ABSTRACT
The spline collocation is a very efficient method of solving boundary problem of differential equations. So far, there are many works such as Gauss collocation, Smoothest spline collocation, Extrapolation collocation etc. The cubic spline collocation enjoys great popularity in every spline collocation. Gauss collocation obtained saturation approximation order O(h 4) by increasing collocation points and decreasing smooth property of spline function but unknown quantity increase one time and collocation points are irrational points. Although the smoothest spline collocation lowers computation quantity, approximation order is only O ( h 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq928.tif"/> . The approximation order of extrapolation collocation is O ( h 4 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq929.tif"/> . but approximation order for k-order derivative is only O ( h 4 − k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq930.tif"/> . In this paper, utilizing the feature that cubic spline function s(x) and its k-fold spline-on-spline function s [k](x) can uniformly approximate f(x) and k-order derivative within O(h 4), we construct the spline-on-spline collocation scheme. Furthmore, the asymptotic expansion of collocation solution and its one-fold spline-on-spline p(x) is obtained and two-fold spline-on-spline function approximation order has O ( h 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq931.tif"/> . In addition, computation quantity is almost the same as common cubic spline collocation when the above scheme is solved by iterative method, so the scheme is efficent and pratical numerical method for solving two-point boundary problem and k-order derivation functions.
