A superconvergent projection method for nonlinear compact operator equations

Grammont, Laurence; Kulkarni, Rekha P.

doi:10.1016/j.crma.2005.11.011

Cited by 11 publications

(3 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also, authors had shown that, the Computational complexities are almost same in modified projection methods as classical projection methods. After that, modified projection methods have been applied in several papers for solving nonlinear type integral equations with smooth kernels (see [13,14] etc). Now, in [24], M. Mandal et.…”

Section: Introductionmentioning

confidence: 99%

Modified Galerkin Method for Derivative Dependent Fredholm–Hammerstein Integral Equations of Second Kind

Kapil Kant,

Payel Das,

Gnaneshwar Nelakanti

et al. 2024

AAMM

View full text Add to dashboard Cite

In this paper, we consider modified Galerkin and iterated modified Galerkin methods for solving a class of two point boundary value problems. The methods are applied after constructing the equivalent derivative dependent Fredholm-Hammerstein integral equations to the boundary value problem. Existence and convergence of the approximate solutions to the actual solution is discussed and the rates of convergence are obtained. Superconvergence results for the approximate and iterated approximate solutions of piecewise polynomial based modified Galerkin method in infinity norm are given. We have also established that iterated modified Galerkin approximation improves over the modified Galerkin solution. Numerical examples are presented to illustrate the theoretical results.

show abstract

Section: Introductionmentioning

confidence: 99%

Modified Galerkin Method for Derivative Dependent Fredholm–Hammerstein Integral Equations of Second Kind

Kapil Kant,

Payel Das,

Gnaneshwar Nelakanti

et al. 2024

AAMM

View full text Add to dashboard Cite

show abstract

“…In [12], the author first introduces multi-projection methods with orthogonal and interpolatory projection operators for linear compact operator equations.In ( [16]), authors proposed Legendre multi-projection method to solve the eigenvalue problem (1.3) and showed that the proposed multi-Galerkin and multi-collocation methods exhibit superconvergence results over the Galerkin and collocation solutions. In [9], authors propose modified projection operator for nonlinear compact operator equation. For linear equations, modified projection operator is not other than multi-projection operator.…”

Section: Introductionmentioning

confidence: 99%

Discrete Legendre multi-projection methods for Fredholm integral equations of the second kind and the corresponding eigenvalue problem

Panigrahi¹

2018

Communications in Numerical Analysis

View full text Add to dashboard Cite

In this paper, the discrete multi-projection methods are proposed to solve Fredholm integral equations of the second kind with the corresponding eigenvalue problem using Legendre polynomial bases. The error bounds for the approximate, iterated approximate solutions for Fredholm integral equations of the second kind are evaluated using sufficiently accurate numerical quadrature rule. It has been shown that iterated Legendre multi-Galerkin solution converges faster than Legendre multi-Galerkin solutions in both infinity and L 2-norm and Legendre multi-projection method converges faster than Legendre projection method. The convergence rates for approximated eigenelements in Legendre multi-projection methods also have been evaluated. Numerical examples are presented to validate the theoretical estimates for both the problems.

show abstract

“…In the collocation method, the above equation is approximated by φ C n − Q n K(φ C n ) = Q n f. This method has been studied extensively in research literature, see Krasnoselskii In Grammont and Kulkarni [7], the following modified projection method is proposed:…”

mentioning

confidence: 99%

Approximate solution of Urysohn integral equations with non-smooth kernels

Kulkarni¹,

Nidhin²

2016

J. Integral Equations Applications

View full text Add to dashboard Cite

Consider a nonlinear operator equation x − K(x) = f , where K is a Urysohn integral operator with a kernel of the type of Green's function and defined on L ∞ [0, 1]. For r ≥ 0, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree ≤ r with respect to a quasi-uniform partition of [0, 1] and consider an interpolatory projection at r + 1 Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are r + 1 and r + 2 + min{r, 1}, respectively. We show that the order of convergence in the iterated modified projection method is 4 if r = 0 and is 2r + 3 if r ≥ 1. This improvement in the order of convergence is achieved while retaining the size of the system of equations that needs to be solved, the same as in the case of the collocation method. Numerical results are given for specific examples. 2010 AMS Mathematics subject classification. Primary 45L10, 65J15, 65R20. Keywords and phrases. Urysohn integral operator, collocation method, Gauss points.The first author would like to thank the Indo-French Centre for Applied Mathematics (IFCAM), Bangalore, India, for the partial support.

show abstract

A superconvergent projection method for nonlinear compact operator equations

Cited by 11 publications

References 7 publications

Modified Galerkin Method for Derivative Dependent Fredholm–Hammerstein Integral Equations of Second Kind

Modified Galerkin Method for Derivative Dependent Fredholm–Hammerstein Integral Equations of Second Kind

Discrete Legendre multi-projection methods for Fredholm integral equations of the second kind and the corresponding eigenvalue problem

Approximate solution of Urysohn integral equations with non-smooth kernels

Contact Info

Product

Resources

About