arXiv subjects

Shashank K. Shukla

Publications and source records attributed to Shashank K. Shukla.

4 recordsLinked to original sources

arXiv2026

A note on superconvergence in projection-based numerical approximations of eigenvalue problems for Fredholm integral operators

This paper studies the eigenvalue problem $K ψ= λψ$ associated with a Fredholm integral operator $K$ defined by a smooth kernel. The focus is on analyzing the convergence behaviour of numerical approximations to eigenvalues and their corresponding spectral subspaces. The interpolatory projection methods are employed on spaces of piecewise polynomials of even degree, using $2r+1$ collocation points that are not restricted to Gauss nodes. Explicit convergence rates are established, and the modified collocation method attains faster convergence of approximation of eigenvalues and associated eigenfunctions than the classical collocation scheme. Moreover, it is shown that the iteration yields superconvergent approximations of eigenfunctions. Numerical experiments are presented to validate the theoretical findings.

math.NA↗

arXiv2026

Projection-based approximations for eigenvalue problems of Fredholm integral operators with Green's kernels

We consider the eigenvalue problem $K x = λx$. Our analysis focuses on the convergence rates of eigenvalue and spectral subspace approximations for compact linear integral operator $K$ with Green's kernels. By employing orthogonal and interpolatory projections at $2r+1$ collocation points (which are not necessarily Gauss points) onto an approximating space of piecewise even degree polynomials, we establish the superconvergence of eigenfunctions under iteration. The modified projection methods achieve a faster convergence rates compared to classical projection methods. The enhancement in convergence rate is verified by numerical examples.

math.NA↗

arXiv2025

Acceleration of convergence in approximate solutions of Urysohn integral equations with Green's kernels

Consider a non-linear operator equation $x - K(x) = f$, where $f$ is a given function and $K$ is a Urysohn integral operator with Green's function type kernel defined on $L^\infty [0, 1]$. We apply approximation methods based on interpolatory projections onto the approximating space $\mathcal{X}_n$, which is the space of piecewise polynomials of even degree with respect to a uniform partition of $[0, 1]$. The approximate solutions obtained from these methods demonstrate enhanced accuracy compared to the classical collocation solution for the same equation. Numerical examples are given to support our theoretical results.

math.NA↗

arXiv2024

A note on improvement by iteration for the approximate solutions of second kind Fredholm integral equations with Green's kernels

Consider a linear operator equation $x - Kx = f$, where $f$ is given and $K$ is a Fredholm integral operator with a Green's function type kernel defined on $C[0, 1]$. For $r \geq 0$, we employ the interpolatory projection at $2r + 1$ collocation points (not necessarily Gauss points) onto a space of piecewise polynomials of degree $\leq 2r$ with respect to a uniform partition of $[0, 1]$. Previous researchers have established that, in the case of smooth kernels with piecewise polynomials of even degree, iteration in the collocation method and its variants improves the order of convergence by projection methods. In this article, we demonstrate the improvement in order of convergence by modified collocation method when the kernel is of Green's function type.

math.NA↗