New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation

In this paper, I introduce a class of super Bell polynomials, which are found to play an important role in the characterization of super supersymmetric equations. An effective approach based on the use of the super Bell polynomials is developed to systematically investigate the bilinearization, Bäcklund transformation, and Lax pair for supersymmetric equations. I take a supersymmetric two‐boson equation to illustrate this procedure. A new bilinear Bäcklund transformation and a Lax pair with both fermionic and bosonic parameters are given. In addition, a kind of exact solitons for the equation are further constructed with the help of the bilinear Bäcklund transformation.     

Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials

In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B‐polynomials) of any degree and for any fractional‐order in terms of B‐polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree‐n polynomial on [0,1] into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

A uniform asymptotic expansion of the single variable Bell polynomials

In this paper, we investigate the uniform asymptotic behavior of the single variable Bell polynomials on the negative real axis, to which all zeros belong. It is found that there exists an ascending sequence {Z_k}₁^∞(−e,0) such that the polynomials are represented by a finite sum of infinite asymptotic series, each in terms of the Airy function and its derivative, and the number of series under this sum is 1 in the interval (−∞,Z₁) and k+1 in [Z_k,Z_k+1), k1. Furthermore, it is shown that an asymptotic expansion, also in terms of Airy function and its derivative, completed with error bounds, holds uniformly in (−∞,−δ] for positive δ.     

A look-ahead strategy for the implementation of some old and new extrapolation methods

Sequence transformations, used for accelerating the convergence, are related to biorthogonal polynomials. In the particular cases of theG-transformation and the Shanks transformation (that is the -algorithm of Wynn), there is a connection with formal orthogonal polynomials. In this paper, this connection is exploited in order to propose a look-ahead strategy for the implementation of these two transformations. This strategy, which is quite similar to the strategy used for treating the same type of problems in Lanczos-based methods for solving systems of linear equations, consists in jumping over the polynomials which do not exist, thus avoiding a division by zero (breakdown) in the algorithms, and over those which could be badly computed (near-breakdown) thus leading to a better numerical stability. Numerical examples illustrate the procedure.     

Limit cycles of generalized Liénard equations

The generalized Liénard equations of the form:

where F, g, and h are polynomials, are examined. It has been found that the results given by Blows, Lloyd and Lynch [1–5] for Liénard equations hold also for the generalized systems. A new result is also presented within this article.     

On the connectedness of numerical range of matrix polynomials

An investigation on nonconnectedness of numerical range for monic matrix polynomials L(λ) is undertaking here. The distribution of eigenvalues of L(λ) to the components of numerical range and some other algebraic properties are also presented.     

Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation

Transformations of the measure of orthogonality for orthogonal polynomials, namely Freud transformations, are considered. Jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations. Dynamics of the coefficients are described by generalized Toda equations. The classical Toda lattice equations are the simplest special case of dynamics of the coefficients under the Freud transformation of the measure of orthogonality. Also dynamics of Hankel determinants, its minors and other notions corresponding to the orthogonal polynomials are studied.     

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Q_n} associated with the inner product

, where p(x) = (1 − x)(1 + x)^β is the Jacobi weight function, ,β> − 1, A₁,B₁,A₂,B₂0 and p, q P, the linear space of polynomials with real coefficients. The hypergeometric representation (₆F₅) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q_n(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.     

Intrinsically parallel solution of systems of linear partial differential equations

We give here an intrinsically parallel algorithm, which solves systems of linear partial differential equations. The mathematical foundations of the algorithm rely upon a particular representation of polynomials on a structure called “hypercube”, introduced by Beauzamy-Frot-Millour, and use Bombieri's scalar product. This scalar product also permits a detailed study of the stability of the algorithm. Boundary conditions and compatibility conditions are handled by the algorithm in an intrinsically parallel manner. This algorithm has been implemented on a Connection Machine CM5, at the “Etablissement Technique Central de l'Armement” (Arcueil, France). We give here several numerical examples.     

The location of zeros of polynomials

Let A be a complex n×n matrix. p an equilibrated vectonal norm and x(A) the spectrial abscissa of A. Then, it is known [5] x(A)≤x(γ_p(A)) where γp is the matricial logarithmic derivative induced by p. We will make use of the above inequality to obtain regions in the plane which contain the zeros of complex polynomials.     

Covariance of Lax Pairs and Integrability of the Compatibility Condition

We study the joint covariance of Lax pairs (LPs) with respect to Darboux transformations (DT). The scheme is based on comparing general expressions for the transformed coefficients of a LP and its Frechet derivative. We use the compact expressions of the DT via a version of non-Abelian Bell polynomials. We show that the so-called binary version of Bell polynomials forms a convenient basis for specifying the invariant subspaces. Some nonautonomous generalizations of KdV and Boussinesq equations are discussed in this context. We consider a Zakharov–Shabat-like problem to obtain restrictions at a minimal operator level. The subclasses that allow a DT symmetry (covariance at the LP level) are considered from the standpoint of dressing-chain equations. The cases of the classical DT and binary combinations of elementary DTs are considered with possible reduction constraints of the Mikhailov type (generated by an automorphism). Examples of Liouville–von Neumann equations for the density matrix are considered as illustrations.     

An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations

In this article, we proposed an auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations. This method is called the Auxiliary Laplace Parameter Method (ALPM). The nonlinear terms can be easily handled by the use of Adomian polynomials. Comparison of the present solution is made with the existing solutions and excellent agreement is noted. The fact that the proposed technique solves nonlinear problems without any discretization or restrictive assumptions can be considered as a clear advantage of this algorithm over the numerical methods.     

Orthogonal polynomials and differential equations in neutron-transport and radiative-transfer theories

There is a set of orthogonal polynomials {g_n(x)} which plays a relevant role in the treatment of the case of anisotropic scattering in neutron-transport and radiative-transfer theories. They appear also in the spherical harmonics treatment of the isotropic scattering. These polynomials are orthogonal with respect to a weight function which is continuous in the interval [−1, + 1] and has a finite number of symmetric Dirac masses. Although some other structural properties of these polynomials (e.g., the three-term recurrence relation) as well as some properties of their zeros have been published, much more need to be known. In particular, neither the second-order differential equation nor the density of zeros (i.e., the number of zeros per unit of interval) of the polynomial g_n(x) have been found. Here we obtain the second-order differential equation in the case that these polynomials are hypergeometric, so leaving open the general case. Furthermore, the exact expressions of the moments around the origin of the density of zeros of g_n(x) are given in the general case. The asymptotic density of zeros is also pointed out. Finally, these polynomials are shown to belong to the Nevai's class.     

On the jordan form of the tensor product over fields of prime characteristic

Let A, B denote the companion matrices of the polynomials x^m,xⁿ over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A ^TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established.     

Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Rational solutions and special polynomials associated with the generalized K ₂ hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations including the Fordy-Gibbons equation. Differential-difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations Γ and −2Γ is established. Properties of the polynomials are studied. Differential-difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found.     

Spectral properties of solutions of hypergeometric-type differential equations

The second-order differential equation σ(x)y″ + τ(x)y′ + λy = 0 is usually called equation of hypergeometric type, provided that σ, τ are polynomials of degree not higher than two and one, respectively, and λ is a constant. Their solutions are commonly known as hypergeometric-type functions (HTFs). In this work, a study of the spectrum of zeros of those HTFs for which , v , and σ, τ are independent of ν, is done within the so-called semiclassical (or WKB) approximation. Specifically, the semiclassical or WKB density of zeros of the HTFs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy. Applications to the Gaussian and confluent hypergeometric functions as well as to Hermite functions are shown.     

Harmonic polynomials and their application in solving problems of the mathematical theory of cracks

We consider new spectral relations for harmonic polynomials of two variables. The basic relations of this type are the integral relations in a disk, which make it possible to use these polynomials to construct the solution of certain singular equations of Newtonian potential type, of both first and second kind, given on the disk. By use of these results we determine the eigenvalues of certain two-dimensional integral equations of second kind.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 88–93.     

Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials

It has been shown in Ferreira et al. [Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials, Adv. in Appl. Math. 31(1) (2003) 61–85], López and Temme [Approximations of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999) 131–146; The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001) 623–633] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations. In Ferreira et al. [Limit relations between the Hahn polynomials and the Hermite, Laguerre and Charlier polynomials, submitted for publication] we have established new asymptotic connections between the fourth level and the two lower levels. In this paper, we continue with that program and obtain asymptotic expansions between the fourth level and the third level: we derive 16 asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials. From these expansions, we also derive three new limits between those polynomials. Some numerical experiments show the accuracy of the approximations and, in particular, the accuracy in the approximation of the zeros of those polynomials.     

On the integrability of the (1+1)-dimensional and (2+1)-dimensional Ito equations

Under investigation in this paper are the (1+1)-dimensional and (2+1)-dimensional Ito equations. With the help of the Bell polynomials method, Hirota bilinear method and symbolic computation, the bilinear representations, N-soliton solutions, bilinear Bäcklund transformations and Lax pairs of these two equations are obtained, respectively. In particular, we obtain a new bilinear form and N-soliton solutions of the (2+1)-dimensional Ito equation. The bilinear Bäcklund transformation and Lax pair of the (2+1)-dimensional Ito equation are also obtained for the first time. Copyright © 2014 John Wiley & Sons, Ltd.     

Extending the notions of companion and infinite companion to matrix polynomials

The notion of infinite companion matrix is extended to the case of matrix polynomials (including polynomials with singular leading coefficient). For row reduced polynomials a finite companion is introduced as the compression of the shift matrix. The methods are based on ideas of dilation theory. Connections with systems theory are indicated. Applications to the problem of linearization of matrix polynomials, solution of systems of difference and differential equations and new factorization formulae for infinite block Hankel matrices having finite rank are shown. As a consequence, any system of linear difference or differential equations with constant coefficients can be transformed into a first order system of dimension n = deg det D.