Differential-operator Equations Research Articles

Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.

Criteria for the Uniqueness of a Solution to a Differential-Operator Equation with Non-Degenerate Conditions

In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and B. The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers, the criterion for the uniqueness of the boundary value problem formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required.

The general solution of the non-homogeneous Euler–Cauchy operator-differential equation with Neumann boundary conditions using the Laplace transform

Many researchers are interested in topics related to solutions of differential equations with variable coefficients. Here for the first time, we study the nonhomogeneous second-order Euler–Cauchy operator-differential equation. In this paper, we apply the Laplace transform method to find the general solution of the Euler–Cauchy equation with Neumann boundary conditions. We show that this method is powerful in finding the general solution of the Euler–Cauchy equation in terms of generalized exponential functions.

Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems

This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Its main theorem establishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states in d≥3 space dimensions. Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic symmetrizers is assumed. The dissipation is characterized by algebraic conditions implying the uniform decay of all Fourier modes at the reference state. On a technical level, the theory developed herein uses para-differential operators as its main tool. Apparently being the first to apply such operators in the context of global-in-time existence of small solutions for quasi-linear hyperbolic systems, the present work contains a new version of the strong Gårding inequality. In the context of theoretical physics, the theorem applies to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids such as notably that of Bemfica, Disconzi and Noronha (2018) [2].

On the completeness of a system of decreasing elementary solutions for one class of fourth-order operator-differential equations on the semi-axis

In this work, we study a homogeneous fourth-order elliptic operator-differential equation with an unbounded operator under non-homogeneous boundary conditions on the semi-axis. We obtain sufficient conditions for the completeness of decreasing elementary solutions of the equation under study in the space of all its solutions from a fourth-order Sobolev type space.

Operator-difference schemes on non-uniform grids for second-order evolutionary equations

Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.

Discrete Approximation of Solutions of the Cauchy Problem for a Linear Homogeneous Differential-Operator Equation with a Caputo Fractional Derivative in a Banach Space

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order α ∈ (0, 1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented.

On natural invariants and equivalence of differential operators

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of order k≥2 on an n-dimensional manifold, n≥2, and apply the results to the equivalence problem of such operators.

Boundary-Value Problems for the System of Operator-Differential Equations in Banach and Hilbert Spaces

We establish necessary and sufficient conditions for the existence of solutions of linear and nonlinear boundary-value problems in Hilbert and Banach spaces and present a convergent iterative procedure for finding the solutions in the nonlinear case.

On invariants and equivalence of differential operators under Lie pseudogroups actions

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms.

Degenerate Third Order Differential-operator Equations

Degenerate Third Order Differential-operator Equations

On paracomposition and change of variables in paradifferential operators

On paracomposition and change of variables in paradifferential operators

Solvability of boundary-value problem with bounded operator in boundary conditions for fourth-order elliptic operator-differential equation

In the paper, we derive sufficient conditions for well-posed and unique solvability of a boundary-value problem with a bounded operator in the boundary conditions for a class of fourth-order elliptic operator-differential equations. At the same time, we make estimates for the norms of operators of intermediate derivatives closely related to the solvability conditions, which are expressed using the properties of the operator coefficients of the boundary-value problem.

On conditions of regular solvability for two classes of third-order operator-differential equations in a fourth-order Sobolev-type space

In this paper, we study two classes of operator-differential equations of the third order with a multiple characteristic, considered on the whole axis. We introduce the concept of a smooth regular solution of order 1 and obtain sufficient conditions for the "smoothly" regular solvability of these equations.

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

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