Optimal interpolation grids for accurate numerical solutions of singular ordinary differential equations

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Title: Optimal interpolation grids for accurate numerical solutions of singular ordinary differential equations
Author: Margitus, Michael
Abstract: Researchers are interested in three types of error: experimental error, truncation error, and interpolation error. This thesis will study the last. Given a differential equation g"(x) = f(x) and a fixed number of interpolation grid points, an optimation problem is formulated to minimize the difference between f(x) and its interpolating function, thus reducing the error between the actual solution and the approximated solution of the ODE. Using the Nelder-Mead Simplex Method, the optimal distribution of grid points that will minimize the error between the solution g and its approximated solution will be found. This technique will then be applied to the one dimensional light scattering equation gxx = E²x/R. Using the Nelder-Mead Method, the optimal interpolation grid for a given number of grid points will be found. These numerical computations will ultimately be used to give guidance to experimenters on where to take measurements for the Rayleigh Ratio R.
Record URI: http://hdl.handle.net/1850/10823
Date: 2009-05

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