Improving Petrov–Galerkin elements via Chebyshev polynomials and solving Fredholm integral equation of the second kind by them

Akhavan, S.; Maleknejad, K.

doi:10.1016/j.amc.2015.08.128

Cited by 3 publications

(4 citation statements)

References 10 publications

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“…We present in Table 13 the corresponding absolute errors R F n , R R n and R RF n respectively for this example. We compare our results with the results of Petrov-Galerkin elements via Chebyshev polynomials described in [2], for k ¼ 1 and n ¼ 10.…”

Section: Examplementioning

confidence: 73%

“…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].…”

Section: Numerical Examplesmentioning

confidence: 99%

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

Mennouni

Zaouia

2017

Math Sci

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Three main contributions are presented in this paper. First, the septic quasi-interpolants are calculated with all their coefficients. Second, we explore the results to solve a generalized and broad class of Fredholm integral equations of the second kind. Finally, we present three degenerate kernel methods; the latter is a combination of the two previously established methods in the literature. Moreover, we provide a convergence analysis and we give new error bounds. Finally, we exhibit some numerical examples and compare them with previous results in the literature.

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Section: Examplementioning

confidence: 73%

Section: Numerical Examplesmentioning

confidence: 99%

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

Mennouni

Zaouia

2017

Math Sci

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“…Besides, the spectral methods have enormous rates of convergence, they also have high level of reliability. The spectral method were divided into four classifications: collocation [5,6,26,27], tau [8,20,24,41], Galerkin [14,15,23] and Petrov-Galerkin [4,29] method. The main idea of the spectral methods is to express the solution of the problem as a finite sum of given basis of functions (orthogonal polynomials or combination of orthogonal polynomials) and then to choose the coefficients in order to minimize the difference between the exact and the numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

Doha

Abdelkawy

Amin

et al. 2019

NAMC

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This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE.

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“…In this method FIE is first converted into a system of linear equations. Later, Akhavan and Maleknejad [12] improved the Petrov-Galerkin elements using Chebyshev polynomials which eliminates some restrictions and improves accuracy from the previous method. Recently, Rostami and Maleknejad [13] used Galerkin method with Franklin wavelet as the basis for numerical approximation of two-dimensional FIE.…”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods

Molla

Saha

2019

GANIT: J. Bangladesh Math. Soc.

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In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25

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Improving Petrov–Galerkin elements via Chebyshev polynomials and solving Fredholm integral equation of the second kind by them

Cited by 3 publications

References 10 publications

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

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

Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods

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