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CONVERGENCE IN RELAXATION SPECTRUM RECOVERY

Part of: Integral transforms, operational calculus Fredholm integral equations

Published online by Cambridge University Press:  02 November 2016

R. J. LOY ,
F. R. DE HOOG  and
R. S. ANDERSSEN
R. J. LOY
Affiliation:
Mathematical Sciences Institute, Australian National University, John Dedman Building 27, Union Lane, Canberra, ACT 2601, Australia email [email protected]
F. R. DE HOOG
Affiliation:
CSIRO Data61, GPO Box 664, Canberra, ACT 2601, Australia email [email protected]
R. S. ANDERSSEN*
Affiliation:
CSIRO Data61, GPO Box 664, Canberra, ACT 2601, Australia email [email protected]
*
✉ [email protected]
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Abstract

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Because of its practical and theoretical importance in rheology, numerous algorithms have been proposed and utilised to solve the convolution equation $g(x)=(\text{sech}\,\star h)(x)\;(x\in \mathbb{R})$ for $h$ , given $g$ . There are several approaches involving the use of series expansions of derivatives of $g$ , which are then truncated to a small number of terms for practical application. Such truncations can only be expected to be valid if the infinite series converge. In this note, we examine two specific truncations and provide a rigorous analysis to obtain sufficient conditions on $g$ (and equivalently on $h$ ) for the convergence of the series concerned.

Keywords

relaxation spectrumconvolution equation

MSC classification

Primary: 44A35: Convolution
Secondary: 45B05: Fredholm integral equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 121 - 132
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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