REGULAR FREDHOLM INTEGRAL EQUATIONS OF FIRST KIND AND CHAIN LEAST SQUARES METHOD

Publish Year: 1397
نوع سند: مقاله کنفرانسی
زبان: English
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شناسه ملی سند علمی:

ICBVPA01_028

تاریخ نمایه سازی: 5 آذر 1397

Abstract:

Recently, in order to improve the e ciency of least squares method in numericalsolution of sti and ill-posed problems, the chain least squares method is given ina recurrent process by Babolian et al. The main purpose of this new method is toincrease the e ciency of the least squares method in numerical solution of ill-posedfunctional and physical equations. Determining the least squares of a given functionin an arbitrary set is often an ill-posed problem. In chain least squares method, byde ning arti cial constraint and using Lagrange multipliers method, the attempt is toturn n-dimensional least squares problems into (n-1) ones, in a way that the conditionnumber of the corresponding system with (n-1)-dimensional problem be low. At rst,the new method is introduced for 2 and 3-term basis, then the presented method isgeneralized for n-term basis.In this article, the aim is to solve the regularized Fredholm integral equations ofthe rst kind (with Tikhonov approach) by chain least squares method. TikhonovRegularization (sometimes named Tikhonov-Phillips regularization) is a e cient ap-proach to deal with linear ill-posed functional equations. Because solving regularizedFredhom integral equations of rst kind for small parameter of regularization is con-sidered as an ill-posed problem, it is tried to demonstrate the e ciency of the chainleast squares method compared with ordinary least squares method by presentingsome numerical examples. The numerical results show that the chain least squaresmethod is more e cient and stable than the ordinary least squares method and ap-plicable to other ill-posed functional equations.According to the presented results, the best value for parameter of regularization ofFredholm integral equations of the rst kind is achieved. The numerical results showthat chain least squares method can be applied for other ill-posed problems.

Keywords:

Chain least squares , Tikhonov regularization , Fredholm integral equation of the rst kind

Authors

A Abdollahi

Department of Mathematics, Maragheh Branch, Islamic Azad University, Maragheh, Iran