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Beyond fractional derivatives: local approximation of other convolution integrals

Singh, Satwinder Jit and Chatterjee, Anindya (2010) Beyond fractional derivatives: local approximation of other convolution integrals. In: Proceedings of the royal society a-mathematical physical and engineering sciences, 466 (2114). pp. 563-581.

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Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time t needs O(t(2)) computations owing to the repeated evaluation of integrals over intervals that grow like t. Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled in finite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives (Singh & Chatterjee 2006 Nonlinear Dyn. 45, 183-206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with O(t) computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e. g. in stability analyses.

Item Type: Journal Article
Publication: Proceedings of the royal society a-mathematical physical and engineering sciences
Publisher: Royal Society
Additional Information: Copyright for this article belongs to Royal Society.
Keywords: fractional derivative; integrodifferential; convolution; decaying kernel
Department/Centre: Division of Mechanical Sciences > Mechanical Engineering
Date Deposited: 09 Feb 2010 11:30
Last Modified: 19 Sep 2010 05:54
URI: http://eprints.iisc.ac.in/id/eprint/25503

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