Special Matrix Functions Research Articles

Multi‐soliton solutions of the three‐level coupled Schrödinger Maxwell‐Bloch equations via the Riemann‐Hilbert approach

In this paper, a study of the coupled Schrödinger Maxwell‐Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three‐level atoms, is presented via the Riemann‐Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann‐Hilbert problem is established successfully through a standard dressing procedure via the Riemann‐Hilbert approach, and then the potential function related to the solution of the Riemann‐Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann‐Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann‐Hilbert problem without reflection for the coupled Schrödinger Maxwell‐Bloch equation, and the multi‐soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters.

Some Relations on the rRs(P,Q,z) Matrix Function

In this paper, we derive some classical and fractional properties of the rRs matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials.

Euler-type integrals for the generalized hypergeometric matrix function

Abstract The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 {|z|=1} . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.

From Fractal Behavior of Iteration Methods to an Efficient Solver for the Sign of a Matrix

Investigating the fractal behavior of iteration methods on special polynomials can help to find iterative methods with global convergence for finding special matrix functions. By employing such a methodology, we propose a new solver for the sign of an invertible square matrix. The presented method achieves the fourth rate of convergence by using as few matrix products as possible. Its attraction basin shows larger convergence radii, in contrast to its Padé-type methods of the same order. Computational tests are performed to check the efficacy of the proposed solver.

Refined decay bounds on the entries of spectral projectors associated with sparse Hermitian matrices

Spectral projectors of Hermitian matrices play a key role in many applications, such as electronic structure computations. Linear scaling methods for gapped systems are based on the fact that these special matrix functions are localized, which means that the entries decay rapidly away from the main diagonal or with respect to more general sparsity patterns. The relation with the sign function together with an integral representation is used to obtain new decay bounds, which turn out to be optimal in an asymptotic sense. The influence of isolated extremal eigenvalues on the decay properties is also investigated and a superexponential behaviour is predicted.

Further investigations on the two variables second Appell hypergeometric matrix function

Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second Appell hypergeometric matrix functions. Precisely, we derive certain mathematical properties such as limit formulae of Humbert hypergeometric matrix functions, recursion formulae using contiguous matrix relations, some differentiation and summation formulae. We also find general differential operators which induce some differential equations. This enriches the theory of special matrix functions. The obtained results are believed to be newly presented.

Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels, in this article we derive the formulas for Fourier cosine and sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms several results are obtained, which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.

On A Matrix Hypergeometric Differential Equation

In this paper we consider a matrix Hypergeometric differential equation, which are special matrix functions and solution of a specific second order linear differential equation. The aim of this work is to extend a well known theorem on Hypergeometric function in the complex plane to a matrix version, and we show that the asymptotic expansions of Hypergeometric function in the complex plane ” that are given in the literature are special members of our main result. Background and motivation are discussed.

On Hankel transforms of generalized Bessel matrix polynomials

The present article deals with the evaluation of the Hankel transforms involving Bessel matrix functions in the kernel. Moreover, these transforms are associated with products of certain elementary functions and generalized Bessel matrix polynomials. As applications, many useful special cases are discussed. Further, the current results are more general to the previous one. In addition to, these results are yielded to more results in the modern integral transforms with special matrix functions.

A Note on Relative Controllability of Higher Order Linear Delayed Discrete Systems

This article is concerned with relative controllability of higher order linear discrete delayed systems with a single delay. By a formula that uses special matrix functions (discrete delayed matrix sine and cosine) to compute the solution of the initial problem, a criterion is found for relative controllability. In terms of special matrix functions, a control function is constructed to solve the problem of relative controllability. It is shown that the transformation of a given system into a higher dimensional system without delays leads to a nonequivalent problem of relative controllability. The results are illustrated by an example.

Lie algebras of matrix difference differential operators and special matrix functions

We discuss certain models of irreducible representations of Lie algebra sl(2,C) from the special matrix functions point of view. These models are constructed in terms of matrix differential operators and matrix difference differential operators and are connected through a matrix integral transformation. In the process, we find new matrix function identities involving one and two variable special matrix functions.

On the incomplete second Appell hypergeometric matrix functions

ABSTRACT In this paper, we introduce the incomplete second Appell hypergeometric matrix functions in terms of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, recurrence relation, recursion formula and differentiation formula of the incomplete second Appell hypergeometric matrix functions. This enriches the theory of special matrix functions. The results listed here are believed to be new.

Models of Lie algebra sl(2,C) and special matrix functions by means of a matrix integral transformation

In this paper, we present four models of irreducible representations of special complex Lie algebra s l ( 2 , C ) from the special matrix functions point of view. These models, which involve differential operators, are transformed into matrix difference–differential operators using an integral transformation motivated by the integral representation of beta matrix function. We also obtain the matrix identities involving one or two variable special matrix functions.

Computation of matrix gamma function

Matrix functions have a major role in science and engineering. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other special matrix functions, is the matrix gamma function. This research article focus on the numerical computation of this function. Well-known techniques for the scalar gamma function, such as Lanczos, Spouge and Stirling approximations, are extended to the matrix case. This extension raises many challenging issues and several strategies used in the computation of matrix functions, like Schur decomposition and block Parlett recurrences, need to be incorporated to make the methods more effective. We also propose a fourth technique based on the reciprocal gamma function that is shown to be competitive with the other three methods in terms of accuracy, with the advantage of being rich in matrix multiplications. Strengths and weaknesses of the proposed methods are illustrated with a set of numerical examples. Bounds for truncation errors and other bounds related with the matrix gamma function will be discussed as well.

Further results on the generalised hypergeometric matrix functions

In recent years, various results of the special matrix functions have been established in many papers. In the present paper, we have developed certain properties involving generalised hypergeometric matrix functions, such as, integral representations and reduction formulae. Also, open problems concerning the generalised hypergeometric matrix functions are stated.

Special matrix functions: characteristics, achievements and future directions

ABSTRACTIn recent years, special matrix functions and polynomials of a real or complex variable have been in a focus of increasing attention leading to new and interesting problems. In this work, we present matrix space analogues to generalized some functions and polynomials in the framework of matrix setting. Many of the special matrix functions and polynomials are constructed along standard procedures. Recently published papers are also surveyed and we list the most essential ones.

New kind of special matrix functions and generating functions

In this paper, a new kind of special matrix functions is introduced and some properties of these special matrix function are established. Further, some generating functions for the 3-variable Hermite matrix-based Laguerre polynomials involving the special matrix function are derived by using operational methods.

Глобальная экспоненциальная устойчивость для дифференциальных систем с нелинейными задержками

Березанский Леонид, отделение математики, Университет имени Бен-Гуриона, Беэр-Шева, Израиль; brznsky@cs.bgu.ac.il. Leonid Berezansky, brznsky@cs.bgu.ac.il Ben-Gurion University of the Negev, Beer-Sheva, Israel

On the incomplete hypergeometric matrix functions

By means of the familiar incomplete gamma matrix functions \(\gamma (A,x)\) and \(\Gamma (A,x)\), we introduce the incomplete Pochhammer matrix symbols that lead us to a generalization and decomposition of the incomplete hypergeometric matrix functions (IHMFs). Some properties such as a matrix differential equation, integral expressions and recurrence relations of IHMFs are given. Besides, connections between these matrix functions and other special matrix functions are investigated.

Control of Oscillating Systems with a Single Delay

Systems are considered related to the control of processes described by oscillating second-order systems of differential equations with a single delay. An explicit representation of solutions with the aid of special matrix functions called a delayed matrix sine and a delayed matrix cosine is used to develop the conditions of relative controllability and to construct a specific control function solving the relative controllability problem of transferring an initial function to a prescribed point in the phase space.