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Article Citation - WoS: 33Citation - Scopus: 46Higher-Order Self-Adjoint Boundary-Value Problems on Time Scales(Elsevier Science Bv, 2006) Anderson, Douglas R.; Guseinov, Gusein Sh.; Hoffacker, JoanIn this study, higher-order self-adjoint differential expressions on time scales and their associated self-adjoint boundary conditions are discussed. The symmetry property of the corresponding Green's functions is shown, together with specific formulas of Green's functions for select time scales. (c) 2005 Elsevier B.V. All rights reserved.Article Citation - WoS: 1Citation - Scopus: 3Spectral Approach To Derive the Representation Formulae for Solutions of the Wave Equation(Hindawi Publishing Corporation, 2012) Guseinov, Gusein Sh.Using spectral properties of the Laplace operator and some structural formula for rapidly decreasing functions of the Laplace operator, we offer a novel method to derive explicit formulae for solutions to the Cauchy problem for classical wave equation in arbitrary dimensions. Among them are the well-known d'Alembert, Poisson, and Kirchhoff representation formulae in low space dimensions.Article Citation - WoS: 12Citation - Scopus: 16The Laplace Transform on Isolated Time Scales(Pergamon-elsevier Science Ltd, 2010) Bohner, Martin; Guseinov, Gusein Sh.Starting with a general definition of the Laplace transform on arbitrary time scales, we specify the Laplace transform on isolated time scales, prove several properties of the Laplace transform in this case, and establish a formula for the inverse Laplace transform. The concept of convolution is considered in more detail by proving the convolution theorem and a discrete analogue of the classical theorem of Titchmarsh for the usual continuous convolution. (C) 2010 Elsevier Ltd. All rights reserved.Article Citation - WoS: 1Citation - Scopus: 1Solving an initial boundary value problem on the semiinfinite interval(Tubitak Scientific & Technological Research Council Turkey, 2016) Atalan, Ferihe; Guseinov, Gusein Sh.We explore the sign properties of eigenvalues and the basis properties of eigenvectors for a special quadratic matrix polynomial and use the results obtained to solve the corresponding linear system of differential equations on the half line subject to an initial condition at t = 0 and a condition at t = infinity.Article On the Resolvent of the Laplace-Beltrami Operator in Hyperbolic Space(Cambridge Univ Press, 2015) Guseinov, Gusein Sh.In this paper, a detailed description of the resolvent of the Laplace-Beltrami operator in n-dimensional hyperbolic space is given. The resolvent is an integral operator with the kernel (Green's function) being a solution of a hypergeometric differential equation. Asymptotic analysis of the solution of this equation is carried out.Article Citation - WoS: 55Citation - Scopus: 68The h-laplace and q-laplace Transforms(Academic Press inc Elsevier Science, 2010) Bohner, Martin; Guseinov, Gusein Sh.Starting with a general definition of the Laplace transform on arbitrary time scales, we specify the particular concepts of the h-Laplace and q-Laplace transforms. The convolution and inversion problems for these transforms are considered in some detail. (c) 2009 Elsevier Inc. All rights reserved.Article Citation - WoS: 47Citation - Scopus: 63The Convolution on Time Scales(Hindawi Publishing Corporation, 2007) Bohner, Martin; Guseinov, Gusein Sh.The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. Via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. As an extensive example, we consider the q-difference equations case. Copyright (c) 2007 M. Bohner and G. Sh. Guseinov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Article Citation - WoS: 15Citation - Scopus: 18Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians(Natl Acad Sci Ukraine, inst Math, 2009) Guseinov, Gusein Sh.In this paper, the concept of generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.Article Citation - WoS: 2Citation - Scopus: 4On a Discrete Inverse Problem for Two Spectra(Hindawi Ltd, 2012) Guseinov, Gusein Sh.A version of the inverse spectral problem for two spectra of finite-order real Jacobi matrices (tridiagonal symmetric matrices) is investigated. The problem is to reconstruct the matrix using two sets of eigenvalues: one for the original Jacobi matrix and one for the matrix obtained by deleting the last row and last column of the Jacobi matrix.Article Citation - WoS: 3Citation - Scopus: 3Inverse Spectral Problem for Finite Jacobi Matrices With Zero Diagonal(Taylor & Francis Ltd, 2015) Aydin, Ayhan; Guseinov, Gusein Sh.In this study, the necessary and sufficient conditions for solvability of an inverse spectral problem about eigenvalues and normalizing numbers for finite-order real Jacobi matrices with zero diagonal elements are established. Anexplicit procedure of reconstruction of the matrix from the spectral data consisting of the eigenvalues and normalizing numbers is given. Numerical examples and error analysis are provided to demonstrate the solution technique of the inverse problem. The results obtained are used to justify the solving procedure of the finite Langmuir lattice by the method of inverse spectral problem.
