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Two-Point Boundary Value Problems

Two-Point Boundary Value Problems - Two-Point Boundary...

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10/27/2010 Math Modeling II, Fall 2010, Dr. Xin Li, Mathematics, UCF 1 Two-Point Boundary Value Problems Cubic Splines Method 10/27/2010 What is a spline function? y A spline function is a piecewise polynomial function. y Definition: A function S(x) is a cubic spline function on interval [a,b] if (i) There is a partition of [a,b]: a=x 0 <x 1 <…<x n =b such that the restriction of S(x) on each subinterval [x j-1 ,x j ] is a cubic polynomial; (ii) S(x) has a continuous second derivative on [a b] (ii) S(x) has a continuous second derivative on [a,b]. y We will only consider the uniform partition when: x j -x j-1 =(b-a)/n =h , i=1,2,…,n.
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10/27/2010 Math Modeling II, Fall 2010, Dr. Xin Li, Mathematics, UCF 2 Representation of splines y For a given interval [a,b], {x 0 ,…,x n }, the collection of all cubic splines form a linear space: (i) the sum of two splines is again a spline; (ii) the scalar multiple of any spline is also a spline. y For a fixed partition on [a,b], {x 0 ,…,x n }, (i) the collection of all cubic splines form a linear sub-space; (ii) the dimension of this subspace is n+3.
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