Singular Differential Equations and their Applications for Modeling Strongly Oscillating Processes

The normal system of ordinary differential equations, whose right-hand sides are the ratios of linear and nonlinear positive functions, is considered. A feature of these ratios is that some of their denominators can take on arbitrarily small nonzero values. (Thus, the modules of the corresponding derivatives can take arbitrarily large value.) In the sequel, the constructed system of differential equations is used to model strongly oscillating processes (for example, processes determined by the rhythms of electroencephalograms measured at certain points in the cerebral cortex). The obtained results can be used to diagnose human brain diseases

How Can We Manage Repairing a Broken Finite Vibrating String? Formulations of the Problem

Three approaches to solve a well-known IBVP posed for vibrating composite string with piece-wise constant properties have been applied. The main issue of the stufy is the number of the matching conditions to be imposed for the solution to the IBVP to be obtained

Qualitative Analysis of an Optimal Sparse Control Problem for Quasi-Linear Parabolic Equation with Variable Order of Nonlinearity

In this work, we study a sparse optimal control problem involving a quasilinear parabolic equation with variable order of nonlinearity as a state equation and with a pointwise control constraints. We show that in the case if the cost functional contains the terminal term of the tracking type, the proposed optimal control problem is ill-posed, in general. In view of this, we provide a sufficiently mild relaxation of the proposed problem and establish the existence of optimal solutions for the relaxed version. Using the compensated compactness technique and the consept of variational convergence of minimization problems, we study the attainability of optimal pairs to the relaxed problem by optimal solutions of the special approximating problems. We also discuss the optimality conditions for approximating problems and provide their substantiation

Model of Non-Isothermal Consolidation in the Presence of Geobarriers and the Total Approximation Properties of its Finite Element Solutions

The boundary value problem for the system of quasi-linear parabolic equations in the presence of integral conjugation conditions is considered. The boundary value problem is a mathematical model of the process of non-isothermal filtration consolidation of the soil mass which contains a thin geobarrier. Geobarriers exposed to non-isothermal conditions are a component of waste storage facilities. The change of hydromechanical and thermal properties the geobarriers, as well as the phenomenon of thermal osmosis, require modification of both the equations in the mathematical model and the conjugation conditions. The finite element method is used to find approximate solutions of the corresponding system of quasi-linear parabolic equations. The existence and uniqueness of the approximate generalized solution is proved. The accuracy of finite element solutions in the sense of total approximation are also estimated. The differences in the values of pressure and temperature distributions for the classical case and the case considered in the article were analyzed on the test model example

Univariate Time Series Analysis with Hyper Neural ODE

Neural ordinary differential equations (NODE) are ordinary differential equations whose right-hand side is determined by a neural network. Hyper NODE (hNODE) is a special type of neural network architecture, which is aimed at creating such NODE system that regulates its own parameters based on known input data. The article uses a new approach to the study of one-dimensional time series, the basis of which is the hNODE system. This system takes into account the relationship between the input data and its latent representation in the network and uses an explicit parametrization when controlling the latent flow. The proposed model is tested on artificial time series of data. The influence of some activation functions (besides sigmoid and hyperbolic tangent) on the quality of the forecast is also considered

Can a Finite Degenerate ‘String’ Hear Itself? Numerical Solutions to a Simplified IBVP

Some discrete models for a simplified (compared to that published earlier in JODEA, 28 (1) (2020), 1 – 42) initial boundary value problem for a 1D linear degenerate wave equation, posed in a space-time rectangle and solved earlier exactly (JODEA, 30 (1) (2022), 89 – 121), have been considered. It has been demonstrated that the correct evaluation of the degenerate grid flux can be possible

Nonlinear Parabolic Variational Inequalities with Variable Time-Delay in Time Unbounded Domains

We research the well-posedness of the problem without initial condition for nonlinear parabolic variational inequalities with variable time-delay. To justify our results, we impose some assumptions on the solution behavior and growth of the data-in as time variable tends to −∞. Also, we obtain estimates for weak solutions of this problem

Stability w.r.t. Disturbances for the Global Attractor of Multi-Valued Semiflow Generated by Nonlinear Wave Equation

The paper investigates the issue of stability with respect to external disturbances for the global attractor of the wave equation under conditions that do not ensure the uniqueness of the solution to the initial problem. Under general conditions for nonlinear terms, it is proved that the global attractor of the undisturbed problem is locally stable in the sense of ISS and has the AG property with respect to disturbances

Optimal Control Problem for Non-linear Degenerate Parabolic Variation Inequality: Solvability and Attainability Issues

We investigate the optimal control problem with respect to coefficients of the degenerate parabolic variational inequality. Since problems of this type can have the Lavrentieff effect, we consider the optimal control problem in a class of so-called Hadmissible solutions. We substantiate the attainability of H-optimal pairs via optimal solutions of some nondegenerate perturbed optimal control problems

The Analytical View of Solution of the First Boundary Value Problem for the Nonlinear Equation of Heat Conduction with Deviation of the Argument

In this article, for the first time, the first boundary value problem for the equation of thermal conductivity with a variable diffusion coefficient and with a nonlinear term, which depends on the sought function with the deviation of the argument, is solved. For such equations, the initial condition is set on a certain interval. Physical and technical reasons for delays can be transport delays, delays in information transmission, delays in decision-making, etc. The most natural are delays when modeling objects in ecology, medicine, population dynamics, etc. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (solid bodies, liquids, etc.) is modeled by heat conduction equations. The Green’s function of the first boundary value problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated, and the formula for the solution is established