# Kazem GhanbariSahand University of Technology | SUT · Faculty of Sciences

Kazem Ghanbari

PhD

## About

54

Publications

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264

Citations

Introduction

Kazem Ghanbari currently is full professor at the Faculty of Sciences, Sahand University of Technology. Kazem does research in Linear Algebra (Inverse Eigenvalue Problems) Differential Equations. Their current project is 'Solvability of fractional differential, difference and integral equations. Moreover the area of Isospectrality of Matrix flows and Inverse Sturm-Liouville problems.

## Publications

Publications (54)

Abstract This paper deals with a typical system of Caputo fractional difference equations. Using the Guo–Krasnosel’skii fixed point theorem, we find a parameter interval for which at least one positive solution of the system exists. We give two examples to illustrate the results.

In this article we investigate existence and nonexistence results for some regular fractional Sturm-Liouville problems. We find the eigenvalues intervals of this problem may or may not have a positive solution. Some sufficient conditions for existence and nonexistence of positive solutions are given. Further, we study some special properties of pos...

This paper focuses on providing a novel high-order algorithm for the numerical solution of a class of linear semi-explicit fractional order integro-differential algebraic equations with weakly singular kernels using the discrete Legendre collocation method. For this purpose, we investigate the existence and uniqueness as well as the regularity prop...

This paper presents a spectral Tau method based on Müntz–Jacobi basis functions for approximating the solutions of fractional order Volterra integro-differential algebraic equations. We examine the solvability of the considered equation, and we show that the exact solutions suffer from a singularity at the origin. Numerical solvability and converge...

This paper is devoted to a high-order Legendre collocation approximation for solving fractional-order
linear semi-explicit differential algebraic equations numerically. We discuss existence, uniqueness, and regularity
results and conclude that the solutions typically suffer from a singularity at the origin. Moreover, we show that the
representation...

In this paper, we provide an approximate approach based on the Galerkin method to
solve a class of nonlinear fractional differential algebraic equations. The fractional derivative operator in the Caputo sense is utilized and the Generalized Jacobi functions are employed as trial functions. The existence and uniqueness theorem as well as the asympto...

In this paper, first we introduce new definition of conformable fractional derivatives and study their algebraic properties. As application, we consider certain classes of conformable differential linear systems subject to impulsive effects and establish qualitative behavior of the nontrivial solutions such as stability, disconjugacy, nonexistence,...

This paper, deals with Lyapunov inequalities of conformable fractional boundary value problems on an N-dimensional spherical shell. Applicability of these Lyapunov inequalities will be examined by establishing the disconjugacy as a nonexistence criterion for nontrivial solutions, lower bound estimation for eigenvalues of the corresponding fractiona...

In this paper, we study two classes of fractional half-linear boundary value problems subject to the Dirichlet boundary conditions. The golden aims of this paper can be summarized as follows. First, we introduce extended theory of the conformable fractional calculus and its basic analysis. In the next level using the Green function technique, we ob...

The main research line of this paper is concerned with the existence and uniqueness of solutions for a certain class of coupled systems of Caputo type fractional Δ-difference boundary value problems at resonance. To this aim, we use coincidence degree theory to obtain existence results and impose growth controlling conditions on nonlinearities, uni...

In this paper, we consider a new study about fractional Δ - difference equations. We consider two special classes of Sturm-Liouville problems equipped with fractional Δ -difference operators. In couple of steps, the Lyapunov type inequalities for both classes will be obtained. As application, some qualitative behaviour of mentioned fractional probl...

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given by dA/dt = [A(u) - A(l),A], A(0) = A(0), where A is a real n x n matrix (not necessarily symmetric), [A, B] = AB - BA is the matrix commutator (also known as the Lie bracket), A is the strictly upper triangular pa...

Of concern is studying solvability of the hybrid systems of quadratic fractional integral equations. To this aim applying hybrid fixed point theory due to Dhag e, existence of at least one positive solution for mentioned systems via so called D-Lipschitzian mappings will be concluded. We illustrate the obtained results by presenting an example .

In this paper, we establish the solvability of coupled systems of two-point fractional ∆-difference boundary value problems. To this aim we use the nonlinear alternative of Leray-Schauder and Krasnoselskii-Zabreiko fixed point theorems for existence results and by imposing Lipschitzian conditions on nonlinearities uniqueness of solutions will be co...

In this paper, we study the solvability of coupled hybrid systems of quadratic fractional integral equations. Applying hybrid fixed point theory, due to B. C. Dhage, the existence of at least one positive solution for mentioned systems is proved. At the end, illustrating obtained results, an example is given.

In this paper, we study the existence of solutions for a coupled system of two-point fractional $\nabla$-difference boundary value problems of the form
\begin{equation*}
\begin{split}
&\left(\begin{array}{l}
\nabla_{a^{+}}^{\alpha}u(t)\\
\nabla_{a^{+}}^{\beta}v(t)
\end{array}\right)+
\left(\begin{array}{l}
f(t,v(t)\\
g(t,u(t))
\end{array}\right)=0,...

In this paper, we will introduce some new results about Lyapunov type inequalities by studying the given fractional order Sturm-Liouville problems and fractional order Hamiltonian systems. We state and prove Lyapunov type inequalities for fractional S-L problems equipped by fractional Riemann-Liouville derivatives. In this paper for establishing fr...

In this paper, we study the existence of solutions for a coupled system of two-point fractional $\nabla$-difference boundary value problems of the form
\begin{equation*}
\begin{split}
&\left(\begin{array}{l}
\nabla_{a^{+}}^{\alpha}u(t)\\
\nabla_{a^{+}}^{\beta}v(t)
\end{array}\right)+
\left(\begin{array}{l}
f(t,v(t)\\
g(t,u(t))
\end{array}\right)=0,...

This paper deals with study about fractional order impulsive Hamiltonian systems and fractional impulsive Sturm-Liouville type problems derived from these systems. The main purpose of this paper devotes to obtain so called Lyapunov type inequalities for mentioned problems. Also, in view point on applicability of obtained inequalities, some qualitat...

In this paper, we study coupled systems of the half-linear boundary value problems involving left sided Caputo fractional derivatives. The main goal of this paper is restricted to the existence verification of positive solutions for mentioned fractional boundary value problems. To this aim we use nonlinear alternative of Leray-Schauder and Krasnose...

In this paper, we find matrix representation of a class of sixth order Sturm-Liouville problem (SLP) with separated, self-adjoint boundary conditions and we show that such SLP have finite spectrum. Also for a given matrix eigenvalue problem HX = λV X, where H is a block tridiagonal matrix and V is a block diagonal matrix, we find a sixth order boun...

This paper illustrates how classical integration methods for differential equations
on manifolds can be modified in order to preserve certain geometric properties of the
exact
ow. Runge-Kutta-Munthe-Kass method is considered and some examples are
shown to verify the efficiency of the method.

Of concern is studying solvability results for multiple positive solutions of a certain fractional order coupled hybrid system. To this aim we will apply the generalized hybrid fixed point theorem for simplicity and 2-D Leray-Schauder fixed point theorem for multiplicity results.

In this paper, we consider a generalized inverse eigenvalue problem of the form , where and are both pentadiagonal matrices. We propose an algorithm for reconstructing a pentadiagonal matrix . Let be leading principal submatrix of . Given , a pentadiagonal matrix , distinct real numbers , , and real vectors , , , where , we construct pentadiagonal...

In this paper, we consider a generalized inverse eigenvalue problem [Inline formula], where [Inline formula] is a Jacobi matrix and [Inline formula] is a positive definite tridiagonal matrix. Let [Inline formula] be [Inline formula] leading principal submatrix of [Inline formula]. Given [Inline formula], two vectors [Inline formula], [Inline formul...

In this paper, we propose an algorithm for constructing a pentadiagonal matrix with given prescribed three spectra. Sufficient conditions for solvability of the problem are given. We generate an algorithmic procedure to construct the solution matrices and we given a numerical example illustrating the construction algorithm.

Gladwell [Gladwell GML. Minimal mass solutions to inverse eigenvalue problems. Inverse Probl. 2006;22:539-551] found an explicit solution for the inverse eigenvalue problem of an in-line system of masses and springs with one given spectrum corresponding to the fixed-free end conditions, and a minimal mass condition. In this paper we formulate and s...

In this paper, we consider a coupled system of nonlinear fractional differential
equations (FDEs), such that both equations have a particular perturbed
terms. Using Leray-Schauder fixed point theorem, we investigate the existence
and multiplicity of positive solutions for this system.

This paper concerns about existence and multiplicity results of the given
fractional order boundary value problem �. The main results involve
some well known nonlinear analysis techniques to prove our claimed results. In order to
illustrate the main results, at the end of each technique, an example is represented.

A matrix function F(A) is said to be skew-symmetric operator if (F(A)) T = -F(A). It is well known that [5] if F(A) is skew-symmetric then the solution A(t) of (Equation) maintains the spectrum of Ao, where [A, B] = AB - BA. The matrix A is said to be centrosymmetric if JAJ = A, where J is the matrix with ones on the secondary diagonal and zeros el...

In this work, we consider the certain fractional BVP
. Using a fixed point theorem for operators on a cone, we
obtain sufficient conditions for the existence of positive solution of the above
BVP. At the end, example is presented illustrate the main results.

In this paper we investigate famiLies of sixth-order Sturm-Liouville equations having the same spectrum. We factorize the Sturm-Liouville operator as the product of a third order linear differential operator and its adjoint. By reversing the order of the factors we obtain another sixth-order Sturm-Liouville operator which is isospectral with the in...

Isospectral matrix
flows are dynamical systems evolving
on space of matrices, which have specific properties that their
solutions maintain the eigenvalues of the initial matrix. These
flows
are of particular interest to numerical analysis because of their
connection to algorithms where the eigenvalues of a certain matrix
remain fixed throughout t...

In this work, we consider the following BVP
\begin{equation*}
^{c}D_{0}^{\alpha}u(t)=\lambda g(t)f(t,u(t))~;\qquad t\in
(0,1)~,~\alpha\in(2,3)
\end{equation*}
$$\hspace{-4cm}u(0)+u^{'}(0)=0 $$
$$\hspace{-4cm}u(1)+u^{'}(1)=0 $$
\begin{equation*}
au^{''}(0)+bu^{''}(1)=0 ~;\qquad a>0~,b\leq0~,a+b>0
\end{equation*}
where $^{c}D_...

In this paper, we consider a coupled system of nonlinear fractional differential equations (FDEs), such that both equations have a particular perturbed terms. Using Leray-Schauder fixed point theorem, we investigate the existence and multiplicity of positive solutions for this system.

In this article we find sufficient conditions for existence and multiplicity
of positive solutions for an m-point nonlinear fractional boundary-value
problem on an infinite interval. Moreover, we prove that the set of positive
solutions is compact. Nonexistence results for the boundary-value problem
also are obtained.

In this article we find sufficient conditions for existence and multiplicity
of positive solutions for an m-point nonlinear fractional boundary-value
problem on an infinite interval. Moreover, we prove that the set of positive
solutions is compact. Nonexistence results for the boundary-value problem
also are obtained.

In this paper we consider a generalized inverse eigenvalue problem J nX=λC nX, where J n is a Jacobi matrix and C n is a nonsingular diagonal matrix that may be indefinite. Let J k be k×k leading principal submatrix of J n. Given C n, two vectors X 2=(X k+1,X k+2, ⋯,X n) T, Y 2=(y k+1,y k+2,⋯,y n) T∈ℝ n-k, two distinct real numbers λ, μ, we constru...

Let $D$ be a connected bounded domain in ${{\mathbb{R}}^{n}}$ . Let $0\,<\,{{\mu }_{1}}\,\le \,{{\mu }_{2}}\,\le \,\cdots \,\le \,{{\mu }_{k}}\,\le \,\cdots $ be the eigenvalues of the following Dirichlet problem:
$$\left\{ \begin{align} & {{\Delta }^{2}}u(x)\,+\,V(x)u(x)\,=\,\mu \rho (x)u(x),x\in \,D \\ & u{{|}_{\partial D}}\,=\,\frac{\partial u}{...

We consider the boundary value problem D α u(t)+λa(t)f(u)=0for0<t<1,u(0)=u(1)=0, where 1<α≤2 is a real number and D α is the standard Riemann-Lioville differentiation. We determine the values of λ, the so-called eigenvalues, for which the boundary value problem has at least one positive solution under specific conditions on f.

Applications of the jet space analysis to isospectral beams are considered. Using the jet space analysis, we show that the principal equations related to the beam equation have unique solutions.

In this paper, we give a survey of well-known works of constructing an specific tridiagonal matrix using prescribed data. Then we focus on two well-known methods namely, m-functions and Lanczos algorithm. We describe the first method for a generalized inverse eigenvalue problem and the second one for a classic inverse eigenvalue problem. We show th...

In this paper we present a study on the analogous properties of discrete and continuous Sturm-Liouville problems arising in matrix analysis and differential equations, respectively. Green's functions in both cases have analogous expressions in terms of the spectral data. Most of the results associated to inverse problems in both cases are identical...

In this paper we generalize the main results of Bai, Wang and Ge (Electron J. Diff. Eqns. 6 (2004), 1 - 8) by considering a general type of boundary value problem associated with a nonlinear fractional differential equation. We obtain sufficient conditions for the existence of at least three positive solution with corresponding upper and lower boun...

We denote the spectrum of an square matrix A by σ(A), and that of the matrix obtained by deleting the first i rows and columns of A by σ
i
(A). It is known that a symmetric pentadiagonal oscillatory (SPO) matrix may be constructed from σ, σ
1 and σ
2. The pairs σ, σ
1 and σ
1, σ
2 must interlace; the construction is not unique; and the conditions o...

The free undamped infinitesimal transverse vibrations of a thin straight beam are modelled by a forth-order differential equation. This paper investigates the families of fourth-order systems which have one spectrum in common, and correspond to four different sets of end-conditions. The analysis is based on the transformation of the beam operator i...

If A∈Mn is totally positive (TP), we determine the maximum open interval I around the origin such that, if μ∈I, then A−μI is TP. If A is TP, μ∈I and A−μI=LU, then B defined by B−μI=UL is TP, and has the same total positivity interval I. If A is merely nonsingular and totally nonnegative (TN), or oscillatory, there need be no such interval in which...

We study the inverse generalized eigenvalue problem (IGEP) Ax = λBx, in which A is a Jacobi matrix with positive off-diagonal
entries ci>0, and B = diag(b1,b2,...,bN), where
bi≠0 for i = 1,2,...,N. We use the concept of
m-functions to solve the IGEP, which corresponds to the
continuous case of the inverse problem.

We study a generalized inverse eigenvalue problem (GIEP), Ax=ÃŽÂ»Bx, in which A is a semi-infinite Jacobi matrix with positive off-diagonal entries ci>0, and B=Ã¢Â€Â‰diagÃ¢Â€Â‰(b0,b1,Ã¢Â€Â¦), where biÃ¢Â‰Â 0 for i=0,1,Ã¢Â€Â¦. We give an explicit solution by establishing an appropriate spectral function with respect to a given set of spectral data.

## Projects

Projects (3)

In this project, we are interested in study of the fractional order differential, difference, integral and sum equations consisting of linear, half-linear and non-linear boundary value problems and coupled systems subject to various classes of boundary conditions such as multi-point, integral and strip ones considering the possibility of nonlinearity for boundary conditions. The main tool of this project to reach the solutions of the mentioned problems concerns with the fixed point theory and integral transformations.