Related papers: Differentiation by integration with Jacobi polynom…
This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
Without question regarding its pivotal significance, the computation of function derivatives carries substantial weight within a multitude of engineering and applied mathematical fields. These encompass optimization, the development of…
We propose a spectral method by using the Jacobi functions for computing eigenvalue gaps and their distribution statistics of the fractional Schr\"{o}dinger operator (FSO). In the problem, in order to get reliable gaps distribution…
This paper presents a method for the accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The methods uses spin-weighted spherical harmonics in the angular directions and…
As one myth of polynomial interpolation and quadrature, Trefethen [30] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation…
We investigate the problem of numerical differentiation of bivariate functions from weighted Wiener classes using Chebyshev polynomial expansions. We develop and analyze a new version of the truncation method based on Chebyshev polynomials…
This paper introduces a scalar numerical differentiator, represented as a system of nonlinear differential equations of any high order. We derive the explicit solution for this system and demonstrate that, with a suitable choice of…
The key difficulty to develop efficient high-order methods for integrating stochastic differential equations lies in the calculations of the multiple stochastic integrals. This letter suggests a scheme to compute the stochastic integrals…
The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytical Hamiltonian systems via the differential Galoisian obstruction. In this paper we give a new Morales-Ramis type theorem on the…
This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit…
We discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and…
For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving…
In this paper, we propose algorithms to compute differential Chow forms for prime differential ideals which are given by their characteristic sets. The main algorithm is based on an optimal bound for the order of a prime differential ideal…
In this contribution, we introduce the multiplicative Jacobi polynomials that arise as one of the solutions of the multiplicative Sturm-Liouville equation \begin{equation*} \frac{d^*}{dx}\left( e^{(1-x^2)\omega(x)}\odot \frac{d^*y}{dx}…
We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to the classical weight function for the Jacobi polynomials together with point masses at both…
We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$…
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…