convergence of the numerical solutions of equations with the help of compact schemes of high orders on the layer-resolving grids. By using the transfinite interpolation methodology or numerical solutions to the Beltrami and diffusion equations in a control metric, built by the coordinate transformations eliminating the solution singularities, the developed technology can be generalized to the solution of multi-dimensional equations with boundary and interior layers.

The algorithms for solving the inverse source problem for the production-destruction type systems of nonlinear ordinary differential equations with measurement data in the form of time series are presented. The sensitivity operator and its discrete analogue on the basis of adjoint equations are constructed. This operator binds the perturbations in the unknown parameters of the model to those of the measured values. The operator allows one to construct a family of quasi-linear operator equations linking the required unknown parameters and the data of the inverse problem. The Newton-Kantorovich type method with right-hand side r- pseudoinverse matrices is used to solve the equations. The algorithm is applied to solving the inverse source problem for the atmospheric impurities transformation model.

This paper is concerned with investigating and solving the mixed initial boundary value problem for the heat conduction equation. The statement of the problem includes the three intervals: the first one (from 0 → T₁) describes heating the combustion chamber, the second (from T₁ → T₂) - cooling the chamber and a slower cooling of its wall. Finally, the third interval describes natural cooling of the chamber wall when the chamber has the temperature coinciding with that of environment. The validity of the application of the Fourier transform with respect to this problem has been proved. This made possible to transform the governing equation to the ordinary differential equation. By using the resulting equation, the inverse boundary value problem for the heat conduction equation by applying the nonlinear method of projection regularization was solved and the error of approximate solution was obtained.

One of widespread approaches to solving the Cauchy problem for the Laplace equation is to reduce it to the inverse problem. As a rule, an iterative procedure to solve the latter is used. In this study, an efficient direct method for the numerical solution of the inverse problem in the rectangular form is described. The main idea is based on the expansion of the desired solution with respect to a basis consisting of eigenfunctions of a difference analogue of the Laplace operator.