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# A Fictitious Time Integration Method for the Numerical Solution of the Fredholm Integral Equation and for Numerical Differentiation of Noisy Data, and Its Relation to the Filter Theory

Chein-Shan Liu1, Satya N. Atluri2

Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liuchei.nshan@msa.hinet.net
Center for Aerospace Research & Education, University of California, Irvine

Computer Modeling in Engineering & Sciences 2009, 41(3), 243-262. https://doi.org/10.3970/cmes.2009.041.243

## Abstract

The Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri (2008a) is employed here to solve a system of ill-posed linear algebraic equations, which may result from the discretization of a first-kind linear Fredholm integral equation. We rationalize the mathematical foundation of the FTIM by relating it to the well-known filter theory. For the linear ordinary differential equations which are obtained through the FTIM (and which are equivalently used in FTIM to solve the ill-posed linear algebraic equations), we find that the fictitous time plays the role of a regularization parameter, and its filtering effect is better than that of the Tikhonov and the exponential filters. Then, we apply this new method to solve the problem of numerical differentiation of noisy data [such as finding da/dN in fatigue, where a is the measured crack-length and N is the number of load cycles], and the inversion of the Abel integral equation under noise. It is established that the numerical method of FTIM is robust against the noise.

## Keywords

Liu, C., Atluri, S. N. (2009). A Fictitious Time Integration Method for the Numerical Solution of the Fredholm Integral Equation and for Numerical Differentiation of Noisy Data, and Its Relation to the Filter Theory. CMES-Computer Modeling in Engineering & Sciences, 41(3), 243–262. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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