JournalTOCs

Computational Mathematics and Mathematical Physics
Journal Prestige (SJR): 0.421

Citation Impact (citeScore): 1
Number of Followers: 5

Hybrid journal (It can contain Open Access articles)
ISSN (Print) 1555-6662 - ISSN (Online) 0965-5425
Published by Springer-Verlag

[2469 journals]

The Avalos–Triggiani Problem for the Linear Oskolkov System and a
System of Wave Equations
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  Abstract: The Avalos–Triggiani problem for a system of wave equations and the linear Oskolkov system is investigated. The method proposed by G. Avalos and R. Triggiani is used to prove a theorem on the existence of a unique solution to the Avalos–Triggiani problem. The underlying mathematical model involves the linear Oskolkov system describing the flow of an incompressible viscoelastic Kelvin–Voigt fluid of zero order and a vector wave equation describing a structure immersed in the fluid.
  PubDate: 2022-03-01
A Version of Closing the System of Moment Equations of an Arbitrary Order
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  Abstract: A method for closing the system of moment equations of order greater than two that does not use the approximation function of speed distribution of molecules is proposed. The moments closing this system are built as combinations of moments of lower orders. Systems of moment equations up to the sixth order, inclusive, are constructed. Using the shock wave profile problem as an example, it is shown that an increase in the order of the system of moment equations does not result in improving the solution. This is explained by the fact that the terms of the closing moment equations that cannot be expressed in terms of lower moments introduce an error comparable in magnitude with the magnitude of the closing moment. The solutions are verified using the model kinetic equation of polyatomic gases.
  PubDate: 2022-03-01
Quadrature Formula for a Double Layer Potential in the Case of the
Helmholtz Equation
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  Abstract: A quadrature formula for a double layer potential is derived in the case of the Helmholtz equations with a continuous density given on a smooth closed or open surface. The quadrature formula ensures higher numerical accuracy than a standard quadrature formula, which has been confirmed by numerical tests. An advantage of the new formula is especially noticeable near the surface, where the standard formula diverges rapidly, while the new one ensures acceptable numerical accuracy for points separated from the surface by distances comparable to or larger than the integration step size.
  PubDate: 2022-03-01
Optimal Control Problems for Complex Heat Transfer Equations with Fresnel
Matching Conditions
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  Abstract: A class of optimal control problems for a system of nonlinear elliptic equations simulating radiative heat transfer with Fresnel matching conditions on the surfaces of discontinuity of the refractive index is considered. Based on estimates for the solution of the boundary value problem, the solvability of the optimal control problems is proved. The existence and uniqueness of the solution of a linearized problem with the matching conditions is analyzed, and the nondegeneracy of the optimality conditions is proved. As an example, a control problem with boundary observation is considered and the relay-like character of the optimal control is shown.
  PubDate: 2022-03-01
Monotone Decomposition of the Cauchy Problem for a Hyperbolic Equation
Based on Transport Equations
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  Abstract: For the Cauchy problem for a hyperbolic equation, a multiplicative approach is developed: a monotone decomposition of the problem is constructed since the hyperbolic operator can be represented by a product of transport operators. The problem for the hyperbolic equation is reduced to a system of problems for transport equations—transport in the direction of the axis \(x\) and transport in the opposite direction of the axis \(x\) . Conditions for the monotonicity of each problem for the transport equations and for the entire multiplicative problem are found. Such a decomposition of the Cauchy problem based on transport problems solved one after the other significantly simplifies the solution of the hyperbolic equation, and the problems for the transport equations are monotone thus ensuring the monotonicity of the decomposition of the Cauchy problem for the hyperbolic equation.
  PubDate: 2022-03-01
Lagrange Interpolation and the Newton–Cotes Formulas on a Bakhvalov Mesh
in the Presence of a Boundary Layer
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  Abstract: Application of a Lagrange polynomial on a Bakhvalov mesh for the interpolation of a function with large gradients in an exponential boundary layer is studied. The problem is that the use of a Lagrange polynomial on a uniform mesh for interpolation of such a function can lead to errors of order \(O(1),\) despite the smallness of the mesh size. The Bakhvalov mesh is widely used for the numerical solution of singularly perturbed problems, and the analysis of interpolation formulas on such a mesh is of interest. Estimates of the error of interpolation by a Lagrange polynomial with an arbitrary number of interpolation nodes on a Bakhvalov mesh are obtained. The result is used to estimate the error of the Newton–Cotes formulas on a Bakhvalov mesh. The results of numerical experiments are presented.
  PubDate: 2022-03-01
Features of Numerical Reconstruction of a Boundary Condition in an Inverse
Problem for a Reaction–Diffusion–Advection Equation with Data on the
Position of a Reaction Front
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  Abstract: A new approach to the reconstruction of a boundary condition in an inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation with data on the reaction front position is proposed. The problem is solved via gradient minimization of a cost functional with an initial approximation chosen by applying asymptotic analysis methods. The efficiency of the proposed approach is demonstrated by numerical experiments.
  PubDate: 2022-03-01
Monotone Schemes of Conditional Approximation and Arbitrary Order of
Accuracy for the Transport Equation
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  Abstract: An initial-boundary value problem for the one-dimensional transport equation with a constant coefficient \(a > 0\) is approximated by a usual explicit explicit monotone difference scheme of traveling calculation “upwind scheme”. Under a Courant-type condition, it is proved that the scheme has an arbitrary \(k\) th order of accuracy for smooth solutions. Assuming the existence of weakly discontinuous solutions, the results are generalized to multidimensional equations. Monotone finite difference schemes for equations with variable coefficients and for first-order semilinear hyperbolic equations are constructed with the use of a special Steklov averaging with respect to nonlinearity. The efficiency of the considered methods is illustrated by numerical results.
  PubDate: 2022-03-01
Integral Representation of the Solution to the Nonstationary Lamb Problem
in the Case of a Limiting Poisson Ratio
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  Abstract: The nonstationary Lamb problem for an elastic half-space with Poisson’s ratio taking a limiting value of \(1{\text{/}}2\) is considered. In the axially symmetric case, the solution is represented in the form of a repeated improper integral. The inner integral over the vertical line in the complex plane is reduced to a sum of residues and a sum of several integrals of a real variable. An estimate of the solution is obtained for large values of the polar radius.
  PubDate: 2022-03-01
Smooth Solution of the Second Initial-Boundary Value Problem for a Model
Parabolic System in a Semibounded Nonsmooth Domain on the Plane
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  Abstract: The second initial-boundary value problem for a second-order Petrovskii parabolic system with constant coefficients in a semibounded plane domain with a nonsmooth lateral boundary is considered. The uniqueness of a solution to this problem in the class \({{C}^{{2,1}}}(\Omega ) \cap {{\mathop C\limits_0 }^{{1,0}}}(\bar {\Omega })\) is proved. The minimum condition on the boundary function under which the solution of the problem belongs to \({{\mathop C\limits_0 }^{{2,1}}}(\bar {\Omega })\) is investigated. A constructive solution is obtained by applying the boundary integral equation method.
  PubDate: 2022-03-01
Semi-Implicit and Semidiscrete Difference Schemes for Solving a
Nonstationary Kinetic Equation of Thermal Radiative Transfer and Energy
Equation
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  Abstract: Semi-implicit and semidiscrete difference schemes of higher order accuracy are proposed for solving kinetic equations of thermal radiative transfer and the energy equation by applying a modified splitting method. A feature of the schemes is that thermal radiative transfer is computed using explicit or implicit difference schemes approximating a usual transport equation. The radiation–matter interaction is computed using implicit difference schemes in the semi-implicit case and using analytical solutions of ordinary differential equations in the semidiscrete case. The difference schemes of higher order accuracy are constructed relying on the second-order Runge–Kutta method. Solutions are found without using outer iterations with respect to the collision integral or matrix inversion. The solution algorithms for difference equations are well suited for parallelization. The method can naturally be generalized to multidimensional problems.
  PubDate: 2022-03-01
On Normal Modes of a Waveguide
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  Abstract: Electromagnetic waves propagating in a waveguide with a constant simply connected cross section \(S\) are considered under the condition that the material filling the waveguide is characterized by permittivity and permeability varying smoothly over the cross section \(S\) but constant along the waveguide axis. On the walls of the waveguide, the perfect conductivity conditions are imposed. It is shown that any electromagnetic field in such a waveguide can be represented via four scalar functions: two electric and two magnetic potentials. If the permittivity and permeability are constant, then the electric potentials coincide with each other up to a multiplicative constant, as do the magnetic potentials. Maxwell’s equations are written in the potentials and then in the longitudinal field components as a pair of integro-differential equations splitting into two uncoupled wave equations in the optically homogeneous case. The general theory is applied to the problem of finding the normal modes of the waveguide, which can be formulated as an eigenvalue problem for a self-adjoint quadratic pencil. At small perturbations of the optically homogeneous filling of the waveguide, the linear term of the pencil becomes small. In this case, mode hybridization occurs already in the first order and the phase deceleration indices of normal modes leave the real and imaginary axes only in the second order.
  PubDate: 2022-03-01
Cauchy Problem for a New Aggregation–Fragmentation Model in the Case of
Equal Reaction Rate Constants
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  Abstract: The existence and uniqueness of a solution to the Cauchy problem for a new aggregation–fragmentation model in the case of equal reaction rate constants are proved. The eigenstates of the right-hand side operator that correspond to real eigenvalues are studied, and an evolution operator is constructed.
  PubDate: 2022-03-01
Modeling Flexible Manipulators without Inverting Their Mass Matrices
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  Abstract: In modern scientific literature, much attention is paid to optimal modeling of flexible dynamic systems. The importance of such studies is dictated by the ever-increasing demand of high-precision robotic manipulators and automatic mechanisms in the theory of control. This demand consists in the need for continuous adjustment of the movement of their effectors in real time, taking into account the compliance of the constituent links of these systems. The generalized Newton–Euler method formulated in this connection provides a reliable platform for the subsequent construction of modifications that accelerate the dynamic analysis of different classes of elastodynamic systems faster. A version of this method designed for optimal modeling of flexible manipulators is proposed that does not use the well-known procedure of inverting their mass matrices.
  PubDate: 2022-03-01
On the Congruence Centralizers of a Block Diagonal Matrix and the
Horn–Sergeichuk Matrix
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  Abstract: The congruence centralizer of a matrix \(A\) is the set of matrices \(X\) that satisfy the relation \(X\text{*}{\kern 1pt} AX = A\) . The concept of a direct sum and the Horn–Sergeichuk matrix are two constituents in the description of the canonical form of an arbitrary complex matrix with respect to Hermitian congruences. In this paper, we eliminate unnecessary assumptions and inaccuracies in the available descriptions of the congruence centralizers of block diagonal matrices and the Horn–Sergeichuk matrix.
  PubDate: 2022-02-01
Mathematical Modeling of Hazardous Natural Phenomena in a Shallow Basin
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  Abstract: This paper is devoted to the construction and analysis of coupled mathematical models of hydrophysics and biological kinetics used for predicting hazardous natural phenomena occurring in shallow basins. The propagation and transformation of aquatic organisms is affected by such physical factors as three-dimensional spatial motion of water taking into account the advective transfer and microturbulent diffusion, spatially inhomogeneous distribution of temperature, salinity, and oxygen. Biogenic pollutants cause algae growth, including toxic and harmful ones; this growth can cause hazardous phenomena in the basin, including eutrophication and suffocation phenomena. A three-dimensional mathematical model of hydrodynamics is constructed and used for calculating the water flow velocity field. To investigate hazardous phenomena in a shallow basin related to suffocation phenomena in it, a three-dimensional spatially inhomogeneous ichthyological model of commercial fish dynamics is developed. Models of observations parameterized on the basis of stoichiometric relations, Monod, Michaelis–Menten, and Mitscherlich–Baule laws that describe the consumption and accumulation of nutrients by phytoplankton and commercial detritophagous fish and the growth of aquatic organisms depending on the spatial distribution of salinity, temperature, and oxygen regimen are considered. To calibrate and verify the models, constantly updated ecological databases obtained, in particular, in field research of the Sea of Azov and Taganrog Bay are used. To improve the accuracy of predictive simulation, the field data is filtered using the Kalman algorithm. As a result of processing the hydrological data, salinity and temperature isolines in the surface layer are obtained; for this purpose, a recognition algorithm is used. Using interpolation and superposition of domain boundaries, more detailed depth, salinity, and temperature maps for the Sea of Azov are obtained. Numerical methods for solving the formulated problems that are based on finite difference schemes taking into account the degree of filling of the computation domain control cells are developed. These methods are implemented on high-performance computers, and they decrease the numerical solution error and reduce the computation time by several fold. Based on the numerical implementation of the developed models, hazardous natural phenomena in shallow basins (related to the propagation of harmful pollutants), eutrophication, and algae bloom, which causes suffocation phenomena, are reconstructed.
  PubDate: 2022-02-01
Primal–Dual Newton Method with Steepest Descent for the Linear
Semidefinite Programming Problem: Newton’s System of Equations
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  Abstract: The linear semidefinite programming problem is considered. It is proposed to solve it using a feasible primal–dual method based on solving the system of equations describing the optimality conditions in the problem by Newton’s method. The selection of Newton’s displacement directions in the case when the current points of the iterative process lie on the boundaries of feasible sets is discussed. The partition of the space of symmetric matrices into subspaces is essentially used.
  PubDate: 2022-02-01
Invariant Curves of Some Discrete Dynamical Systems
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  Abstract: The classical problem on construction of continuous iterations of an analytical map is considered as a problem on construction of invariant curves of discrete dynamical systems. Such systems are often studied as reductions of continuous dynamical systems (Poincarè map). The existence of analytical invariant curves in a discrete dynamical system implies (locally) the existence of an additional analytical first integral in the continuous dynamical system. However, the proofs of existence of such integrals are extremely rare, since these proofs are usually based on convergence of formal power series representing these curves. We give some examples of discrete dynamical systems invariant curves of which are given by a fortiori divergent series but are analytical nonetheless. In particular, we give an example of an integrable discrete dynamical system which has chaotic trajectories.
  PubDate: 2022-02-01
Stochastic Algorithms for Solving the Dirichlet Boundary Value Problem for
Certain Second-Order Elliptic Equations with Discontinuous Coefficients
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  Abstract: Stochastic algorithms for solving the Dirichlet boundary value problem for a second-order elliptic equation with coefficients having a discontinuity on a smooth surface are considered. It is assumed that the solution is continuous and its normal derivatives on the opposite sides of the discontinuity surface are consistent. A mean value formula in a ball (or an ellipsoid) is proposed and proved. This formula defines a random walk in the domain and provides statistical estimators (on its trajectories) for finding a Monte Carlo solution of the boundary value problem at the initial point of the walk.
  PubDate: 2022-02-01
Fast Solution Algorithm for a Three-Dimensional Inverse Multifrequency
Problem of Scalar Acoustics with Data in a Cylindrical Domain
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  Abstract: A new algorithm for stable solution of a three-dimensional scalar inverse problem of acoustic sensing of an inhomogeneous medium in a cylindrical domain is proposed. Data for its solution is the complex amplitude of the wave field measured outside the acoustic inhomogeneities in the cylindrical layer. With the help of the Fourier transform and Fourier series, the inverse problem is reduced to a set of one-dimensional Fredholm integral equations of the first kind. Next, the complex amplitude of the wave field is computed in the inhomogeneity region and the desired sonic velocity field is found in this region. When run on a moderate-performance personal computer, the algorithm takes tens of seconds to solve the inverse problem on rather fine three-dimensional grids. The accuracy of the algorithm is analyzed numerically as applied to test inverse problems at different frequencies, and the stability of the algorithm with respect to data perturbations is investigated.
  PubDate: 2022-02-01