

2021 year, number 4
Jmeih N. Abou, Arwadi T. El, S. Dib
Beirut Arab University, Beirut, Lebanon
Keywords: finite element scheme, a priori error analysis, dynamical boundary conditions, DirichlettoNeumann semigroup
Abstract >>
This study aims to implement a numerical scheme in order to find the eigenvalues of the DirichlettoNeumann semigroup. This can be used to check its positivity for noncircular 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.

S. Guemar^{1}, H. Guebbai^{2}, S. Lemita^{3}
^{1}UniversitÃ© Yahia Fares MÃ©dÃ©a. PÃ´le urbain, MÃ©dÃ©a, AlgÃ©rie ^{2}University of Guelma, Guelma, AlgÃ©rie ^{3}'Ecole Normale SupÃ©rieure de Ouargla, CitÃ© Ennacer, Ouargla, AlgÃ©rie
Keywords: Volterra equation, integrodifferentialfractional equation, fixed point, nonlinear equation, product integration method
Abstract >>
In this paper, we study a nonlinear integrodifferential 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.

S. Korotov^{1}, M. KÅ™á½·Å¾ek^{2}
^{1}Western Norway University of Applied Sciences, Bergen, Norway ^{2}Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic
Keywords: Zhang tetrahedra, dihedral angle, measure of degeneracy, red refinement, maximum angle condition
Abstract >>
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.

V.A. Litvinov^{1}, V.V. Uchaikin^{2}
^{1}Barnaul Law Institute of the Ministry of Internal Affairs of Russia, Barnaul, Russia ^{2}Ulyanovsk State University, Ulyanovsk, Russia
Keywords: inverse problems, diffusion equation, operators, fractional derivatives
Abstract >>
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 timedependence of diffusivity and determination of fractional time and spacederivatives order in anomalous diffusion equation is demonstrated. There is shown a possibility of sufficient accuracy with insignificant computational expanses.

Ch. Liu^{1}, T. Hou^{2}, Zh. Weng^{3}
^{1}Hunan University of Science and Engineering, Yongzhou, China ^{2}Beihua University, Jilin, China ^{3}Huaqiao University, Quanzhou, China
Keywords: nonlinear parabolic equations, PP mixed finite element method, a priori error estimates, square integrable function space
Abstract >>
In this paper, we consider P ^{2}_{0} 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.

I.N. Medvedev^{1,2}
^{1}Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia ^{2}Novosibirsk State University, Novosibirsk, Russia
Keywords: exponential transformation algorithm, trajectory branching of the Markov chain, majorant crosssection method (Woodcock tracking), gammaradiation transfer, variance of the weighted estimator, computation costs, stochastic medium
Abstract >>
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 gammaray transport in an inhomogeneous medium. These algorithms were applied to a maximum (majorant) crosssection method or the Woodcock tracking which is extremely efficient for the simulation in an inhomogeneous medium. On an example of gammaray 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.

M. Uddin, A. Khan
University of engineering and technology, Peshawar, Pakistan
Keywords: oscillatory kernels of convolution type, Volterra integral equations, Laplace transform, inverse Laplace transform, Numerical method
Abstract >>
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 nonoscillatory 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.

