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A. A. Alikhanov

Publications and source records attributed to A. A. Alikhanov.

7 recordsLinked to original sources
arXiv2019

About one method of parallelization of calculations during the reconstruction of a tomographic image

A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel numeration method for two-dimensional problems, allows us to divide the initial system of algebraic equations into two independent systems, which enables us to carry out the calculation in parallel. The dimension of the matrices of the resulting systems is 4 times less than the dimension of the original matrix, and these matrices are less dispersed.

math.NA↗
arXiv2014

A new difference scheme for the time fractional diffusion equation

In this paper we construct a new difference analog of the Caputo fractional derivative (called the $L2$-$1_σ$ formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and forth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid $L_2$ - norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by the numerical calculations carried out for some test problems.

math.NA↗
arXiv2014

A difference method of solving the Steklov nonlocal boundary value problem of the second kind for the time-fractional diffusion equation

We consider difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters $α$, $β$ and $γ$. By the method of energy inequalities, for the solution of the difference problem, we obtain a priori estimates, which imply the stability and convergence of these difference schemes. The obtained results are supported by the numerical calculations carried out for some test problems.

math.NA↗
arXiv2014

Stability and convergence of difference schemes approximating a two-parameter nonlocal boundary value problem for time-fractional diffusion equation

Difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters $α$ and $β$ are considered. By the method of energy inequalities, for the solution of the difference problem, we obtain a priori estimates, which imply the stability and convergence of these difference schemes.

math.NA↗
arXiv2014

Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation

Solutions of the Dirichlet and Robin boundary value problems for the multi-term variable-distributed order diffusion equation are studied. A priori estimates for the corresponding differential and difference problems are obtained by using the method of the energy inequalities. The stability and convergence of the difference schemes follow from these a priory estimates. The credibility of the obtained results is verified by performing numerical calculations for test problems.

math.NA↗
arXiv2012

Boundary value problems for the diffusion equation of the variable order in differential and difference settings

Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case. The credibility of the obtained results is verified by performing numerical calculations for a test problem.

math.NA↗