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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 22
Adaptive Numerical Methods for IntegroDifferential Equations of Fractional Order E. Cuesta
Department of Applied Mathematics, University of Valladolid, Spain E. Cuesta, "Adaptive Numerical Methods for IntegroDifferential Equations of Fractional Order", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 22, 2006. doi:10.4203/ccp.84.22
Keywords: fractional, integrals, convolution, variable, quadratures, adaptive.
Summary
Integrodifferential equations of fractional order have recently
showed highly interesting in connections with many fields of
science [3,7,8]. Infact, we
consider linear equations whose prototype is
where and , which can be written in a equivalent way as Properties concerning to the continuous solution of (59) can be found in the literature, (for example [8] and references cited there), also in a more general framework [9]. In this paper we focus on the numerical solution of (59). In this way, one can easily understand that the key to these numerical methods consists in obtaining good enough quadratures to approximate the fractional integrals as the one in (59) or (60). Discrete solutions of (59) have been widely studied in the literature (see [1,2,6]) but unfortunately the high computational cost of the algorithm involving convolution equations, and more precisely quadratures of convolution type, sometimes makes its implementation in applications useless. Thus, the effort from the numerical point of view, must concentrate in obtaining fast and efficient quadratures in order to obtain numerical schemes which implementation could be affordable in practical applications. The goal of our paper is the obtaining of a numerical scheme based on a variable step size quadrature which represents an extension of the quadratures considered in [1,2,6]. The obtaining of these numerical methods is based in writing (60) in a suitable way which allows us to use classical schemes for initial value problems. The quadrature considered here is based on the variable step size backward Euler method and read as The main result of our paper provides a priori error estimates for the proposed numerical scheme. At the same time, these estimates show that, if constant step size is considered, then the numerical method exhibits, as expected in view of [1], a first order of convergence. In the paper we show numerical experiments illustrating the theoretical results. Open problems keep still open as, for instance, the choice of suitable or even optimal, time step settings. References
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