DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods

Okayama, Tomoaki; Tanaka, Ken’ichiro; Matsuo, Takayasu; Sugihara, Masaaki

doi:10.1007/s00211-013-0540-x

Cited by 14 publications

(12 citation statements)

References 17 publications

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“…and define h as Eq. (11). Let Y be the solution of the system of linear equations (14), and let us write it as Y = [y 1,−N , y 1,−N+1 , .…”

Section: Se-sinc-collocation Methodsmentioning

confidence: 99%

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Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems

Okayama

2017

Bit Numer Math

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A Sinc-collocation method has been proposed by Stenger, and he also gave theoretical analysis of the method in the case of a 'scalar' equation. This paper extends the theoretical results to the case of a 'system' of equations. Furthermore, this paper proposes more efficient method by replacing the variable transformation employed in Stenger's method. The efficiency is confirmed by both of theoretical analysis and numerical experiments. In addition to the existing and newly-proposed Sinc-collocation methods, this paper also gives similar theoretical results for Sinc-Nyström methods proposed by Nurmuhammad et al. From a viewpoint of the computational cost, it turns out that the newly-proposed Sinc-collocation method is the most efficient among those methods.

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“…and define h as Eq. (11). Let Y be the solution of the system of linear equations (14), and let us write it as Y = [y 1,−N , y 1,−N+1 , .…”

Section: Se-sinc-collocation Methodsmentioning

confidence: 99%

“…let N be a positive integer, and let h be selected by the formula (11). Then there exists a constant C which is independent of N, such that…”

Section: Theorem 3 (Okayama Et Al [10 Theorem 29])mentioning

confidence: 99%

Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems

Okayama

2017

Bit Numer Math

Self Cite

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“…Our aim here is just to review some theoretical aspects in order to refine the choice of the parameters that allow faster convergence, even if, still of type (3). As for the double-exponential transform, widely investigated in [12,13,14,15,16] for general scalar functions, in this work we show how to improve the existing error estimates for the function λ −α . We also extend the analysis to operators, showing that it is possible to reach a convergence rate of type exp −c n ln n .…”

Section: Introductionmentioning

confidence: 99%

Exponentially convergent trapezoidal rules to approximate fractional powers of operators

Aceto¹,

Novati²

2021

Preprint

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In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a single-exponential and a double-exponential transform of the integrand function. For the first approach our aim is only to review some theoretical aspects in order to refine the choice of the parameters that allow a faster convergence. As for the double exponential transform, in this work we show how to improve the existing error estimates for the scalar case and also extend the analysis to operators. We report some numerical experiments to show the reliability of the estimates obtained.

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“…After that, several researchers used Sinc methods with the double exponential transformation to solve some problems related to ordinary differential equations [22] and integral equations [23, 24]. In [25] was developed an approximate formula for fractional derivatives by means of Sinc method with the double exponential transformation. Also, in [26] Sinc methods were employed for solving a nonlinear fractional integro‐differential equation of pantograph type.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

Babaei

Banihashemi

Cattani

2020

Numerical Methods Partial

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The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro-partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc-collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.

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DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods

Cited by 14 publications

References 17 publications

Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems

Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems

Exponentially convergent trapezoidal rules to approximate fractional powers of operators

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

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