Superconvergent Nyström method for Urysohn integral equations

Allouch, C.; Sbibih, D.; Tahrichi, M.

doi:10.1007/s10543-016-0629-6

Cited by 10 publications

(13 citation statements)

References 19 publications

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“…Remark 1 There are integrals in the nonlinear system equivalent to the approximate equation ( 2) and in the evaluation of the iterated solution (3). We refer to [4] for more implementation details of the method. In practice, these integrals are evaluated numerically using quadrature rules of a convergence order d chosen such that the original convergence orders of the method be maintained.…”

Section: )mentioning

confidence: 99%

“…Then, in the case where π n is given by the interpolatory projector at r Gauss points in each subinterval of the partition, we have proved in [4], the following error estimates for the superconvergent Nyström solution and its iterated version :…”

Section: Introductionmentioning

confidence: 97%

“…We choose π n to be the interpolatory projector at r distinct points in each subinterval of the partition. In [4], we have introduced a superconvergent Nyström method to solve the Urysohn integral equation (1). It consist to solve the following approximate equation…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Error Expansion of the Iterated Superconvergent Nyström Method for Urysohn Integral Equations

Allouch¹,

Medbouhi

Sbibih

et al. 2021

Int. J. Appl. Comput. Math

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For a nonlinear Urysohn integral equation with smooth kernel, a superconvergent Nyström method proposed recently in Allouch et al. (BIT Numer Math 258:10777-10790, 2017), based on an interpolatory projection onto a space of piecewise polynomials of degree ≤ r , is shown to have convergence of order 4r for its iterated version. In this paper, a general theorem dealing with asymptotic error expansion for the iterated superconvergent Nyström solution is proved. The Richardson extrapolation is then applied to obtain a more accurate solution of order 4r + 2. Numerical results are given to illustrate this improvement of the convergence order.

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Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

Asymptotic Error Expansion of the Iterated Superconvergent Nyström Method for Urysohn Integral Equations

Allouch¹,

Medbouhi

Sbibih

et al. 2021

Int. J. Appl. Comput. Math

View full text Add to dashboard Cite

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“…In [11], the Nyström method was introduced for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces. Recently, the Nyström method also has been applied to the Urysohn integral equations [12], Volterra integro-differential equations [13], Mellin convolution equations [14], and multifrequency oscillatory systems [15]. Influenced by the work mentioned above, the goal of this work is to develop the Nyström method for problem (1) and provide rigorous error analysis for the method.…”

Section: Introductionmentioning

confidence: 99%

Iterative Method for a System of Nonlinear Fredholm Integral Equations

Wang

2021

Mathematical Problems in Engineering

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In this paper, the iteration method is proposed to solve a class of system of Fredholm-type nonlinear integral equations. First, the existence and uniqueness of solution are theoretically proven by the fixed-point theorem. Second, the approximation solution method is given by using the appropriate integration rule. The error analysis for the approximated solution with the exact solution is discussed for infinity-norm, and the rates of convergence are obtained. Furthermore, an iteration algorithm is constructed, and the convergence of the proposed numerical method is rigorously derived. Finally, some numerical examples are given to illustrate the theoretical results.

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“…Integral equations occur in different fields of mathematical physics, engineering and mechanics. In particular, many problems like physical applications, potential theory and electrostatics are reduced to the solution of nonlinear integral equations, as noticed in [5] and references therein. One of the most important kinds of nonlinear integral equations is the Urysohn one, defined in the following form…”

Section: Introductionmentioning

confidence: 99%

Projection methods based on spline quasi-interpolation for Urysohn integral equations

Dagnino,

Dallefrate,

Remogna

2018

Preprint

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In this paper we propose projection methods based on spline quasiinterpolating projectors of degree d and class C d−1 on a bounded interval for the numerical solution of nonlinear integral equations. We prove that they have high order of convergence 2d + 2 if d is odd and 2d + 3 if d is even. We also present the implementation details of the above methods. Finally, we provide numerical tests, that confirm the theoretical results. Moreover, we compare the theoretical and numerical results with those obtained by using a collocation method based on the same spline quasi-interpolating projectors.

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Superconvergent Nyström method for Urysohn integral equations

Cited by 10 publications

References 19 publications

Asymptotic Error Expansion of the Iterated Superconvergent Nyström Method for Urysohn Integral Equations

Asymptotic Error Expansion of the Iterated Superconvergent Nyström Method for Urysohn Integral Equations

Iterative Method for a System of Nonlinear Fredholm Integral Equations

Projection methods based on spline quasi-interpolation for Urysohn integral equations

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