Numerical Solution of Linear Integral Equations of the First Kind. Calculation of Molecular Weight Distributions from Sedimentation Equilibrium Data

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Abstract
A new method is described for the numerical solution of g(x)= ∫ abK(x, t)f(t)dtfor f(t) when g(x) is experimental data of limited accuracy. Under these conditions, the well-known least-squares and algebraic methods often give poor solutions. The new method is an optimum combination of these two. It has the stability of the least-squares method to errors in g(x), but still does not assume a functional form for f(t). The problem of the determination of the molecular weight distributions (MWD) of ideal polydisperse solutes from sedimentation equilibrium data is discussed in detail. Results of the analyses of sets of computer-simulated experimental data containing pseudorandom errors show that the method can give good estimates of the MWD from data accurate to only two significant figures. This accuracy should be readily attainable experimentally. The method should be useful in the analysis of data from many other types of experiments, particularly scattering, where the presently used Fourier methods are often unsatisfactory because they require the extrapolation of data in order to evaluate the Fourier transforms.

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