مجله موجکها و جبر خطی
سال پنجم شماره 3 (Winter and Spring 2018)

In this paper, we apply the local discontinuous Galerkin method for solving fractional ordinary differential equations, in general.  In this method, choosing the (natural) numerical upwind flux enables us to solve the initial value problems for ordinary fractional equations interval by interval and forward in time. This means that we require to solve a low-order (k+1) × (k+1) system of equations locally in each subinterval, and there is no need to solve the global system; Here k is the degree of the basis functions in each subinterval. To implement the method, we consider the (local) basis functions as the (shifted) Legendre polynomials. This, in turn, makes some of the coefficient matrices in the system of equations sparse and thus accelerates the computations. Also the stability in the infinity norm and the error estimation of the method are discussed. Finally, with a series of linear and nonlinear examples, we show the efficiency and, in particular, the accuracy of the local discontinuous Galerkin method for fractional differential equations.

Recently, K-fusion frames are introduced as an extension of discrete K-frames which are suitable tools for some problems in sampling theory where can be analyzed by fusion frames. In this paper, we obtain several equalities and inequalities on K-fusion frames and Parseval K-fusion frames, these results generalize and improve some important equalities and inequalities on discrete and fusion frames.

In the present paper, we prove that the N-soliton solutions of K-P equation found via inverse Scattering Transform (IST) are equivalent to Wronskian of some functions. This equation is one of the fundamental equations in soliton theory and is a general form of KDV equation. During the proof of Hirota derivative, vondermond matrices and Plackre relation are applied.

According to the application of fusion and g-fusion frames in data processing, iterative reconstruction of the frame is of particular importance when a member of the frame is removed. In this paper, we present a way of reconstruction of g-fusion frames and introduce the error operator and we get the upper bound for it. Also, the approximation operator for these frames will be introduced and we study some results about them.

In this paper, using the eigenvalues of positive matrices and the numerical Popoviciu inequality, we present this inequality for the trace of positive matrices. Moreover, considering matrix functions of negative power, we obtain some matrix inequalities of Popoviciu type. The results of this paper are reverses of some known matrix inequalities.

In this paper, using  polynomial numerical hulls of matrices, the convergence of GMRES restarted method for linear equations systems in which their coefficients are block companion matrices of monic matrix polynomials, are investigated. The relationship between the polynomial numerical hulls of a matrix polynomial and the polynomial numerical hulls of its companion linearization are also studied.