Towards the accuracy of iterative numerical methods for fuzzy Hammerstein–Fredholm integral equations

In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra–Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also provide successive iterations of the solution) and suitable cubature formulas for the numerical approximations. We discuss in detail a method using Picard iteration and the two-dimensional composite trapezoidal rule, giving convergence conditions and error estimates. The paper concludes with numerical experiments and a discussion of the methods proposed.

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Numerical Methods for Solving Fredholm Integral Equations of Second Kind

Abstract and Applied Analysis ◽

10.1155/2013/426916 ◽

2013 ◽

Vol 2013 ◽

pp. 1-17 ◽

Cited By ~ 7

Author(s):

S. Saha Ray ◽

P. K. Sahu

Keyword(s):

Integral Equation ◽

Numerical Methods ◽

Integral Equations ◽

Applied Mathematics ◽

Fredholm Integral Equations ◽

Nonlinear Fredholm Integral Equations ◽

Linear And Nonlinear ◽

Interdisciplinary Effort

Integral equation has been one of the essential tools for various areas of applied mathematics. In this paper, we review different numerical methods for solving both linear and nonlinear Fredholm integral equations of second kind. The goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations.

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Numerical Algorithms of Optimal Complexity for Weakly Singular Volterra Integral Equations

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2006-0027 ◽

2006 ◽

Vol 6 (4) ◽

pp. 436-442 ◽

Cited By ~ 4

Author(s):

A.N. Tynda

Keyword(s):

Numerical Methods ◽

Integral Equations ◽

Numerical Algorithms ◽

Volterra Integral Equations ◽

Volterra Equations ◽

Fredholm Integral Equations ◽

Optimal Complexity ◽

Weakly Singular ◽

Different Types ◽

Singular Kernels

AbstractIn this paper we construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the ”block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods.

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Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the SpacesHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X)

Abstract and Applied Analysis ◽

10.1155/2013/607204 ◽

2013 ◽

Vol 2013 ◽

pp. 1-20

Author(s):

İsmet Özdemir ◽

Ali M. Akhmedov ◽

Ö. Faruk Temizer

Keyword(s):

Banach Space ◽

Numerical Methods ◽

Banach Spaces ◽

Integral Equations ◽

Singular Integral ◽

Integral Operators ◽

Fredholm Integral Equations ◽

Bounded Linear ◽

Linear Integral ◽

Linear Integral Operators

The spacesHα,δ,γ((a,b)×(a,b),ℝ)andHα,δ((a,b),ℝ)were defined in ((Hüseynov (1981)), pages 271–277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419–2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the setsHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X)by taking an arbitrary Banach spaceXinstead ofℝ, and we show that these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms∥·∥α,δ,γand∥·∥α,δ. Besides, the bounded linear integral operators on the spacesHα,δ,γ((a,b)×(a,b),X)andHα,δ((a,b),X), some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the formf(s)=ϕ(s)+λ∫abA(s,t)f(t)dt,f(s)=ϕ(s)+λ∫abA(t,s)f(t)dtandf(s,t)=ϕ(s,t)+λ∫abA(s,t)f(t,s)dtare investigated in these spaces by analytical methods.

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