Related papers: On first and second order linear Stieltjes differe…
We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that the n-th power of the function satisfies an (n+1)-th order differential equation and give a simple method for…
This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative…
Existence, uniqueness and stability of the solutions of linear stochastic evolution equations are investigated. The results obtained are used to prove theorems on solvability of linear second order stochastic partial differential equations…
In this paper we present some applications of the Stieltjes constants including, for example, new derivations of Binet's formulae for the log gamma function and the evaluation of some integrals related to the Barnes multiple gamma…
This paper deals with linear stochastic partial differential equations with variable coefficients driven by L\'{e}vy white noise. We first derive an existence theorem for integral transforms of L\'{e}vy white noise and prove the existence…
In this work we study differential problems in which the reflection operator and the Hilbert transform are involved. We reduce these problems to ODEs in order to solve them. Also, we describe a general method for obtaining the Green's…
We derive a priori estimates for second order derivatives of solutions to a wide calss of fully nonlinear elliptic equations on Riemannian manifolds. The equations we consider naturally appear in geometric problems and other applications…
We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a…
For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of…
The Riccati equation method is used for study the behavior of solutions of the systems of two linear first order ordinary differential equations. All types of oscillation and regularity of these system are revealed. A generalization of…
This paper treat determinacy of strong moment problems in part I and indeterminacy of strong moment problems in part II. This paper is a summary of the following papers: [1] Ald\'en. E., Determinacy of Strong Moment Problems. [2] On…
We establish a link between the basic properties of the discriminant of periodic second-order differential equations and an elementary analysis of Herglotz functions. Some generalizations are presented using the language of self-adjoint…
We employ some results about continued fraction expansions of Herglotz-Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral…
This note is concerned with an important for modelling question of existence of solutions of stochastic partial differential equations as proper stochastic processes, rather than processes in the generalized sense. We consider a first order…
We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a second $\alpha$-order fractal differential equation with constant coefficients across different scenarios. We…
For a second-order elliptic equation of nondivergence form in the plane, we investigate conditions on the coefficients which imply that all strong solutions have first-order derivatives that are Lipschitz continuous or differentiable at a…
In this paper, we analyze a second-order differential equation with a piecewise constant argument and reflection coupled to periodic boundary conditions. Our main contribution is the construction of the related Green's function and a…
The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to other topics. Though existence and uniqueness of solutions are established for long, we present several new aspects which…