Green’s functions for stationary problems with nonlocal boundary conditions

In this paper, we investigate Green’s functions for various stationary problems with nonlocal boundary conditions. We express the Green’s function per Green’s function for a problem with classical boundary conditions. This property is illustrated by various examples. Properties of Green’s functions with nonlocal boundary conditions are compared with those for classical problems. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-73/09.     

Rothe’s method for an integrodifferential equation with integral conditions

This work deals with an integrodifferential equation with initial, Newman and integral conditions. The existence and uniqueness of the weak solution in an appropriate sense is proved by the method of Rothe.     

Jensen–Steffensen inequality for diamond integrals,its converse and improvements via Green function and Taylor’s formula

In this paper we define the Jensen–Steffensen inequality and its converse for diamond integrals. Then we give some improvements of these inequalities using Taylor’s formula and the Green function. We investigate bounds for the identities related to improvements of the Jensen–Steffensen inequality and its converse.     

Bounds for canonical Green’s function at cusps

In this article, we derive bounds for the canonical Green’s function defined on a noncompact hyperbolic Riemann surface, when evaluated at two inequivalent cusps.     

Generalized Green’s functions for the second-order discrete problems with nonlocal conditions

In this paper, we investigate a generalized discrete Green’s function that describes the general least squares solution of every second-order discrete problem with two nonlocal conditions. We develop the problem where the necessary and sufficient existence condition of ordinary discrete Green’s function is not satisfied. Some examples are also presented.     

Positivity conditions and bounds for Green’s functions for higher order two-point BVP

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.     

Generalization of Taylor’s formula and differential transform method for composite fractional q-derivative

The Ramanujan Journal - In the present paper, we first establish a generalized q-Taylor’s formula involving composite fractional q-derivative. Next, we define the generalized q-differential...     

Stochastic Nicholson’s blowflies delay differential equation with regime switching

In this paper, we investigate the global existence of almost surely positive solution to a stochastic Nicholson’s blowflies delay differential equation with regime switching, and give the estimation of the path. The results presented in this paper extend some corresponding results in Wang et al. Stochastic Nicholson’s blowflies delayed differential equations, Appl. Math. Lett. 87 (2019) 20–26 .     

Fast algorithms for implementation of the Green’s function

This paper is devoted to new fast algorithms for implementation of the Green’s function for the Helmholtz operator in high-frequency regions in periodic and helical structures.     

On Ramanujan’s large argument formula for the Gamma function

The aim of this paper is to improve some approximation formulas of Ramanujan type discussed by E.A. Karatsuba [J. Comput. Appl. Math. 135 (2001), 225–240].     

Kolmogorov equation and large-time behaviour for fractional brownian motion driven linear sde’s

We consider a stochastic process X _t ^x which solves an equation

A new bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation

The complexity of computing the Galois group of a linear differential equation is of general interest. In a recent work, Feng gave the first degree bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation. This bound is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group (see Definition 2.7) and is sextuply exponential in the order of the differential equation. In this paper, we use Szántó’s algorithm of triangular representation for algebraic sets to analyze the complexity of computing the Galois group of a linear differential equation and we give a new bound which is triple exponential in the order of the given differential equation.     

The two-time Green’s function and the diagram technique

Based on constructing the equations of motion for the two-time Green’s functions, we discuss calculating the dynamical spin susceptibility and correlation functions in the Heisenberg model. Using a Mori-type projection, we derive an exact Dyson equation with the self-energy operator in the form of a multiparticle Green’s function. Calculating the self-energy operator in the mode-coupling approximation in the ferromagnetic phase, we reproduce the results of the temperature diagram technique, including the correct formula for low-temperature magnetization. We also consider calculating the spin fluctuation spectrum in the paramagnetic phase in the framework of the method of equations of motion for the relaxation function.     

Constructing approximate Green’s function for a vector equation for the electric field using the variational iteration method

In this work, we use He’s variational iteration method (VIM) to find approximate Green’s functions for a vector equation for the electric field with anisotropic dielectric permittivity and magnetic permeability. We present numerical examples which show that an approximate solution of an initial value problem (IVP) for a vector equation can be obtained by using these approximate Green’s functions.     

Dependence of Green’s function of an e-dichotomous differential equation with matrix projector in the space\mathfrak{M} on parameters

In the space of bounded number sequences, we investigate an e-dichotomous differential equation with matrix projector. We prove that Green's function of this equation exists and present conditions under which it is continuous and differentiable as a function of the parameters appearing in the original equation. Kamenets-Podol'skii Pedagogical Institute, Kamenets-Podol'skii. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 748–753, June, 1994.     

Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution

Abstract

In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1???ba (resp., ba) being pseudo core invertible while 1???ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1???ba (resp., ba) is given in terms of 1???ab (resp., ab). Inspired by the above idea, Jacobson’s lemma for Moore-Penrose inverse is considered.     

Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

We prove the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$

for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.

     

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain \(D \subset \mathbb {R}^d\) with \(d \ge 2\), for every (measurable) uniformly elliptic tensor field a and for almost every point \(y \in D\), there exists a unique Green’s function centred in y associated to the vectorial operator \(-\nabla \cdot a\nabla \) in D. This result implies the existence of the fundamental solution for elliptic systems when \(d>2\), i.e. the Green function for \(-\nabla \cdot a\nabla \) in \(\mathbb {R}^d\). In the second part, we introduce a shift-invariant ensemble \(\langle \cdot \rangle \) over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for \(\langle |G(\cdot ; x,y)|\rangle \), \(\langle |\nabla _x G(\cdot ; x,y)|\rangle \) and \(\langle |\nabla _x\nabla _y G(\cdot ; x,y)|\rangle \). These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.