Home – Home – Jornals – Numerical Analysis and Applications 2021 number 4

This study aims to implement a numerical scheme in order to find the eigenvalues of the Dirichlet-to-Neumann semigroup. This can be used to check its positivity for non-circular domains. This generalized scheme is analyzed after studying the case of the unit ball, in which an explicit representation for the semigroup was obtained by Peter Lax. After analyzing the generalized scheme, we checked its convergence through numerical simulations that were performed using FreeFem++ software.

In this paper, we study a nonlinear integro-differential Volterra equation with a fractional Caputo derivative. Based on techniques derived from a study of classical Volterra equations, namely Picard's iterative sequence and the product integration method, we propose a complete analytical and numerical study of this equation. Our study is closed by the development of two numerical examples.

We analyse red refinements of tetrahedral partitions and prove that the measure of degeneracy of some produced tetrahedra may tend to infinity if refinements are constructed in an inappropriate way. The maximum angle condition is shown to be violated in these cases as well.

The work considers problem of reconstruction of differential equations parameters, describing anomalous diffusion processes, on the base of known solutions. As a tool, is used the variational interpolation method elaborated by the authors earlier. The reconstruction time-dependence of diffusivity and determination of fractional time- and space-derivatives order in anomalous diffusion equation is demonstrated. There is shown a possibility of sufficient accuracy with insignificant computational expanses.

In this paper, we consider P ²₀- P ¹ mixed finite element approximations of a class of nonlinear parabolic equations. The backward Euler scheme for temporal discretization is used. Firstly, a new mixed projection is defined and the related a priori error estimates are proved. Secondly, optimal a priori error estimates for pressure variable and velocity variable are derived. Finally, a numerical example is presented to verify the theoretical results.

The paper presents the algorithm of exponential transformation (biasing) and its randomized modification with branching of a Markov chain trajectory for solving the problems of gamma-ray transport in an inhomogeneous medium. These algorithms were applied to a maximum (majorant) cross-section method or the Woodcock tracking which is extremely efficient for the simulation in an inhomogeneous medium. On an example of gamma-ray transport through a thick water slab containing a random amount of air or Al balls, the numerical study of the above algorithms in comparison with the standard simulation algorithm is performed.

In the present work, a numerical scheme is constructed for the approximation of a class of Volterra integral equations of the convolution type with highly oscillatory kernels. The proposed numerical technique transforms the Volterra integral equations of the convolution type into simple algebraic equations. By an inverse transform the problem is converted into an integral representation in the complex plane, and then computed by a suitable quadrature formula. The numerical scheme is applied for a class of linear and nonlinear Volterra integral equations of the convolution type with highly oscillatory kernels, and some of the obtained results are compared with the methods available in the literature. The main advantage of the present scheme is the transformation of a highly oscillatory problem to a non-oscillatory and simple problem. So a large class of a similar type of integral equations having kernels of a highly oscillatory type can be very effectively approximated.