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We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…

We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These…

A system of first-order differential equations for a particle with nonzero mass and spin $S = 1$ is constructed. As distinct from the Proca-Duffin-Kemmer (PDK) equations, the system has the form of the dynamical equation…

A class of first order linear impulsive differential equation with continuous and piecewise constant arguments is studied. Sufficient conditions for the oscillation of the solutions are obtained.

In this article we address some issues related to the initial value problems for a rotating shallow water hyperbolic system of equations and the diffusive regularization of this system. For initial data close to the solution at rest, we…

This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical…

For ordinary differential equations in the complex domain, a central problem is to understand, in a given equation or class of equations, those whose solutions do not present multivaluedness. We consider autonomous, first-order, quadratic…

In this article, the order of some classes of fractional linear differential equations is determined, based on asymptotic behavior of the solution as time tends to infinity. The order of fractional derivative has been proved to be of great…

We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail…

An initial-boundary value problem for a subdiffusion equation with an elliptic operator $A(D)$ in $\mathbb{R}^N$ is considered. The existence and uniqueness theorems for a solution of this problem are proved by the Fourier method.…

We consider the variational discretization of a linear-quadratic optimal control problem with pointwise control and state constraints. In order to allow for a Fr\'echet smooth norm, the problem is reformulated by means of a reflexive…

We examine the analytic extension of solutions of linear, constant-coefficient initial-boundary value problems outside their spatial domain of definition. We use the Unified Transform Method or Method of Fokas, which gives a representation…

We investigate a class of fourth-order differential systems with instantaneous and non-instantaneous impulses. Our technical approach is mainly based on a variant of Clark's theorem without the global assumptions. Under locally subquadratic…

A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is…

Partial orders have been used to model several experimental setups, going from classical thermodynamics and general relativity to the quantum realm with its resource theories. In order to study such experimental setups, one typically…

The paper considers the initial-boundary value problem for equation $D^\rho_t u(x,t)+ (-\Delta)^\sigma u(x,t)=0$, $\rho\in (0,1)$, $\sigma>0$, in an N-dimensional domain $\Omega$ with a homogeneous Dirichlet condition. The fractional…

For a discrete function $f\left( x\right) $ on a discrete set, the finite difference can be either forward and backward. If $f\left( x\right) $ is a sum of two such functions $f\left( x\right) =f_{1}\left( x\right) +f_{2}\left( x\right) $,…

We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…

Complex systems are composed of a large number of simple components connected to each other in the form of a network. It is shown that, for some network configurations, the equivalent dynamic behavior of the system is governed by an…

We introduce a new class of singular partial differential equations, referred to as the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First, we…