Superconvergence Results for Weakly Singular Fredholm–Hammerstein Integral Equations

Mandal, Moumita; Nelakanti, Gnaneshwar

doi:10.1080/01630563.2018.1561468

Cited by 12 publications

(4 citation statements)

References 17 publications

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“…Next, we state the following lemma for weakly singular kernels, which can be proved in the same way as in Mandal and Nelakanti (2019a), that helps us to prove our next lemma.…”

mentioning

confidence: 94%

“…There are various numerical methods such as collocation, Galerkin, multi-collocation, multi-Galerkin, Nyström, Petrov-Galerkin, degenerate kernel methods, etc. available in the literature to solve the single Fredholm-Hammerstein integral equation (see, Atkinson 1997;Panigrahi 2019;Das et al 2015Das et al , 2016Xu 1991, 1996;Kaneko et al 1997Kaneko et al , 1992Kumar 1987;Kumar and Sloan 1987;Mandal and Nelakanti 2019a). The Galerkin method is a well established numerical method (see, Atkinson and Potra 1987;Atkinson 1997;Das et al 2015Das et al , 2014Das et al , 2016Mandal and Nelakanti 2019b) to solve the single nonlinear Fredholm integral equation.…”

Section: Introductionmentioning

confidence: 99%

“…In Chen et al (2007), Kant and Nelakanti (2020b), Kant and Nelakanti (2020a), , Long et al (2009), Mandal and Nelakanti (2018), Nahid and Nelakanti (2019) multi-projection (multi-collocation and multi-Galerkin) methods are proposed for solving single Fredholm integral equation and obtained superconvergence rates. In Mandal and Nelakanti (2019a), Mandal et al applied the multi-Galerkin and iterated multi-Galerkin methods for solving single weakly singular Fredholm-Hammerstein integral equation and obtained the improved superconvergence rates for iterated multi-Galerkin method. To the best of our knowledge, there is not much literature available in the study of convergence analysis for system of nonlinear Fredholm-Hammerstein integral equations.…”

Section: Introductionmentioning

confidence: 99%

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Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

Chakraborty

Nelakanti

2021

Comp. Appl. Math.

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In this article, we apply projection methods and their iterated versions to approximate the solution of system of Fredholm-Hammerstein integral equations with both smooth and weakly singular kernels of algebraic and logarithmic type using the piecewise polynomial basis functions. We show that the iterated Galerkin approximate solution converges to the exact solution faster than the Galerkin approximate solution for both smooth and weakly singular algebraic and logarithmic type kernels. We improve these results further in iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results. KeywordsSystem of Fredholm-Hammerstein integral equations • Galerkin method • Multi-Galerkin method • Smooth kernels • Weakly singular kernels • Superconvergence results Mathematics Subject Classification 45G15 • 45B05 • 65R20 B Samiran Chakraborty

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“…Next, we state the following lemma for weakly singular kernels, which can be proved in the same way as in Mandal and Nelakanti (2019a), that helps us to prove our next lemma.…”

mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

Chakraborty

Nelakanti

2021

Comp. Appl. Math.

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“…In attempt of improving the accuracy of the approximate solutions, projection methods are used to solve Fredholm integral equations. Several results on different projection methods to solve nonlinear Fredholm integral equations can be found in literature (see [11,12,17,19,21,22,25,26]). Classical projection methods such as Galerkin, collocation methods for Fredholm Hammerstein integral equations with smooth as well as weakly singular kernels were developed and superconvergence was obtained by several authors (see [9, 11, 12, 17-19, 21, 22, 26]) etc.).…”

Section: Introductionmentioning

confidence: 99%

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

Kant,

Das,

null

et al. 2024

AAMM

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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.

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Galerkin and multi-Galerkin methods for weakly singular Volterra–Hammerstein integral equations and their convergence analysis

Kant

Nelakanti

2020

Comp. Appl. Math.

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In this paper, we consider the Galerkin and iterated Galerkin methods to approximate the solution of the second kind nonlinear Volterra integral equations of Hammerstein type with the weakly singular kernel of algebraic type, using piecewise polynomial basis functions based on graded mesh. We prove that the Galerkin method converges with the order of convergence O(n −m), whereas iterated Galerkin method converges with the order O(n −2m) in uniform norm. We also prove that in iterated multi-Galerkin method the order of convergence is O(n −3m). Numerical results are provided to justify the theoretical results.

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Superconvergence Results for Weakly Singular Fredholm–Hammerstein Integral Equations

Cited by 12 publications

References 17 publications

Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

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

Galerkin and multi-Galerkin methods for weakly singular Volterra–Hammerstein integral equations and their convergence analysis

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