Bijaya Laxmi Panigrahi

Legendre multi-projection methods for solving eigenvalue problems for a compact integral operator

In this paper, we consider the M-Galerkin and M-collocation methods for solving the eigenvalue problem for a compact integral operator with smooth kernels, using Legendre polynomial bases. We obtain error bounds for the eigenvalues and the gap between the spectral subspaces in both Legendre M-Galerkin and Legendre M-collocation methods. We also obtain superconvergence results for the eigenvalues and iterated eigenvectors in both L^2 and infinity norm. We illustrate our results with numerical examples.

Discrete multi-projection methods for eigen-problems of compact integral operators

Abstract In this paper, a discrete multi-projection method is developed for solving the eigenvalue problem of a compact integral operator with a smooth kernel. We propose a theoretical framework for analysis of the convergence of these methods. The theory is then applied to establish super-convergence results of the corresponding discrete Galerkin method, collocation method and their iterated solutions. Numerical examples are presented to illustrate the theoretical estimates for the error of these methods.

Wavelet Galerkin method for eigenvalue problem of a compact integral operator

Abstract We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by the Galerkin method using wavelet bases. By truncating the Galerkin operator, we obtain a sparse representation of a matrix eigenvalue problem. We prove that the error bounds for the eigenvalues and for the distance between the spectral subspaces are of the orders O ( n μ - 2 nr ) and O ( μ - nr ) , respectively, where μ−n denotes the norm of the partition and r denotes the order of the wavelet basis functions. By iterating the eigenvectors, we show that the error bounds for the eigenvectors are of the order O ( n μ - 2 nr ) . We illustrate our results with numerical results.

Legendre multi-Galerkin methods for Fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem

Abstract In this paper, we consider Legendre multi-Galerkin methods to solve Fredholm integral equations of the second kind with weakly singular kernel and the corresponding eigenvalue problem. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind in both L 2 and infinity-norm. We also establish error bounds of approximated eigenelements with exact eigenelements in the eigenvalue problem of a compact integral operator with weakly singular kernel in both infinity and L 2 -norm. Numerical examples are presented to prove the theoretical estimates.

Hybrid collocation methods for eigenvalue problem of a compact integral operator with weakly singular kernel

Abstract In this paper, we consider the hybrid collocation methods to solve the eigenvalue problem of a compact integral operator with weakly singular kernels of algebraic and logarithmic type. We obtain the global convergence rates for eigenvalues, the gap between the spectral subspaces and iterated eigenvectors. The numerical examples are presented to verify the theoretical estimates and also shown that this method is computationally useful in comparison to other methods.

Richardson Extrapolation of Iterated Discrete Galerkin Method for Eigenvalue Problem of a Two Dimensional Compact Integral Operator

We consider approximation of eigenelements of a two-dimensional compact integral operator with a smooth kernel by discrete Galerkin and iterated discrete Galerkin methods. By choosing numerical quadrature appropriately, we obtain superconvergence rates for eigenvalues and iterated eigenvectors, and for gap between the spectral subspaces. We propose an asymptotic error expansions of the iterated discrete Galerkin method and asymptotic error expansion of approximate eigenvalues. We then apply Richardson extrapolation to obtain improved error bounds for the eigenvalues. Numerical examples are presented to illustrate theoretical estimate.

Error analysis of Jacobi spectral collocation methods for Fredholm-Hammerstein integral equations with weakly singular kernel

ABSTRACT The Jacobi spectral collocation method is being proposed to solve Fredholm-Hammerstein integral equations with the weakly singular kernel and smooth solutions. By using the appropriate quadrature rule, the integral operator is approximated by the discrete operator, which gives rise to the Jacobi collocation method for Fredholm-Hammerstein integral equations with the weakly singular kernel. The convergence analysis for the approximated solution with the exact solution is being discussed for both weighted -norm and infinity-norm. Numerical examples are presented to validate the theoretical estimate.