Differential-operator Equations Research Articles

Solution Uniqueness Criteria in a Time-Nonlocal Problem for the Operator Differential Equation $$l(\cdot )\!-\!A$$ with the Tricomi Operator $$A$$

Solution Uniqueness Criteria in a Time-Nonlocal Problem for the Operator Differential Equation $$l(\cdot )\!-\!A$$ with the Tricomi Operator $$A$$

On a nonlocal problem for the first-order differential-operator equations

In this work, we study the spaces of generalised elements identified with formal Fourier series and constructed via a non-negative self-adjoint operator in Hilbert space. The spectrum of this operator is purely discrete. For a differential-operator equation of the first order, we formulate a nonlocal multipoint by time problem if the corresponding condition is satisfied in a positive or negative space that is constructed via such operator; such problem can be treated as a generalisation of an abstract Cauchy problem for the specified differential-operator equation. The correct solvability of the aforementioned problem is proven, a fundamental solution is constructed, and its structure and properties are studied. The solution is represented as an abstract convolution of a fundamental solution with a boundary element. This boundary element is used to formulate a multipoint condition, and it is a linear continuous functional defined in the space of main elements. Furthermore, this solution satisfies multipoint condition in a negative space that is adjoint with a corresponding positive space of elements.

On Small Motions of Hydraulic Systems Containing Viscoelastic Fluid

In this paper, we study two problems describing small motions of partially dissipative hydraulic systems. The first problem concerns small motions of a hydraulic system consisting of a viscoelastic fluid and a barotropic gas located above the fluid; the second concerns small motions of the hydraulic system “viscoelastic fluid–ideal fluid–ideal fluid” in an immovable vessel. Using the operator approach developed in previous works of the authors, we reduce both problems to the Cauchy problem for a differential-operator equation in a Hilbert space and prove a theorem on the solvability of the problem on an arbitrary finite time interval.

Splitting Schemes for One Class of Operator Differential Equations

Splitting Schemes for One Class of Operator Differential Equations

Uniqueness Criteria for Solving a Time Nonlocal Problem for a High-Order Differential Operator Equation l(·)—A with a Wave Operator with Displacement

This article presents a criterion for the uniqueness of the solution of a problem nonlocal in time for a differential-operator equation with a symmetric operator part on space variables. The symmetry of the operator part of the operator-differential equation guarantees the existence of good basic properties of its system of root elements. The spectral properties of the symmetric operator part make it possible not only to prove the necessity of the criterion formulated by us, but also to substantiate their sufficiency. In contrast to previously known works, in this work the semiboundedness of the symmetric part of the differential-operator equation can be violated. In this article, the differential-operator equation is represented as the difference of two commuting operators. The uniqueness of the solution is guaranteed when the spectra of the commuting operators do not intersect. It is important that only one of the operators should be symmetrical.

Problems on Small Motions of Systems of Two Viscoelastic Fluids in Fixed Vessels

In this paper, we study the problem on small motions of two Oldroyd viscoelastic incompressible fluids contained in a fixed vessel. By means of the operator approach, we reduce the original initial-boundary problem to the Cauchy problem for an operator-differential equation in a Hilbert space and prove the well-posed solvability of the problem on an arbitrary time interval. We deduce the equation for normal oscillations of the considered hydraulic system (the Krein generalized operator pencil).

Conditions for the existence of smooth solutions for a class of fourth order operator-differential equations

. In the paper, we consider a fourth-order operator-differential equation on the entire axis, the main part of which has a multiple characteristic, and we introduce the concept of its ”smoothly” regular solvability. We find exact values of the norms of intermediate derivatives operators in a Sobolev-type space and indicate their connection with the conditions for the solvability of the equation under study. Note that the conditions found for ”smoothly” regular solvability are sufficient and are imposed only on the operator coefficients of the operator-differential equation under consideration

Randomised one-step time integration methods for deterministic operator differential equations

Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065–1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.

Asymptotic behaviour of negative eigenvalues of an operator differential equation

In this work, we find the asymptotic formulas for the sum of the negative eigenvalues smaller than ?? (? > 0) of a self-adjoint operator L defined by the following differential expression ?(y)=?(p(x)y?(x))??Q(x)y(x) with the boundary condition y(0) = 0 in the space L2(0,?;H).

Колебания системы твёрдых тел, частично заполненных вязкими жидкостями, под действием упругодемпфирующего устройства

In this paper, we investigate the linearized two-dimensional problem on small motions of a system of rigid bodies partially filled with viscous incompressible fluids and connected in series by springs. The first and last bodies are attached by springs to two supports with a given law of motion. The trajectory of the system is perpendicular to the direction of gravity, and the damping forces acting on the hydrodynamical system are generated by the friction of bodies against a stationary horizontal support. For the described system, the law of total energy balance is formulated. Using the orthogonal projection method and a number of auxiliary boundary problems, the original initial-boundary value problem is reduced to the Cauchy problem for a first-order differential-operator equation in the orthogonal sum of some Hilbert spaces. The properties of operator matrices, which are coefficients of the obtained differential equation, are investigated. A theorem on the unique solvability of the resulting Cauchy problem on the positive semi-axis is proved. On the basis of the proved theorem, sufficient conditions for the existence of a strong with respect to time solution of an initial-boundary value problem describing the evolution of a hydrodynamical system, are found. From a mathematical point of view, the system under consideration is a finite-dimensional perturbation of the well-known S.G. Krein’s problem on small motions of a viscous fluid in an open vessel.

On a Uniqueness Theorem for the Solution of a Boundary Value Problem for a Differential-Operator Equation with the Tricomi Operator

On a Uniqueness Theorem for the Solution of a Boundary Value Problem for a Differential-Operator Equation with the Tricomi Operator

Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators

<p style='text-indent:20px;'>Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations.</p>

Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces

We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of 
 linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator
 interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation
 in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2\times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_\Theta$. We suppose that the weight $\Theta$ satisfies some regularity assumption.
 Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2\left((-T, T); l^2)$ belongs to $C^2\left((-T, T); l^2_\Theta\right)$. 
 And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.

Complex structures for Klein–Gordon theory on globally hyperbolic spacetimes

We develop a rigorous method to parametrize complex structures for Klein–Gordon theory in globally hyperbolic spacetimes that satisfy a completeness condition. The complex structures are conserved under time-evolution and implement unitary quantizations. They can be interpreted as corresponding to global choices of vacuum. The main ingredient in our construction is a system of operator differential equations. We provide a number of theorems ensuring that all ingredients and steps in the construction are well-defined. We apply the method to exhibit natural quantizations for certain classes of globally hyperbolic spacetimes. In particular, we consider static, expanding and Friedmann–Robertson–Walker spacetimes. Moreover, for a huge class of spacetimes we prove that the differential equation for the complex structure is given by the Gelfand–Dikki equation.

Nonlocal multipoint problem for a differential equation of $2n$-th order with operator coefficients

In the article, the spectral properties of a multipoint problem for a differential operator equation of order $2n$ are studied. The operator of the problem has an infinite number of multiple eigenvalues. Each multiple eigenvalue corresponds to a finite set of root functions. A commutative group of transmutation operators is constructed. Each element of the group corresponds to the isospectral perturbation of the problem operator with antiperiodic conditions. The conditions for the existence and uniqueness of the solution are established for the selected family of multipoint problems, and this solution is constructed too.

Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators

Abstract Many non-local processes are modeled using mathematical models that include fractional powers of elliptic operators. The approximate solution of stationary problems with fractional powers of operators is most often based on rational approximations introduced in various versions for a fractional power of the self-adjoint positive operator. The purpose of this work is to use such approximations for the approximate solution of nonstationary problems. We consider Cauchy problems for the first and second order differential-operator equations in finite-dimensional Hilbert spaces. Estimates for the proximity of an approximate solution to an exact one are obtained when specifying the absolute and relative errors of the approximation of the fractional power of the operator. We construct splitting schemes based on the additive representation with a rational approximation of the operator’s fractional power. The stability and accuracy of factorized two-level additive operator-difference schemes for the first order evolution equation and three-level schemes for a second order equation are established.

Existence Results for a Class of Nonlinear Hadamard Fractional with p -Laplacian Operator Differential Equations

Existence Results for a Class of Nonlinear Hadamard Fractional with p -Laplacian Operator Differential Equations

An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems

In this paper, we consider a nonhomogeneous differential operator equation of first order u ′ t + A u t = f t . The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u 0 = Φ or u T = Φ . We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.

Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity

Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity

Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equations

Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equations