The Discrete Multi-Projection Method for Fredholm Integral Equations of the Second Kind

Abstract

In this paper, discrete multi-projection methods are developed for solving the second kind Fredholm integral equations. We propose a theoretical framework for analysis of the convergence of these methods. The theory is then applied to establish super-convergence results for 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.

Superconvergence results for linear second-kind Volterra integral equations

“…In this section, we propose multi-Galerkin (M-Galerkin) and iterated multi-Galerkin (iterated M-Galerkin) methods (see [8,10,12]) for solving (1.1) and obtain the superconvergence results. To do this, we define the multi-projection operator K M n by…”

“…In [10], multi-projection and in [8], discrete multi-projection methods were proposed to solve Fredholm integral equations of second kind and showed that under the same assumptions as that of classical Galerkin and collocation methods, the proposed multi-Galerkin and multi-collocation methods exhibit superconvergence results over iterated Galerkin and iterated collocation methods. In this paper, we consider the Galerkin method and its iterated version to approximate the solution of Volterra integral equations of the form (1.1) with a smooth kernel using piecewise polynomial basis functions.…”

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

“…We know from Proposition 3.1 and Proposition 3.2 in [7] that the operator P n satisfies the conditions (H1) and (H2), and the approximation operator K n satisfies the conditions (H3) and (H4).Using the approximate operators P n and K n , the approximate scheme (2.4) leads to the discrete M-Galerkin method: finding k n 2 rðK M n Þ n f0g and / n 2 X with k/ n k = 1 such that K M n / n ¼ k n / n : ð3:8Þ…”

“…(1.2) gives the Galerkin and collocation methods. Using the notations in [7], the operator K can be written as the following matrix form:…”

Discrete septic spline quasi-interpolants for solving generalized Fredholm integral equation of the second kind via three degenerate kernel methods

“…Denote by R F n , R R n and R RF n error terms for the above three septic spline degenerate kernel method, respectively. We compare our methods with other methods such as discrete Galerkin methods and discrete collocation methods given in [8], Nyström methods given in [5], Iteration methods given in [4] and PetrovGalerkin elements via Chebyshev polynomials described in [2].…”