the main purpose of this article is to present an approximate solution for the two-dimensional nonlinear volterra integral equations using legendre orthogonal polynomials. first, the two-dimensional shifted legendre orthogonal polynomials are defined and the properties of these polynomials are presented. the operational matrix of integration and the product operational matrix are introduced. th...

A new method of obtaining exact solution for the rate matrix R in the Matrix-Analytic method in case of the phase state of dimension two is proposed. The method is based on symbolic solution of the determinental polynomial equation, and obtaining a linear matrix equation for the unknown rate matrix R by Cayley–Hamilton theorem. The method is applied to analyze the EnergyPerformance tradeoff of ...

We prove a comparison theorem for the solutions of a rational matrix difference equation, generalizing the Riccati difference equation, and existence and convergence results for the solutions of this equation. Moreover we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing or almost stabilizing solution. AMS Classification: 39A10, 93E03.

In this paper, we propose a method to approximate the solution of a linear Fredholm integro-differential equation by using the Chebyshev wavelet of the first kind as basis. For this purpose, we introduce the first Chebyshev operational matrix of integration. Chebyshev wavelet approximating method is then utilized to reduce the integro-differential equation to a system of algebraic equations. Il...

This paper offers necessary and sufficient condition for delay-dependent asymptotic stability of linear discrete systems with multiple time delay. The criterion is derived by Lyapunov’s direct method and two matrix equations have been obtained: discrete Lyapunov matrix equation for nondelay systems and polynomial matrix equation. Obtained stability condition does not possess conservatism but re...

This paper considers the solution to a class of the second-order Sylvester matrix equation EVF2 −AVF −CV = BW . Under the controllability of the matrix triple (E,A,B), a complete, general, and explicit parametric solution to the second-order Sylvester matrix equation, with the matrix F in a diagonal form, is proposed. The results provide great convenience to the analysis of the solution to the ...

We consider the algebraic Riccati equation for which the four coefficient matrices form an M -matrix K. When K is a nonsingular M -matrix or an irreducible singular M -matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M ...

We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equation AXB + CXHD = F. When the matrix equation is consistent over reflexive matrix X, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be d...

We consider the nonlinear matrix equation X = Q+A (I⊗X−C)A (0 < δ ≤ 1), where Q is an n× n positive definite matrix, C is an mn×mn positive semidefinite matrix, I is the m×m identity matrix, and A is an arbitrary mn×n matrix. We prove the existence and uniqueness of the solution which is contained in some subset of the positive definite matrices under the condition that I ⊗Q > C. Two bounds for...