Fast collocation methods for Volterra integral equations of convolution type

Conte, Dajana; Prete, Ida Del

doi:10.1016/j.cam.2005.10.018

Cited by 38 publications

(13 citation statements)

References 9 publications

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“…The algorithm presented in [9] has been used as the foundation for several methods [10,13] and extended to allow variable step size [11]. To begin the discussion we outline the scheme proposed in [9].…”

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confidence: 99%

A Kernel Compression Scheme for Fractional Differential Equations

Baffet¹,

Hesthaven²

2017

SIAM J. Numer. Anal.

109

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Abstract. The nonlocal nature of the fractional integral makes the numerical treatment of fractional differential equations expensive in terms of computational effort and memory requirements. In this paper we propose a method to reduce these costs while controlling the accuracy of the scheme. This is achieved by splitting the fractional integral of a function f into a local term and a history term. Observing that the history term is a convolution of the history of f and a regular kernel, we derive a multipole approximation to the Laplace transform of the kernel. This enables the history term to be replaced by a linear combination of auxiliary variables defined as solutions to standard ordinary differential equations. We derive a priori error estimates, uniform in f , and obtain estimates on the number of auxiliary variables required to satisfy an error tolerance. The resulting formulation is discretized to produce a time stepping method. The method is applied to some test cases to illustrate the performance of the scheme.

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confidence: 99%

A Kernel Compression Scheme for Fractional Differential Equations

Baffet¹,

Hesthaven²

2017

SIAM J. Numer. Anal.

109

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“…The method extends the algorithm proposed in [9] to variable step-size schemes for Volterra equations. In [12] a fast collocation method is proposed. In addition to the efficient account of the history term, the kernel compression scheme proposed in [7] has other advantages.…”

Section: Introductionmentioning

confidence: 99%

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

Baffet

Hesthaven

2017

J Sci Comput

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High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel compression scheme for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We study the local truncation error of integral deferred correction schemes for Volterra equations and present numerical results obtained with both implicit and the explicit methods applied to different problems.

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“…As mentioned in [4], the associated costs can be high due to the computation of the 'lag' term which contains the memory aspect of the phenomenon. In this paper, the iterative methodology begins with a realistic initial approximation which has been chosen such that the integral can be expressed as a linear combination of simple integrals.…”

Section: Introductionmentioning

confidence: 99%