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首页> 外文期刊>Computer Modeling in Engineering & Sciences >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
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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

机译：Fredholm积分方程数值解和噪声数据数值微分的虚拟时间积分方法，及其与滤波理论的关系

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摘要

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.
机译：Liu和Atluri（2008a）先前开发的虚拟时间积分方法（FTIM）用于求解不适定的线性代数方程组，该系统可能是由一类线性Fredholm积分方程的离散化产生的。通过将FTIM与著名的滤波器理论联系起来，我们使FTIM的数学基础合理化。对于通过FTIM获得的线性常微分方程（在FTIM中等效用于求解不适定线性代数方程），我们发现虚拟时间起着正则化参数的作用，其滤波效果为比Tikhonov和指数滤波器更好。然后，我们采用这种新方法来解决嘈杂数据的数值微分问题（例如在疲劳中找到da / dN，其中a为测得的裂纹长度，N为载荷循环数），以及噪声下的Abel积分方程。建立了FTIM的数值方法对噪声是鲁棒的。

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