Volume 63, Nº 6 (2023)

A family of first- and second-order accurate noniterative numerical methods is constructed for solving systems of nonlinear Volterra integral equations of the second kind. The methods are examined for A-, L-, and P-stability. The conclusions are illustrated by numerical results obtained for test equations with stiff and oscillating components

Functional identities that hold for deviations from the exact solutions of boundary value and initial boundary value problems with monotone operators are obtained. These identities hold for any function from the corresponding functional class that contains the exact solution of the problem. The left-hand side of an identity is the sum of terms that measure deviation of the approximate solution from the exact one. It is shown that these measures are natural characteristics of the accuracy of approximate solutions. In some cases, the right-hand side of the identity contains only known data of the problem and functions that characterize the approximate solution. Such an identity can be directly used for error control. In other cases, the right-hand side includes unknown functions. However, they can be eliminated to obtain fully computable two-sided bounds. In this case, it is necessary to use special functional inequalities relating the deviation measures to the properties of the monotone operator under consideration. As an example, such bounds and the exact values of the corresponding constants are obtained for a class of problems with the 
-Laplacian operator. It is shown that the identities and the resulting bounds make it possible to estimate the error of any approximation regardless of the method used to obtain it. In addition, they open a way for comparing exact solutions of problems with different data, which makes it possible to evaluate the errors of mathematical models, e.g., those that arise when the coefficients of a differential equation are simplified. In the first part of the paper, the theory and applications concern stationary models, and then the main results are extended to evolutionary models with monotone spatial operators.

For a given boundary value problem, the existence of a solution depending continuously on the boundary conditions is analyzed. Previously, such a fact has been known only for the Cauchy problem, which is a classical result in the theory of differential equations. We prove a similar result for boundary value problems in the case when they are p-regular. In the general case, this result does not hold. Several implicit function theorems are proved in the degenerate case, which is a development of p-regularity theory concerning the existence of a solution to nonlinear differential equations. The results are illustrated by an example of a classical boundary value problem, namely, a degenerate Van der Pol equation is considered, for which the existence of a solution depending continuously on the boundary conditions of the perturbed problem is proved.

The problem of dynamic reconstruction of an unknown boundary input disturbance for a nonlinear system of reaction–diffusion differential equations with distributed parameters is considered. A solution algorithm based on constructions of feedback control theory is presented.

For an optimal control problem with constraints on phase coordinates, an iterative method for finding a numerical solution is considered, based on reduction to a finite-dimensional problem and applying to it a sequential linearization algorithm using a modified Lagrange function. To solve linear auxiliary problems at iterations of the method, the reduced gradient method is used. The efficiency of taking into account phase constraints in the calculation of optimal control is illustrated by the numerical solution of problems from the field of aerodynamics and robotics.

The paper consideres linear systems of ordinary differential equations of arbitrary order with a matrix identically degenerate in the domain of definition at the highest derivative of the desired vector function and with loads in the form of Volterra and Fredholm integral operators. The initial value problems are formulated using projections onto admissible sets of initial vectors. Special attention is paid to systems having singular points on the interval of integration. The concept of a singular point is formalized. Their classification in the case of differential equations is given. A number of examples illustrating the theoretical results are presented.

The paper proves the absence of solutions of semilinear parabolic inequalities and higher order systems with a singular potential and nonlocal sources. The proofs are based on the test function method developed by E. Mitidieri and S.I. Pokhozhaev.

The Cauchy problem for a model nonlinear equation with gradient nonlinearity is considered. We prove the existence of two critical exponents, and, such that this problem has no local-in-time weak (in some sense) solution for , while such a solution exists for , but, for , there is no global-in-time weak solution.

This paper concerns with the initial and boundary value problem for a p-biharmonic wave equation with logarithmic nonlinearity and damping terms. We establish the well-posedness of the global solution by combining Faedo–Galerkin approximation and the potential well method, and derive both the polynomial and exponential energy decay by introducing an appropriate Lyapunov functional. Moreover, we use the technique of differential inequality to obtain the blow-up conditions and deduce the life-span of the blow-up solution.

The properties of modern WENO schemes are examined as applied to large eddy simulation (LES). The WENO-ZM5 scheme with modified smoothness indicators (“upwind”) and the WEN-O‑SYMBOO6 scheme on a symmetric stencil (“symmetric WENO scheme”) are chosen. The schemes are compared on one-dimensional test problems (advection, Hopf, and Burgers equations) with both smooth and discontinuous solutions. The decay of isotropic turbulence is modeled within LES and the results are discussed. The solutions produced by the new schemes are compared with those based on the central difference scheme, the classical WENO5 scheme, and a hybrid scheme. The level of dissipation of the schemes is compared by analyzing their energy spectra. A similar comparison is made between the LES computations of the temporal evolution of a mixing layer, where the profiles of mean velocity and Reynolds stresses are considered in addition to the energy spectrum.