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The paper deals with systems of linear integro-differential equations, including systems with a matrix identically degenerate in the domain at the highest derivative of the unknown vector function. A theorem on the solvability of such systems of equations is proved. On the basis of this theorem, it is shown that the ring of operators corresponding to these systems contains the semigroup of operators for which a left inverse operator from this ring is found. [ABSTRACT FROM AUTHOR]

In this paper, the Chebyshev polynomials method is applied to solve a space-time variable fractional order integro-differential equation. Using operational matrices of Chebyshev polynomials furnished from the Caputo–Prabhakar sense and also suitable collocation points, the variable fractional order integro-differential equation would be converted to the system of algebraic equations. The main aim of the Chebyshev polynomials method is to derive four kinds of operational matrices of Chebyshev polynomials. With such operational matrices, an equation is transformed into the products of several dependent matrices, which can also be viewed as the system of linear equations after dispersing the variables. An error bound is proved for the approximate solution obtained by the proposed method. Finally, some numerical examples are presented to demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a boundary value problem for a system of integro-differential equations arising when modeling the influence of an electric field on a plasma layer. The paper presents an analytical solution of this problem, which is constructed with the use of theory of the Fourier transform of generalized functions and the Gakhov–Muskhelishvili method for solving singular integral equations and the Riemann boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, on the basis of the theory of degenerate semigroups of operators and the contraction mapping theorem, we prove the local unique solvability of initial problems for a class of first-order linear differential operator equations with memory and with degenerate operator multiplying the derivative. The resulting abstract results are used to study initial boundary-value problems for partial integro-differential equations not solvable with respect to the time derivative. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial-value problem for nonlinear first-order impulsive integro-differential equations on time scales T . We establish certain existence criteria by using a fixed-point theorem for operator on cones, under which such problems have aminimal and amaximal solution lying in a corresponding region bounded by their lower and upper solutions. [ABSTRACT FROM AUTHOR]

In the paper, we propose a method for proving the unconditional basis property of eigen- and associated functions of an integro-differential operator defined on a nondense domain of definition. In particular, we obtain a new simpler proof of the unconditional basis property of eigenand associated functions of the spectral Orr-Sommerfeld problem, well-known in hydromechanics, which reduces to the eigenvalue problem for the operator under study. [ABSTRACT FROM AUTHOR]

The one-dimensional integro-differential equation arising in the theory of viscoelasticity with constant density and Lamé coefficients is considered. The direct problem is to determine the displacement function from the initial boundary-value problem for this equation, provided that the initial conditions are zero. The spatial domain is the closed interval [0, l], and the boundary condition is given by the stress function in the form of a concentrated perturbation source at the left endpoint of this interval and as zero at the right endpoint. For the direct problem, we study the inverse problem of determining the kernel appearing in the integral term of the equation. To find it, we introduce an additional condition for the displacement function at x = 0. The inverse problem is replaced by an equivalent system of integral equations for the unknown functions. The contraction mapping principle is applied to this system in the space of continuous functions with weighted norms. A theorem on the global unique solvability is proved. [ABSTRACT FROM AUTHOR]

The inverse problem of determining the solution and the kernel of the integral term in an inhomogeneous two-dimensional integrodifferential wave equation in a rectangular domain is considered. First, the uniqueness of the solution of the direct problem is established using the completeness of the eigenfunction system of the corresponding homogeneous Dirichlet problem for the two-dimensional Laplace operator, and the existence of a solution of the direct problem is proved. Using additional information about the solution of the direct problem, we obtain a Volterra integral equation of the second kind for the kernel of the integral term. The existence and uniqueness of a solution of this equation is proved by the contraction mapping method in the space of continuous functions with a weighted norm. [ABSTRACT FROM AUTHOR]

The Goursat problem for the fractional telegraph equation with Caputo derivatives is studied. An existence and uniqueness theorem for the solution of the problem is proved. [ABSTRACT FROM AUTHOR]

A family of parabolic integro-differential equations with nonlocal diffusion on the circle which have no smooth inertial manifold is presented. [ABSTRACT FROM AUTHOR]

The initial boundary-value problem for systems of fourth-order partial differential equations with two independent variables is considered. By using a new unknown eigenfunction, the problem under consideration is reduced to an equivalent nonlocal problem for a system of second-order hyperbolic-type integro-differential equations with integral conditions. An algorithm for finding an approximate solution of the resulting equivalent problem is proposed, and its convergence is proved. Conditions for the existence of a unique classical solution of the initial boundary-value problem for systems of fourth-order differential equations are established in terms of the coefficients of the system and the boundary matrices. [ABSTRACT FROM AUTHOR]

The uniform exponential stability of a C0-semigroup with generator of a special form is proved. Such semigroups arise in the study of various problems of the theory of viscoelasticity. The proved statement is applied to the study of the asymptotic behavior of solutions in the problem of small motions of a viscoelastic body subject to driving forces of a special form. [ABSTRACT FROM AUTHOR]

Inverse spectral problems for first-order integro-differential operators on a finite interval are studied, the properties of spectral characteristics are established, and uniqueness theorems for solutions of this class of inverse problems are proved. [ABSTRACT FROM AUTHOR]

The article discusses spectral analysis of integro-differential equations in viscoelasticity and thermal physics. Topics discussed include changes the structure of the nonreal spectrum of the operator function caused by the parameter significantly, closeness of the structure of the nonreal spectrum to the spectrum of the wave equation.