卷 63, 编号 11 (2023)

This paper deals with Euler-type integrals and the closely related Lauricella function , which is a hypergeometric function of many complex variables. For  new analytic continuation formulas are found that represent it in the form of Horn hypergeometric series exponentially converging in corresponding subdomains of, including near hyperplanes of the form, . The continuation formulas and identities for  found in this paper make up an effective apparatus for computing this function and Euler-type integrals expressed in terms of it in the entire complex space , including complicated cases when the variables form one or several groups of closely spaced neighbors. The results are used to compute parameters of the Schwarz–Christoffel integral in the case of crowding and to construct conformal mappings of polygons.

The small parameter method allows one to construct solutions of differential equations in the form of power series and has become widespread in mathematical physics. In most cases, these series are asymptotically convergent. The aim of this work is to find conditions for the ordinary convergence of series in powers of a small parameter representing solutions of perturbation theory problems.

We show a connection between lower and upper bounds on the column matrix approximation accuracy and the bounds on the norms of the pseudoinverses of the submatrices of orthogonal matrices. This connection is exploited to derive lower bounds for column approximations accuracy in spectral and Frobenius norms.

The paper considers a class of optimal control problems for a nonlinear parabolic-elliptic system simulating radiative heat transfer with Fresnel matching conditions on surfaces of discontinuity of the refractive index. New estimates for the solution of the initial-boundary value problem are obtained, on the basis of which the solvability of optimal control problems is proved. Non-degenerate first-order optimality conditions are derived. The results are examplified by control problems with final, boundary, and distributed observations.

The paper considers difference schemes with optimal parameters for solving Maxwell’s equations. Using Laguerre transforms, the numerical values of the optimal parameters are determined and differential-difference equations are constructed. Differential-difference equations are solved by the finite-difference method with iterations over small optimal parameters. Optimal second-order difference schemes for one-dimensional and two-dimensional Maxwell’s equations are considered. Optimal parameters of difference schemes are given. It is shown that the use of optimal difference schemes leads to an increase in the accuracy of solution.

A nonclassical fourth-order partial differential equation describing blow-up instability in self-oscillatory systems is studied. Several classes of exact solutions of this equation are constructed. It is shown that these solutions include ones growing to infinity in a finite time, ones bounded globally in time, and ones bounded on any finite time interval, but not globally.

Beam vibrations with allowance for deformation effects in the transverse direction are modeled using a nonlinear Sobolev-type differential equation, for which the Cauchy problem is investigated in the space of continuous functions. Conditions for solution blow-up on a finite time interval are considered.

Nondissipative and weakly dissipative discontinuity structures are considered. A special numerical method for studying periodic waves is used. The location of branches of periodic solutions is investigated. Solitary waves and nondissipative discontinuity structures are sought as limiting solutions. It is found that, in addition to the resonance of long Alfvén waves with short fast and slow magnetosonic waves, there is also resonance with long waves, which leads to the occurrence of hybrid-type solitary waves and hybrid-type discontinuity structures. Partial differential equations are solved to find out if the found structures are actually observed.

This paper is devoted to the use of the multigrid method to accelerate calculations of compressible flows in a stationary formulation on unstructured grids. The multigrid method is used with the construction of a full approximation for each grid level (FAS MG—Full Approximation Scheme Multigrid). In the case of an unstructured grid, such a method can cause difficulties associated both with the construction of grid levels and transition operators between them, and with software implementation in the existing simulation code. The program needs to deal with several different discretizations at once. If the entire data structure, including arrays with grid data, topology, and time integration data, was designed to work on a single grid, then the implementation of the FAS MG can turn into a disaster involving rewriting the entire code. The purpose of this work is to achieve multiple acceleration of calculations at the cost of minimal effort. The problem of implementing the multigrid method on the basis of an existing software package that was not designed to work with several grid levels is solved. The implementation of the multigrid method in an MPI parallel code is carried out in such a way that there is no need to rewrite the program to work with multiple grids at all. Also, difficulties with constructing grid levels for an unstructured grid are avoided; agglomeration of cells is not used, and the number of faces per cell at coarse levels is not increased. In fact, this paper describes how to deploy a FAS MG accelerator in literally a week, even in code that is outdated from the point of view of software architecture.

We develop a fast method for approximating the solution to the generalized eikonal equation in a moving medium. Our approach consists of the following two steps. First, we convert the generalized eikonal equation in a moving medium into a Hamilton–Jacobi–Bellman equation of anisotropic eikonal type for an anisotropic minimum-time control problem. Second, we modify the Neighbor–Gradient Single-pass method (NGSPM developed by Ho et al.), so that it not only suits the converted Hamilton–Jacobi–Bellman equation but also can be faster than original NGSPM. In the case of that Mach number is not comparable than 1, we compare our method and Characteristic Fast Marching Method (CFMM developed by Dahiya) via several numerical examples to show that our method is faster and more accurate than CFMM. We also compare the numerical solutions obtained from our method with the solutions obtained using the ray theory to show that our method captures the viscosity solution accurately even when the Mach number is comparable to 1. We also apply our method to 3D example to show that our method captures the viscosity solution accurately in 3D cases.

The flow around an elongated smooth contour under the free surface of a fluid is considered. The fluid is perfect, incompressible, and heavy. The critical flow branching and flow shedding points are located on the contour. The depth of the contour immersion and its length are given. It is assumed that the velocity magnitude on the free surface is close to its value in the undisturbed flow. A nonlinear approximation of the Bernoulli integral on the free surface associated with logarithm is used. Two auxiliary planes in which the flow domain is a half-plane with an excluded circle and an annular region are used. The complex potential is determined in the first parametric plane using a conformal mapping onto the annular region. A system of equations is derived for finding the defining parameters. This system is solved using the minimization of a functional and an iteration method over two sets of parameters. An algorithm and computer program for solving this system are developed. The hydrodynamic characteristics of a specific hydrofoil are computed. Results for the coefficients of wave drag, lift force, moment, and position of the contour center are analyzed depending on the Froude number and circulation of different signs. Examples of computations of nonlinear waves formed on the free surface at significant Froude numbers are given.

Schrödinger time independent equation arises in Wheeler–DeWitt model of quantum cosmology to develop a wave function for the corrections of cosmologies. In this work, k=0  rödinger–Wheeler–DeWitt model is developed by using canonical transformation to transformed Hamiltonian equation. The exact solutions for flat universe (
) model is presented and then soft computing technique through Levenberg–Marquardt backpropagation neural networks (LMB-NNs) is implemented. The data set of the flat universe model is generated for four different cases of separation constant B  with step size 0.1  by Mathematica and imported into LMB-NNs for training, testing, and validation of the our proposed technique. The performance of LMB-NNs is provided by the validation of mean square error, error histogram, and regression analysis. The Statistical analysis; mean, minimum, maximum, and standard deviation, is also presented.