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 . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with 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.
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Keywords:
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 StyleLin, 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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