and Applied Analysis 3 nonlinear term is treated with the Chebyshev collocation method. The time discretization is a classical Crank-Nicholson-leap-frog scheme. Yuan and Wu 43 extended the Legendre dual-Petrov-Galerkin method proposed by Shen 44 , further developed by Yuan et al. 45 to general fifth-order KdV-type equations with various nonlinear terms. The main aim of this paper is to propose ...

Multimode/multicore fibers are expected to provide an attractive solution to overcome the capacity limit of current optical communication system. In presence of high crosstalk between modes/cores, the squared singular values of the input/output transfer matrix follow the law of the Jacobi ensemble of random matrices. Assuming that the channel state information is only available at the receiver,...

‎‎‎‎‎‎‎‎‎‎‎‎‎This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product‎. ‎The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra integro--differential equations‎ which appear in various fields of science such as physics and engineering. ‎The Operational matr...

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the Eulerian polynomials and the shifted Eulerian polynomials are moment sequences for a simple family of orthogonal polynomials. The coefficient arrays of these families of orthogonal polynomials are shown to be exponential Riordan arrays. Using the theory of orthogonal polynomials we are then able t...

The list of physically motivated urn models that can be solved in terms classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed S. Laplace P. T. Ehrenfest eventually connected with the Krawtchouk Hahn polynomials. This connection was reversed recently case Jacobi where rather contrived, later simpler proposed. Here we consider an going J...

The Volterra delay integral equations have numerous applications in various branches of science, including biology, ecology, physics and modeling of engineering and natural sciences. In many cases, it is difficult to obtain analytical solutions of these equations. So, numerical methods as an efficient approximation method for solving Volterra delay integral equations are of interest to many res...

By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Tra...

For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. The general approach is illustrated by the examples of the classical orthogonal polynomials: Hermite, Jacobi and La...

The bivariate big q-Jacobi polynomials are defined by [3] Pn,k(x, y; a, b, c, d; q) := Pn−k(y; a, bcq , dq; q) y(dq/y; q)k Pk (x/y; c, b, d/y; q) (n ≥ 0; k = 0, 1, . . . , n), where q ∈ (0, 1), 0 < aq, bq, cq < 1, d < 0, and Pm(t;α, β, γ; q) are univariate big q-Jacobi polynomials, Pm(t;α, β, γ; q) := 3φ2 ( q−m, αβq, t αq, γq ∣∣∣∣ q; q) (m ≥ 0) (see, e.g., [1, Section 7.3]). We give structure r...

We obtain the time-dependent correlation function describing the evolution of a single spin excitation state in a linear spin chain with isotropic nearestneighbour XY coupling, where the Hamiltonian is related to the Jacobi matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case, an arbitrary element of the correlation function is expressed in a simple closed form. Its asy...