Find the 3rd derivative using finite difference approach via the Taylor series Using Lagrangian interpolation formula find the value for for the set of points and Using Lagrangian interpolation formula find the value of at for the set of points , , and Given the data points and , construct the quadratic spline interpolation formula as we did class, and using the same constraining boundary condition Below is a table of and -2 0 1 5 5 1 10 1.2 15 2 Set up the equations for a 3rd order Cubic spline () To be able to solve the system of equations, two more pieces of information are required. Note, for as the order of the polynomial approximation is increased the number of constraints/boundary conditions increase. For quadratic spline we had on constraint, which for our in class example ). For a cubic you will have two constraints, afor 4th order polynomial, you would have 4 constraints and so on. Using arbitrary constraints like setting the third derivative in the fourth point to zero may be used. However, the selection of a boundary condition, consisting of a pair of equations, is the commonly used method. The four conditions “natural spline”, “not-a-knot spline”, “periodic spline”, and “quadratic spline”.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the 3^rd derivative using finite difference approach via the Taylor series
Using Lagrangian interpolation formula find the value for for the set of points and
Using Lagrangian interpolation formula find the value of at for the set of points , , and
Given the data points and , construct the quadratic spline interpolation formula as we did class, and using the same constraining boundary condition
Below is a table of and


-2	0
1	5
5	1
10	1.2
15	2

Set up the equations for a 3^rd order Cubic spline ()

To be able to solve the system of equations, two more pieces of information are required. Note, for as the order of the polynomial approximation is increased the number of constraints/boundary conditions increase. For quadratic spline we had on constraint, which for our in class example ). For a cubic you will have two constraints, afor 4^th order polynomial, you would have 4 constraints and so on. Using arbitrary constraints like setting the third derivative in the fourth point to zero may be used. However, the selection of a boundary condition, consisting of a pair of equations, is the commonly used method. The four conditions “natural spline”, “not-a-knot spline”, “periodic spline”, and “quadratic spline”.

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