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A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind

Department of Mathematics, Shantou University, Shantou 515063, China
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Academic Editor: Lokenath Debnath
Mathematics 2017, 5(2), 28; https://doi.org/10.3390/math5020028
Received: 18 December 2016 / Revised: 12 May 2017 / Accepted: 16 May 2017 / Published: 22 May 2017
A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution f TV * . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with f TV * as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach. View Full-Text
Keywords: Fredholm integral equation of the first kind; regularization; piecewise-constant solution; total variation (TV) functional; Modica-Mortola (MM) functional Fredholm integral equation of the first kind; regularization; piecewise-constant solution; total variation (TV) functional; Modica-Mortola (MM) functional
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MDPI and ACS Style

Lin, F.-R.; Yang, S.-W. A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind. Mathematics 2017, 5, 28. https://doi.org/10.3390/math5020028

AMA Style

Lin F-R, Yang S-W. A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind. Mathematics. 2017; 5(2):28. https://doi.org/10.3390/math5020028

Chicago/Turabian Style

Lin, Fu-Rong; Yang, Shi-Wei. 2017. "A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind" Mathematics 5, no. 2: 28. https://doi.org/10.3390/math5020028

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