CHAIN LEAST SQUARES METHOD FOR NUMERICAL SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS

Rately, in order to increase the e ciency of least squares method in numericalsolution of ill-posed problems, the chain least squares method is presented in a recur-rent process by Babolian et al. The main purpose of this new method is to increasethe e ciency of the least squares method in numerical solution of ill-posed functionaland physical equations. Determining the least squares of a given function in an ar-bitrary set is often an ill-posed problem. In chain least squares method, by de ningarti cial constraint and using Lagrange multipliers method, the attempt is to turnn-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 research, The aim is to solve the second order linear di erential equations withregularized singular pointsP(t)y + Q(t)y + r(t)y = f(t); t Є [0; T]; y(0) = y ; y0(0) = y1;by chain least squares method then, the e ciency of both methods are compared bysolving some numerical examples.Although, These functional equation can be solved by the Frobenius series and othernumerical methods, but in this article, the aim is to investigate the e ciency of chainleast squares method. The numerical results show that the chain least squares methodis an accurate and stable method in solving second order di erential equations withregularized singular points. It seems that this new method can be e cient for otherill-posed functional equations.

CHAIN LEAST SQUARES METHOD FOR NUMERICAL SOLUTIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS Keywords:

Chain least squares method , Di erential Equations , singular points