4 research outputs found
A NSFD discretization of two-dimensional singularly perturbed semilinear convection-diffusion problems
Despite the availability of an abundant literature on singularly perturbed problems,
interest toward non-linear problems has been limited. In particular, parameter-uniform
methods for singularly perturbed semilinear problems are quasi-non-existent. In this
article, we study a two-dimensional semilinear singularly perturbed convection-diffusion
problems. Our approach requires linearization of the continuous semilinear problem
using the quasilinearization technique. We then discretize the resulting linear problems
in the framework of non-standard finite difference methods. A rigorous convergence
analysis is conducted showing that the proposed method is first-order parameter-uniform
convergent. Finally, two test examples are used to validate the theoretical findings
On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight
Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight |x|αexp(−cx6),c>0,α>−1. Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials
On certain properties and applications of the perturbed Meixner–Pollaczek weight
This paper deals with monic orthogonal polynomials orthogonal with a perturbation
of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek
polynomials, are described by their weight function emanating from an exponential deformation
of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties
such as moments of finite order, some new recursive relations, concise formulations, differentialrecurrence
relations, integral representation and some properties of the zeros (quasi-orthogonality,
monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials.
Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such
as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–
Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.The NMU Council postdoctoral fellowshiphttps://www.mdpi.com/journal/mathematicsam2022Mathematics and Applied Mathematic
On certain properties of perturbed Freud-type weight: a revisit
In this paper, monic polynomials orthogonal with deformation of the
Freud-type weight function are considered. These polynomials fullfill linear
differential equation with some polynomial coefficients in their holonomic
form. The aim of this work is explore certain characterizing properties of
perturbed Freud type polynomials such as nonlinear recursion relations, finite
moments, differential-recurrence and differential relations satisfied by the
recurrence coefficients as well as the corresponding semiclassical orthogonal
polynomials. We note that the obtained differential equation fulfilled by the
considered semiclassical polynomials are used to study an electrostatic
interpretation for the distribution of zeros based on the original ideas of
Stieltjes.Comment: 17 page