Payel Das

Discrete Legendre spectral projection methods for Fredholm-Hammerstein integral equations

In this paper we discuss the discrete Legendre Galerkin and discrete Legendre collocation methods for Fredholm-Hammerstein integral equations with smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain optimal convergence rates for both discrete Legendre Galerkin and discrete Legendre collocation solutions in both infinity and L 2 -norm. Numerical examples are given to illustrate the theoretical results.

Legendre spectral projection methods for Urysohn integral equations

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n^-^r) in L^2-norm and O(n^1^2^-^r) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n^-^2^r) in both L^2-norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n^-^r) in both L^2-norm and infinity norm, n being the highest degree of the Legendre polynomial employed in the approximation and r being the smoothness of the kernel. We are able to obtain similar superconvergence rates for the iterated Galerkin solution for Urysohn integral equations with smooth kernel as in the case of piecewise polynomial basis functions.

Convergence analysis of discrete legendre spectral projection methods for hammerstein integral equations of mixed type

In this paper, we consider the discrete Legendre spectral Galerkin and discrete Legendre spectral collocation methods to approximate the solution of mixed type Hammerstein integral equation with smooth kernels. The convergence of the discrete approximate solutions to the exact solution is discussed and the rates of convergence are obtained. We have shown that, when the quadrature rule is of certain degree of precision, the rates of convergence for the Legendre spectral Galerkin and Legendre spectral collocation methods are preserved. We obtain superconvergence rates for the iterated discrete Legendre Galerkin solution. By choosing the collocation nodes and quadrature points to be same, we also obtain superconvergence rates for the iterated discrete Legendre collocation solution.

Legendre Spectral Projection Methods for Fredholm---Hammerstein Integral Equations

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equation. The convergence of the approximate solutions to the actual solution is discussed and the rates of convergence are obtained. We are able to obtain similar superconvergence rates for the iterated Legendre Galerkin solution for Hammerstein integral equations with smooth kernel as in the case of piecewise polynomial based Galerkin method.

Discrete Legendre Spectral Galerkin Method for Urysohn Integral Equations

ABSTRACT In this paper, we consider the discrete Legendre spectral Galerkin method to approximate the solution of Urysohn integral equation with smooth kernel. The convergence of the approximate and iterated approximate solutions to the actual solution is discussed and the rates of convergence are obtained. In particular we have shown that, when the quadrature rule is of certain degree of precision, the superconvergence rates for the iterated Legendre spectral Galerkin method are maintained in the discrete case.

Error Analysis of Discrete Legendre Multi-Projection Methods for Nonlinear Fredholm Integral Equations

ABSTRACT In this paper, polynomially-based discrete M-Galerkin and M-collocation methods are proposed to solve nonlinear Fredholm integral equation with a smooth kernel. Using sufficiently accurate numerical quadrature rule, we establish superconvergence results for the approximate and iterated approximate solutions of discrete Legendre M-Galerkin and M-collocation methods in both infinity and L2-norm. Numerical examples are presented to illustrate the theoretical results.

Corrigendum to: ``Convergence analysis of discrete legendre spectral projection methods for hammerstein integral equations of mixed type'' Applied Mathematics and Computation Volume 265, 15 August 2015, Pages 574–601

The authors regret the fact that throughout the paper, we assumed ‖Q n ‖ L 2 ≤ p, where p is a constant independent of n (see estimate (2.45)), which is not true. Although it is true that, ‖Q n u ‖ L 2 ≤ p 1 ‖ u ‖ ∞ , where p 1 is a constant independent of n . Since we used ‖Q n ‖ L 2 ≤ p, in the proof of Theorem 3.8, it is not correct to conclude the results of Theorems 3.8 and 3.9. In this corrigendum, we have modified the Theorem 3.8 and provided the result in the infinity norm. To do this, we made an additional assumption that, for j = 1 , 2 , ∂ j ∂t j ψ (0 , 1) i (t, x (t)) exists and are Lipschitz continuous in x , i.e., for any x 1 , x 2 ∈ R , ∃ d i, 1 , d i, 2 > 0 , i = 1 , 2 , . . . , m such that ∣∣∣∣ ∂ j ∂t j ψ (0 , 1) i (s, x 1 ) − ∂ j ∂t j ψ (0 , 1) i (s, x 2 ) ∣∣∣∣ ≤ d i, j | x 1 − x 2 | , ∀ s ∈ [ −1 , 1] , j = 1 , 2 , ∀ s ∈ [ −1 , 1] , and denote l 3 = max i =1 , 2 , ... ,m d i, 1 , l 4 = max i =1 , 2 , ... ,m d i, 2 . To prove the result of Theorem 3.8 in infinity norm, we have modified the

Convergence Analysis of Legendre Spectral Galerkin Method for Volterra-Fredholm-Hammerstein Integral Equations

In this paper, we analyze the Legendre spectral Galerkin method for a class of nonlinear Volterra-Fredholm mixed-type integral equations. Existence and convergence of the approximate and iterated approximate solutions to the exact solution are discussed and the rates of convergence are obtained. We prove that the iterated approximate solution improves over the approximate solution for Volterra-Fredholm-Hammerstein integral equations with smooth kernels. Also, we obtain optimal order of convergence for the iterated Legendre Galerkin method.