Asymptotic Formulas For Solutions Research Articles

Inverse Sturm-Liouville problem with conformable derivative and transmission conditions

In this paper, we study the inverse problem for Sturm-Liouville problem with conformable fractional differential operators of order $\alpha$, $0.5 < \alpha\leq 1$ and finite number of interior discontinuous conditions. For this aim first, the asymptotic formulas for solutions, eigenvalues and eigenfunctions of the problem are calculated. Then some uniqueness theorems for proposed inverse eigenvalue problem are proved. Finally, the Hald's theorem for conformable Sturm-Liouville problem is developed.

One boundary value problem including a spectral parameter in all boundary conditions

In this paper, asymptotic formulae for solutions and Green's function of a boundary value problem are investigated when the equation and the boundary conditions contain a spectral parameter.

Asymptotic Behavior of Solutions of a Complete Second-Order Integro-Differential Equation

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin-Voigt friction.

Exponential Stability and Asymptotic Periodic Solutions of Linear Integral Equations with Two Delays

Our concern is to consider linear integral equations with two delays. A new result on necessary and sufficient conditions for the exponential stability is presented by careful study of the characteristic equation. Explicit expressions of limits of solutions in the critical case where integral equations lose exponential stability are also given by use of an asymptotic formula for solutions.

Sturm-Liouville Difference Equations Having Special Potentials

In this paper, we present a new approachment for Sturm-Liouville problem having special potentials. We acquire the representations of solutions and asymptotic formulas for solutions with regard to initial conditions. Also, a few applications are given to show the requirement of Sturm-Liouville difference equations having potential function in view of suitability to the spectral theory. The approximate numerical outcomes for the eigenfunctions are compared with each other.

Asymptotic Integration of Certain Differential Equations in Banach Space

We investigate the problem of constructing the asymptotics for weak solutions of certain class of linear differential equations in the Banach space as the independent variable tends to infinity. The studied class of equations is the perturbation of linear autonomous equation, generally speaking, with an unbounded operator. The perturbation takes the form of the family of the bounded operators that, in a sense, decreases oscillatory at infinity. The unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence for the initial equation of the center-like manifold (critical manifold). This manifold is positively invariant with respect to the initial equation and attracts all the trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional ordinary differential system. The asymptotics for the fundamental matrix of this system may be constructed by using the method proposed by the author for asymptotic integration of the systems with oscillatory decreasing coefficients. We illustrate the suggested technique by constructing the asymptotic formulas for solutions of the perturbed heat equation.

Asymptotic integration of a certain second-order linear delay differential equation

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrodinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.

Asymptotic formulae for solutions of linear second-order difference equations

We study asymptotic behaviour of solutions to the (nonoscillatory) linear difference equation , where are positive sequences defined on . We establish sufficient conditions (in terms of regular variation) for all eventually positive solutions to satisfy certain asymptotic formulae. As a by-product, we obtain regular variation of all these solutions and some other of their properties. Various related problems are discussed and several examples are given.

Asymptotic Integration of a Certain Second-Order Linear Delay Differential Equation

We construct some asymptotic formulas for solutions of a certain linear second-order delay differential equation when the independent variable tends to infinity. Two features concerning the considered equation should be emphasized. First, the coefficient of this equation has an oscillatory decreasing form. Second, when the delay equals zero, this equation turns into the so-called one-dimensional Schr¨odinger equation at energy zero with Wigner–von Neumann type potential. Dynamics of the latter is well-known. The question of interest is how the behavior of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. This equation also attracts interest from the standpoint of the theory of oscillations of solutions of functional differential equations. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients. The essence of the method is to construct a so-called critical manifold in the phase space of the considered dynamical system. This manifold is attractive and positively invariant, and, therefore, the dynamics of all solutions of the initial equation is determined by the dynamics of the solutions lying on the critical manifold. The system that describes the dynamics of the solutions lying on the critical manifold is a linear system of two ordinary differential equations. To construct the asymptotics for solutions of this system, we use the averaging changes of variables and transformations that diagonalize variable matrices. We reduce the system on the critical manifold to what is called the L-diagonal form. The asymptotics of the fundamental matrix of L-diagonal system may be constructed by the use of the classical Levinson’s theorem.

Asymptotic integration of the second order differential equation, resonance effect

Abstract There will be presented asymptotic formulas for solutions of the equation y'' + (1 + φ (x))y = 0, 0 < x0 < x < ∞ , where function is small in a certain sense for large values of the argument. Usage of method of L-diagonal systems allows to obtain various forms of solutions depending on the properties of function φ . The main aim will be discussion about the second order differential equations possesing a resonance effect known for Wigner von Neumann potential. A class of potentials generalizing that of Wigner von Neumann will be presented.

Formal adjoint equations and asymptotic formula for solutions of Volterra difference equations with infinite delay

For linear Volterra difference equations with infinite delay, we obtain an explicit asymptotic representation formula of solutions by means of the formal adjoint equation associated with a certain bilinear form.

Asymptotic formulas for solutions of nonlocal elliptic problems

We consider nonlocal elliptic problems in plane domains and obtain asymptotic formulas for solutions in weighted spaces near junction points.

Asymptotics for solutions of periodic difference systemsSupported in part by Fondecyt 1030535, 1080034, DGI-MATH UNAP 2008 and Diufro 120521.

Using dichotomies and periodic conditions, we obtain asymptotic formulas for solutions of a difference system of Poincaré type with periodic coefficients. Some results about the theory of existence of periodic solutions for linear difference systems are presented. At the end, an open problem on the asymptotic spectral representation is proposed.

Riesz potential equations in local L p -spaces

We consider the following equation for the Riesz potential of order one: Uniqueness is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove an existence result and derive an asymptotic formula for solutions near the origin. Our analysis is carried out in local L p -spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted L p -spaces and homogenous Sobolev spaces.

Construction of asymptotic formulas for solutions of singularly perturbed degenerate systems of differential equations

We obtain an asymptotic solution of the Cauchy problem for a singularly perturbed degenerate system of differential equations in the case of a singular limit pencil of matrices.

The asymptotic behavior of solutions for a class of differential equations

A class of (n+1)th-order homogeneous linear differential equations (n > 1) is found for which fundamental solutions can be constructed from fundamental solutions to second-order differential equations. Asymptotic formulas for solutions to equations from this class are presented.

An asymptotic formula for solutions of linear second-order difference equations with regularly behaving coefficients

An asymptotic formula is given for the solutions of a linear second-order difference equation which holds in the case that the coefficients exhibit regular behaviour; in particular, the formula holds in the case that the coefficients are series of powers of the index n. The lowest order asymptotic behaviour is given for both x n and . Examples are given that show that the conditions on the behaviour of the coefficients cannot be weakened too much. The method presented here, which uses matrix sequences, can be extended to other cases.

An approach to optimised calculations of the potential energy surfaces for the case of electron transfer reactions at a metal/solution interface

An effective computational scheme to construct the adiabatic potential energy surfaces (APES) along the reaction coordinates for an electron transfer reaction occurring by two steps at a metal electrode is considered in the framework of the Anderson–Newns model. Two Theorems have been proved which predict the existence of all possible solutions of the Anderson–Newns equations at arbitrary values of the key parameters and point out the region for each solution. Asymptotic formulas for solutions near multiple roots have been derived and combined in an effective way with numerical routines. The analysis of some important properties of the APES, which can be of interest for modelling the electrochemical electron transfer processes, is presented as well. The APES describing the reduction of Zn(II) and In(III) aqua-complexes at a mercury electrode have been built and discussed.

On the asymptotics for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation where We prove the global existence of small solutions , if the initial data u 1 belong to some analytic function space and are sufficiently small. For the coefficients λ j we assume that there exists θ 0 > 0 such that for all and also we suppose that the initial data are such that where ϵ is a small positive constant depending on the size of initial function in a suitable norm. We also find the large time asymptotic formulas for solutions. In the short range region the solution has an additional logarithmic time decay comparing with the corresponding linear case.

Dissipative Elliptic Problems in Domains with Cylindrical Ends, Scattering Matrices, and Radiation Conditions

In a domain \(G \subset \mathbb{R}^{n + 1} \) with cylindrical ends at infinity, we consider a general elliptic dissipative boundary value problem. The coefficients of the imaginary part of the operator of the problem vanish as \(x \to \infty ,x \in G\) The asymptotic behavior of the solutions is expressed in terms of incoming and outgoing “waves” (the amplitudes of such waves can grow at infinity). We introduce an (augmented) scattering matrix and, in terms of this matrix, we compute the number of linearly independent solutions to the homogeneous problem vanishing at infinity with a given rate. We discuss the statement of a problem with the so-called radiation conditions. The natural radiation conditions (only outgoing “waves” occur in asymptotic formulas for solutions) can be applied in any case. Other admissible radiation conditions for the problem under consideration are connected with the natural ones via scattering matrices. Bibliography: 12 titles.