The Lyapunov functional method is applied to a linear parabolic-type equation with homogeneous boundary conditions. Within this framework, a Lyapunov functional is constructed whose derivative along the solutions of the system is a prescribed negative definite quadratic form. A central role in this construction is played by the Lyapunov matrix, whose properties are investigated in detail. In the paper, two definitions of the Lyapunov matrix are proposed. The first one is based on its representation in the form of a series. The second alternative definition relates the matrix to the Green’s function for a corresponding stationary equation. The consistency of the proposed definitions is established, and it is proved that any function satisfying the second definition simultaneously satisfies the first one, thereby confirming the equivalence of the two approaches. An important advantage of the second definition lies in its constructive nature: this makes it possible to derive an explicit analytical representation of the Lyapunov matrix for arbitrary parameters of the boundary value problem. Moreover, it is shown that this approach allows construction of Lyapunov functionals with a prescribed derivative without imposing the requirement of exponential stability. This significantly broadens the scope of potential applications.
Polynomials that are least-deviating from zero play an important role in the theory and practice of using numerical methods. They can be used to solve problems of optimizing the properties of various computational algorithms. So, our work is devoted to the study of polynomials that are least-deviating from zero on the ray $[0,+\infty)$ in the exponential norm. Such a norm for any $\alpha0$ and any infinite subset $K\subset\mathbb{R}$ is defined by the equality $\|P\|_{\alpha,K}=\sup \limits_{x\in K} e^{-|x|^\alpha}|P(x)|$. In this article, we discuss the existence, uniqueness, and characterization of polynomials that deviate least from zero in the norm $\|\cdot\|_{\alpha,[0;+\infty)}$, derive a system of equations that such polynomials must obey, and reformulate the results of Mhaskar and Saff (1984), obtained for the norm $\|\cdot\|_{\alpha,\mathbb{R}}$, according to our kind of norm. Next, we approximately calculate the polynomials of the first and second degrees that deviate least from zero according to the norm $\|\cdot\|_{1,[0;+\ infty)}$. Our method is an alternative to Remez algorithm. In the calculations, we use the principle of contracting mappings, Newton's method and Halley's method. Our results are illustrated by pictures.
The paper is devoted to the study of Volterra equations with an integral operator containing the trace of the unknown solution in the form of its values at certain points of the set. A number of classical boundary value problems of elliptic, hyperbolic, parabolic, and mixed types are reduced to equations with loads. In particular, the Goursat boundary value problem for a hyperbolic type equation is equivalent to a loaded Volterra integral equation of the second kind. The issues of numerical solution of loaded functional equations in integral form have not been sufficiently studied in the literature. Within the framework of this work, the conditions for the existence and uniqueness of solutions to integral equations with local loads are established. A collocation-type numerical method is constructed based on the approximation of the solution by polynomial splines of variable order. The orders of the polynomials that make up the spline are determined adaptively and are consistent with the maximum step in each section of the node grid, which is constructed taking into account the distribution of unknown loads. In the process of discretization, in order to determine the coefficients of the system of equations, the integrals are approximated by Gauss quadrature sums and a general system of linear algebraic equations is formed with respect to all unknown parameters of the spline. The convergence of such an approximation is proved, a number of numerical results are given, confirming the effectiveness of the proposed approach.
The paper considers the problem of automatic optimality control of a distributed discrete information Kalman filter in multisensory networks. The aim of the study is to develop a new method for detecting the loss of optimality of a distributed discrete filter caused by an ubrupt change in the parameters of a dynamic system with a distributed network of sensors. A network with a fully connected topology is considered. A new approach is proposed based on the analysis of the gradient of the negative log likelihood function calculated in terms of a distributed information filter. The method uses sensitivity equations that allow decentralized calculation of the optimality criterion and its gradient in each node of the network, which ensures scalability and fault tolerance of the multisensory system. The main theoretical result is expressions for calculating the optimality criterion in the form of a negative logarithmic likelihood function and its gradient in terms of a distributed discrete filter. The results of computer modeling conducted in MATLAB validate the proposed approach. The developed method can be used in monitoring, motion control and tracking systems using distributed sensor networks. The results obtained show that the proposed approach makes it possible to control the optimality of a discrete filter and, therefore, can be integrated into adaptive algorithms for automatically reconfiguring the parameters of a dynamic system model with a distributed network of sensors.
The paper presents a methodology for numerical solution of two-dimensional gas dynamics problems using geometrically adaptive moving unstructured meshes. Geometric adaptation agrees well with an approach based on highlighting of shock waves and contact discontinuities as solution features. Displacement of internal mesh nodes is found via displacement of boundary nodes. Velocities of discontinuities and other their parameters are determined using Riemann’s problem on a discontinuity breakup. Discretization of initial equations in an integral form is provided. Accuracy increase for the calculation is achieved by determination of pre-breakup flow parameters and by linear or quadratic reconstruction of the flow. In spherically-symmetric problems the algorithm of additional turn of pre-breakup flow velocity is applied. The method is tried out on test problems and applied to modelling of a shock wave that is induced by a spherical charge explosion and propagates over a large distance. Basing on calculation results dependencies of excessive pressure on a distance covered by the wave are obtained. Numerical investigation of the flow structure behind a wave is provided for large distances covered by this wave. Also numerical modelling demonstrated that the wave has N-form that is consequent with earlier results.
The paper studies the inverse problem of the depending on a time variable kernel of the integral term of a multidimensional hyperbolic-type integro-differential equation. First, a direct problem is investigated, assuming that the kernel of the integral term is known. The Fourier method reduces this problem to solving a Volterra-type integral equation of the second kind with respect to the unknown function. A priori estimates for the desired function and for its second-order derivatives are obtained. Next, two inverse problems are studied. The first is determining the memory kernel of a wave process with an integral overdetermination condition. In the second inverse problem, the kernel of the integral term is found from the known solution of the direct problem at some fixed point. In both cases, the inverse problem is reduced to a nonlinear convolution-type Volterra integral equation of the second kind. The method of contracting mappings is used to prove the unique solvability of the posed inverse problems in the space of continuous functions with weighted norms, and an estimate for the conditional stability of the solution is obtained.
When calculating the strength of structural elements, one of the steps is to study the dynamics of these elements under various force loads. In this paper, based on the classical model of free vibrations of an elastic plate, in contrast to previous numerical and analytical studies, an analytical method for studying the dynamics of a concrete slab pinned at its edges is developed. According to the Galerkin method, an approximate solution of the partial differential equation used in the model is found as a linear combination of basis functions. This results in a system of ordinary differential equations for determining the coefficients of this combination. Based on the construction of a Lyapunov-type functional for the partial differential equation and on a Lyapunov function for the system of ordinary differential equations, several methods for determining the error of obtained approximate solution are proposed. Numerical calculations demonstrate the accuracy of error estimates. For this purpose, plots of the difference between the approximation under study and the higher-order approximation are constructed. The best estimate was shown by the method of error determination using the following basis function, whose coefficient was found from the equation obtained in study of the Lyapunov-type functional for the original partial differential equation.
In Russian universities, the problem of distributing the departmental educational load is annually resolved. This problem belongs to the class of combinatorial discrete optimization problems. To solve some problems of this class, it is effective to use the selection sequence optimization algorithm developed by the authors earlier. This article provides a flowchart of this algorithm for the purpose of its visual presentation, which is aimed at further understanding of the material presented. Modeling the dynamics of the algorithm is carried out using a mathematical model built on the basis of one of the varieties of colored Petri nets - that is, on the J-net. The operating logic of this model is described in detail. A reachability tape has been built for this J-net. It contains 269 markings, some of which, representing the main nuances, are presented in the paper in table form. The reachability tape has a significant size even with a minimal non-trivial amount of simulated data, so unattainable markings are rejected by additional analysis of some inequalities sets. Due to the difficulty of contemplative analysis of inequalities’ systems, а software tool has been developed for solving these systems, whose algorithm has polynomial time complexity. Analysis of the reachability tape shows the correct operation of the optimization algorithm. Scientific novelty of the work is that for the first time, a reachability tape for the J-net has been constructed.
The article considers a two-dimensional problem of hydrodynamic escape of the planet’s primordial hydrogen atmosphere as a result of absorption of extreme ultraviolet (EUV) radiation from its host star. As a test case, the model is applied using the parameters of a recently discovered exoplanet TOI-421b, which, according to accepted classification, belongs to so-called class of ''warm mini-Neptunes''. The hydrodynamic parameters are determined by solving the non-stationary Euler and entropy production equations in a spherical coordinate system. The EUV intensity is calculated using the radiation transport equation along parallel rays, with an absorption coefficient proportional to the density of the hydrogen atoms. The numerical method is based on a finite-difference scheme of the modified MacCormack - Runge-Kutta type on a spherical grid with a nonuniform step along the radial direction and a constant step along the spherical angle. The calculation of the radiation intensity at the grid points is perfomed along the characteristics with density interpolation. Steady-state two-dimensional profiles of physical parameters in the upper atmosphere obtained as the result of calculations are presented. An estimate of the dayside atmospheric mass-loss rate and the rate of mass transfer to the nightside under constant external conditions is provided.
This paper examines the mathematical modeling of ventilation systems consisting of deformable air ducts through which an air flow is supplied. Using constructed three- dimensional mathematical model described by a system of partial differential equations, the paper investigates dynamic stability of the elastic wall of an air duct where some gas flows. The Lyapunov dynamic stability criterion is used to study the mechanical system’s stability. To study stability in problems of aerohydroelasticity in compressible and incompressible medium models, Lyapunov-type functionals are constructed for deduced systems of differential equations. By studying these functionals stability conditions are obtained. They ensure that the functional is positive and its time derivative is negative. For a compressible medium model, the dependence between the longitudinal force compressing the plate and the air flow velocity is constructed for specific parameters of the mechanical system. Using the plot constructed, a comparison of the stability conditions for compressible and incompressible medium models is made. It is shown that the medium compressibility has negative effect on the stability of the deformable wall of the air duct and leads to decrease of the stability region.
An analytical framework for synthesizing worst-case external disturbances for linear dynamical systems described by ordinary differential equations is presented in the paper. The study is conducted for three classical functional spaces $L_2, L_{\infty}, L_1$ over a fixed time interval, which corresponds to identifying disturbances with bounded energy, bounded amplitude, and bounded impulse, respectively. Linear elastic mechanical systems are chosen as a illustrative object of analysis, thus providing an intuitive interpretation of the results. A unified performance metric is introduced for quantitative assessment of solutions. This metrics is the ratio of a system's target output (e.g., maximum deviation) to the $L_p$-norm of the disturbance (i.e. the normalized system response). Explicit analytical expressions for the worst-case disturbances and their corresponding performance indices are derived. The interrelations between the indices obtained for different disturbance classes are examined. Numerical simulation results are provided for single- and multiple-degree-of-freedom systems, represented as chains of point masses interconnected by elastic and damping elements, and connected to a movable base.
This paper considers the Full Approximation Scheme (FAS) multigrid method for the Discontinuous Galerkin method with implicit time discretization. The objective of the research is to apply this method to efficient solution of problems governed by nonlinear partial differential equations. A computational algorithm has been developed that implements the Full Approximation Scheme multigrid method using Newton's method and an improved Newton-Krylov method to solve the arising nonlinear equations at each grid level of the multigrid method. This approach significantly improves the efficiency of the algorithm and reduces required computational resources. Numerical experiments were conducted applying both approaches for solving the Hopf equation. The influence of the regularization parameter and of the Courant number on the convergence rate of Newton's method outer iterations was investigated. It has been experimentally demonstrated that the use of the Newton-Krylov method significantly improves the overall performance of the computational process compared to the traditional Newton's method, although both approaches demonstrate a similar order of convergence, approaching second order when using quadratic basis functions.
Longitudinal oscillations of an inhomogeneous chain of linear oscillators coupled by springs are investigated. Both outer springs of the chain are rigidly fixed to immovable supports. The system is subjected to external periodic forces. The inhomogeneity of the chain (the perturbed system) is due to the different stiffness coefficients of the springs. These coefficients deviate slightly from a certain nominal value and depend on dimensionless deviation parameters. Zero values of these parameters correspond to a homogeneous (unperturbed) system. The resonant case is considered when the frequency of the external periodic force coincides with one of the eigenfrequencies of the unperturbed system. To construct an exact periodic solution of the perturbed system, the Lyapunov–Schmidt method is applied. As the problem is linear, this method allows to reduce it to a finite-dimensional algebraic problem of constructing a generalized Jordan chain for a degenerate linear operator. Necessary and sufficient conditions on the dimensionless deviation parameters are obtained, under which the length of such a chain is equal to 1 or 2. For each case, explicit exact formulas for the chain are derived, providing a complete description of the periodic solution. It is shown that for a generalized Jordan chain of length 1, the periodic solution of the perturbed system continuously transforms into a certain periodic solution of the unperturbed system as the small parameter $\varepsilon$ tends to zero. If the length of the generalized Jordan chain is $2$, the periodic solution of the perturbed system possesses a first-order pole at $\varepsilon=0$ and, reduces to a one-parameter family of periodic solutions of the unperturbed system. Numerical simulation was performed for a chain of eight oscillators. Plots of periodic solutions and phase trajectories of the perturbed system are constructed for various values of the small parameter.
The modification of the mother cubic Schoenberg spline is carried out using four cubic Schoenberg splines having finite supports, the sizes of which are smaller compared to the size of the finite support of the mother spline. As a result, eight grid sets of orthogonal cubic Schoenberg splines with real values are constructed. A theorem on the order of approximation of any function of the Sobolev space by linear combinations of constructed orthogonal cubic Schoenberg splines is proved. It is shown that the order of approximation by Schoenberg splines, also modified by Schoenberg splines, is significantly higher than the order of approximation by Schoenberg splines modified by step functions, and coincides with the order of approximation by classical cubic Schoenberg splines. The defect of the modified Schoenberg spline is equal to one, as that of the classical Schoenberg spline. A modified spline is a continuous function in which there are no breaks in the first and second derivatives at the points where the parts of the mother spline and the parts of the splines used for modification meet.
{In this paper, we solve the group classification problem for a nonlinear one-dimensional time-fractional heat conduction equation with full memory and dual-phase-lag, including thermal relaxation and thermal damping. The characteristic times of relaxation processes are assumed to be small enough and therefore a small parameter for fractional differential relaxation terms is introduced. All thermal properties of a medium are considered as functions of temperature. Group classification is performed with respect to groups of approximate point transformations (groups of approximate symmetries) admitted by the equation up to equivalence transformations. We prove that generally the equation admits five-parameter group of approximate transformations, and the cases of its extension to seven- and nine-parameters groups are found. Also, it is shown that the considered nonlinear equation has an infinite approximate symmetry group if the corresponding unperturbed equation is linear. We find that the equation in question always exactly inherits the symmetries of the unperturbed equation. The obtained results make it possible to construct approximately invariant solutions of equation under consideration. In particular, it follows from the classification found that the equation always has a traveling wave solution. The self-similar solutions can be constructed only if the medium thermal properties have power-law dependences on temperature. Ansatzes of these types of solutions are obtained and symmetry reductions of the equation under consideration to the corresponding ordinary fractional differential equations are performed.
In the paper the action of the orthogonal Lie algebra $\mathfrak{o}(V)$ on the exterior powers of a space $V$ is considered for $n$-dimensional vector space $V$ over a perfect field $K$ of characteristic two with a given nondegenerate orthogonal. The exterior algebra is identified with the algebra of truncated polynomials in $n$ variables. The exterior powers of $V$ taken as modules over $\mathfrak{o}(V)$ are identified with homogeneous subspaces of non-alternating Hamiltonian Lie algebra $P(n)$ with respect to the Poisson bracket corresponding to an orthonormal basis of the space $V$ of variables. It is proved that the exterior powers of the standard representation for Lie algebra $\mathfrak{o}(V)$ are irreducible and pairwise nonequivalent. With respect to subalgebra $so(V)$, $n= 2l+1$ or $n= 2l$, there exist $l$ pairwise nonequivalent fundamental representations in the spaces $\Lambda^{r}V$, $r= 1, \ldots, l$. All of them admit a nondegenerate invariant orthogonal form, being irreducible when $n= 2l+1$. When $n= 2l$ the representations of $so(V)$ in $\Lambda^{r}V$, $r= 1, \ldots, l-1$ are irreducible and the space $\Lambda^{l}V$ possesses the only non-trivial proper invariant subspace $M$, which is a maximal isotropic subspace with respect to an invariant form. Two exceptional simple Lie subalgebras $P_{1}(6)$, $P_{2}(6)$ of $P(n)$, of dimension $2^{5}-1$ and $2^{6}-1$, correspondingly, containing the submodule $M$, and exising only in the case of 6 variables, are found.
The problem optimal speed control is investigated in the case when the process is described by a system of linear ordinary differential equations with nonlinear convex restrictions on phase variables and control. By moving from n-dimensional Euclidean space to Hilbert space, the optimal control problem with restrictions on phase variables and control is reduced to an optimal speed problem without restrictions. It is shown that the reachability region in the new space is a convex set. To solve the resulting problem, a modified method of separating hyperplanes is used. One of the key points of this method, on which the convergence speed of the algorithm depends, is finding the normal to the separating hyperplane. In this work, this normal at each iteration is constructed by minimizing a distance-type functional on the convex hull of points supporting the reachability set obtained at previous iterations. After finding the normal to the separating hyperplane, a hyperplane supporting the reachable region is constructed, which is then continuously transferred in increasing time and the first moment in time is found at which the supporting hyperplane reaches the given end point. This moment is taken as the next approximation to the performance time. A theorem is formulated on the convergence of successive approximations in time to the value of the performance time and on the weak convergence of a sequence of controls to an optimal control. The algorithm is tested by solving the problem of external heating of an unlimited plate to a given temperature in a minimal time, taking into account restrictions on tensile and compressive thermal stresses. The results of a computational experiment are presented.
The paper contains sufficient conditions for the action of generalized and linear generalized partial integral Romanovsky operator in the space of continuous functions defined on an n-dimensional parallelepiped. Continuity of these operators is established in case of their action in the space of continuous functions and, in a more general case of continuous kernels of operators with values in the space of measurable and Lebesgue integrable functions. Estimates are obtained for the norms of the operators mentioned. The dependence of the estimate for the norm of a linear Romanovsky type operator with generalized partial integrals on the space dimension and on the norm of continuous kernels of generalized partial integral Romanovsky operators with values in the space of measurable and Lebesgue integrable functions is shown. The properties established are applied to the study of linear generalized partial integral equations of Romanovsky type, in particular, to the study of generalized partial integral equation of n-connected Markov chains.
The article presents a numerical algorithm for studying subsonic viscous chemically active flows in the presence of laser radiation. The process model is described in the Navier – Stokes approximation adjusted for the subsonic flow regime, with addition of source terms corresponding to chemical transformations. An additional ordinary differential equation describing the propagation of laser radiation along the length of the region under study is introduced as well. The computational algorithm is based on splitting by physical processes. This makes it possible to calculate separately changes in concentrations during chemical transformations, convective fluxes, dissipative terms, dynamic pressure deviation and propagation of laser radiation. To account for the dissipative terms (diffusion, viscosity, and thermal conductivity), the local iteration method based on Chebyshev polynomials’ ordering. Due to the possible use of a larger total calculation time step, the software implementation of the constructed algorithm reveals shorter calculation times using the local iteration method for calculating dissipative terms in comparison with the algorithm calculating them based on a scheme with central differences. The algorithm was verified using the example of methane conversion by comparing it with the calculation of the stoichiometric balance of the brutto-reaction, as well as by studying the convergence of the solution on a sequence of thickening grids. Based on the developed algorithm, a numerical study of non-oxidative conversion of methane under the influence of laser radiation in a circular tube was carried out, and graphs of the distribution of the main characteristics of the mixture were obtained.
In this article we propose a method for optimizing the arrangement of approximation nodes and use Runge function as an example to implement this approach. The method is based on the idea of nonlinearity of space along the axes of Cartesian coordinate system. To control the nonlinearity, we use a polynomial function with a parameter uniformly distributed over the segment $[0,1]$. A comparative analysis of the following standard methods of selecting nodes for the approximation of Runge function was carried out: uniformly along the abscissa axis, uniformly along the ordinate axis, uniformly along the curve length, and by Chebyshev's nodes. To compare the Lagrange interpolation polynomials, we estimate the approximation errors of Runge's function. Graphs of the constructed Lagrange's polynomials for five and seven interpolation nodes selected in different ways are presented. To select the optimal arrangement of approximation nodes of the proposed method, we compile an objective function, whose minimization ensures optimal arrangement of nodes $x_i$ along the abscissa axis. The arrangement of approximation nodes along the ordinate axis is determined by calculating the $y_i$ values basing on the original Runge's function. As a result, we found nodes that provide minimal deviations from the original approximated Runge's function. The paper considers cases of five and seven approximation nodes. To visualize the results obtained, we provide graphs of original Runge's function and of its approximation, indicating the optimal nodes found. This method is stable to increasing the number of nodes, whose arrangement is optimized each time and adapted to the original function.
