On Romanovski–Jacobi polynomials and their related approximation results

Abstract

The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F-distribution over the positive real line. We introduce some basic properties of the Romanovski-Jacobi polynomials, the Romanovski-Jacobi-Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski-Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes. KEYWORDS differentiation matrices, finite orthogonal polynomials, Gauss-type quadrature, Romanovski-Jacobi polynomials, spectral methods 1 INTRODUCTION The different families of classical orthogonal polynomials are nowadays part of the basic mathematical machinery of numerous physical, engineering, and mathematical algorithms and methodologies [1-3]. The classical hypergeometric polynomials of Hermite, Laguerre, and Jacobi are infinite types

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

“…consequently, the estimate (48) follows directly from ( 53), (54), and (47). Next, in order to prove (49), we recall the Gamma function's property:…”

“…Equation ( 49) is obtained by combining ( 48) and ( 58). Furthermore, using (49) and the definition of D 1 L yields ( 50) and ( 51), respectively.…”

“…Spectral techniques (see, e.g., previous works [42][43][44][45][46][47][48][49] ) are approaches used in applied mathematics and scientific computing to numerically approximate the solutions of both linear and nonlinear differential and integral equations. The spectral tau, Galerkin, and collocation schemes are three well-known types of spectral techniques.…”

High‐order finite difference/spectral‐Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation

“…The space-fractional Schrödinger equation as special form of fractional Ginzburg-Landau equation was considered extensively in literature [35][36][37][38][39][40][41][42]. However, to the best of our knowledge, the L2 -1 𝜎 /Galerkin spectral method, which is an important method to solve fractional partial differential equations [43][44][45][46][47][48][49][50][51][52], has not been considered for the nonlinear time-space fractional Ginzburg-Landau equation.…”