Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.     

An efficient reproducing kernel method for solving the Allen–Cahn equation

In this paper, an efficient reproducing kernel method combined with the finite difference method and the Quasi-Newton method is proposed to solve the Allen–Cahn equation. Based on the Legendre polynomials, we construct a new reproducing kernel function with polynomial form. We prove that the semi-scheme can preserve the energy dissipation property unconditionally. Numerical experiments are given to show the efficiency and validity of the proposed scheme.     

Indefinite integrals of special functions from inhomogeneous differential equations

A method is presented for deriving integrals of special functions which obey inhomogeneous second-order linear differential equations. Inhomogeneous equations are readily derived for functions satisfying second-order homogeneous equations. Sample results are derived for Bessel functions, parabolic cylinder functions, Gauss hypergeometric functions and the six classical orthogonal polynomials. For the orthogonal polynomials the method gives indefinite integrals which reduce to the usual orthogonality conditions on the usual orthogonality intervals. These indefinite integrals for the orthogonal polynomials appear to be new. All results have been checked with Mathematica.     

Detachment of a Beam Glued to a Rigid Plate

The contact problem of detachment of an elastic beam glued to a rigid plate is considered. A mathematical model is proposed and theoretical results are compared with experimental data.     

Fractal measures and polynomial sampling: I.F.S.–Gaussian integration

Measures generated by Iterated Function Systems can be used in place of atomic measures in Gaussian integration. A stable algorithm for the numerical solution of the related approximation problem – an inverse problem in fractal construction – is proposed. Dedicated to Walter Gautschi.     

Interlacing properties of roots of certain biorthogonal polynomials

In this paper we consider the biorthogonal polynomials with respect to the measure e^-x4-y2+2τxydxdy, and show that their roots interlace. The proof involves showing total nonnegativity of matrices related to Jacobi type matrices.     

On Extrapolation of Electromagnetic Responses in Time and Frequency Domains

The complete electromagnetic responses from conducting objects in time and frequency domains are generated by using their early time and low frequency information. Utilizing two kinds of Hermite polynomials and their Fourier transform, the time-domain signal and its corresponding frequency response can be expressed as a weighted sum of these quantities in an efficient way. The general properties of these two families of Hermite functions are studied, which greatly affect the performance of the proposed method. Due to the performance of the algorithm being sensitive to the choice of the origin and the scaling factor, how to properly choose the initial values of these parameters is considered. An optimal algorithm is also developed to find the above parameters so as to achieve the best performance. A criterion is also provided to assess the sensitivity of the performance. The excellent agreement between the computed results by the proposed method and those obtained by earlier approaches is demonstrated in each case. This work was supported in part by natinal nature science foundatin of China (NO. 60432040).     

一类广义插值在LM^Ba空间中的逼近

先引入了由一列Orlicz空间生成的Ba空间(LMBa)的定义,然后用分数阶α的连续模给出一类广义插值在LBMa空间中逼近阶.     

Properties of convergence for -Bernstein polynomials

In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q⁻¹,…,q⁻ⁿ⁺¹. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), q_n(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if lim_n→∞q_n=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.