TY - JOUR

T1 - 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

AU - Liu, Chein Shan

AU - Atluri, Satya N.

PY - 2009

Y1 - 2009

N2 - The Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri (2008 a) 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.

AB - The Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri (2008 a) 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.

KW - Fictitious Time Integration Method (FTIM)

KW - Filter theory

KW - Ill-posed linear equations

KW - Regularization

UR - http://www.scopus.com/inward/record.url?scp=67049167147&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:67049167147

VL - 41

SP - 243

EP - 261

JO - CMES - Computer Modeling in Engineering and Sciences

JF - CMES - Computer Modeling in Engineering and Sciences

SN - 1526-1492

IS - 3

ER -