Differential Equations最新文献

Abstract

For a nonlinear differential matrix equation, we study a multipoint boundary valueproblem by a constructive method of regularization over the linear part of the equation using thecorresponding fundamental matrices. Based on the initial data of the problem, sufficientconditions for its unique solvability are obtained. Iterative algorithms containing relatively simplecomputational procedures are proposed for constructing a solution. Effective estimates are giventhat characterize the rate of convergence of the iteration sequence to the solution, as well asestimates of the solution localization domain.

Abstract

The paper discusses the features of constructing numerical schemes for solving coefficient inverse problems for nonlinear partial differential equations of the reaction–diffusion–advection type with data of various types. As input data for the inverse problem, we consider (1) data at the final moment of time, (2) data at the spatial boundary of a domain, (3) data at the position of the reaction front. To solve the inverse problem in all formulations, the gradient method of minimizing the target functional is used. In this case, when constructing numerical minimization schemes, both an approach based on discretization of the analytical expression for the gradient of the functional and an approach based on differentiating the discrete approximation of the functional to be minimized are considered. Features of the practical implementation of theseapproaches are demonstrated by the example of solving the inverse problem of reconstructing the linear gain coefficient in a nonlinear Burgers-type equation.

Abstract

We consider initial–boundary value problems for homogeneous parabolic systems withcoefficients satisfying the double Dini condition with zero initial conditions in a semiboundedplane domain with nonsmooth lateral boundary. The method of boundary integral equations isused to prove a theorem on the unique classical solvability of such problems in the space offunctions that are continuous together with their first spatial derivative in the closure of thedomain. An integral representation of the obtained solutions is given. It is shown that thecondition for the solvability of the posed problems considered in the paper is equivalent to thewell-known complementarity condition.

Abstract

We study the feedback control problem for a mathematical model that describes themotion of a viscoelastic fluid with memory along the trajectories of the velocity field. We provethe existence of an optimal control that delivers a minimum to a given bounded and lowersemicontinuous cost functional.

Abstract

In physics, the singular heat equation with the Bessel operator is used to explain the basicprocess of heat transport in a substance with spherical or cylinder symmetry. This paper examinesthe solution of the Cauchy problem for the heat equation with the Bessel operator acting in thespace variable. We obtain some properties of the solution and consider the normalized modifiedBessel function of the first kind.

Abstract

A new methodology for solving inverse dynamics problem is developed. The methodologyis based on using a mathematical model of a dynamical system and robust stabilization methodsfor a system under uncertainty.

Most exhaustively the theory is described for linear finite-dimensionaltime-invariant scalar systems and multiple-input multiple-output systems.

The study shows that with this approach, the zero dynamics of the original systemis of crucial significance. This dynamics, if exists, is assumed to be exponentially stable.

It is established that zero-dynamics, relative order, and the correspondingequations of motion cannot be defined correctly in multiple-input multiple-output systems. Forcorrect inverse transformation of the solution of the problem, additional assumptions have to beintroduced, which generally limits the inverse system category.

Special attention is given to the synthesis of elementary (minimal) inverters, i.e.,least-order dynamical systems that solve the transformation problem.

It is also established that the inversion methods sustain the efficiency with finiteparameter variations in the initial system as well as with uncontrolled exogenous impulses havingno impact on the system’s internal dynamics.

Abstract

We consider a linear-convex control system defined by a set of differential equations withcontinuous matrix coefficients. The system may have control parameters, as well as uncertainties(interference) the possible values of which are subject to strict pointwise constraints. For thissystem, over a finite period of time, taking into account the constraints, we study the problem ofguaranteed hitting the target set from a given initial position despite the effect of uncertainty. Themain stage of solving the problem is the construction of an alternating integral and a solvabilityset. To construct the latter, the greatest computational complexity is the calculation of thegeometric difference between the target set and the set determined by the uncertainty. Atwo-dimensional example of this problem is considered for which a method is proposed for findingthe solvability set without the need to calculate the convex hull of the difference of the supportfunctions of the sets.

Abstract

We study the stability of a modified (with variation in the nonlinearity parameter)“super-twisting” algorithm. The analysis is based on majorizing the trajectories of the system withan arbitrary nonlinearity parameter by the trajectories of systems of the classical “super-twisting”algorithm. Stability conditions for the modified systems are obtained, as well as estimates for thesize of the stability domain depending on system parameters.

Abstract

We study the existence and stability of periodic solutions of the model Navier–Stokesequation in a thin three-dimensional layer depending on the existence and stability of periodicsolutions of a special limit two-dimensional equation.

Abstract

We consider a special system of integral equations of convolution type with a monotoneconvex nonlinearity naturally arising when searching for stationary or limit states in variousdynamic models of applied nature, for example, in models of the spread of epidemics, and provetheorems stating the existence or absence of a nontrivial bounded solution with limits at(pm infty ) depending on the values of these limits and on thestructure of the matrix kernel of the system. We also study the uniqueness of such a solutionassuming that it exists. Specific examples of systems whose parameters satisfy the conditionsstated in our theorems are given.