(358e) Distribution Reconstruction from Moments Via Orthogonal Polynomials

In a previous paper, a set of modified-Laguerre orthogonal polynomials was derived expressly for the purpose of reconstructing stationary or similarity solution distributions from their moments for population balance models involving accretional growth, collisional growth (aggregation) and/or breakage. The known analytical solutions to these problems fall broadly into the modified-gamma distribution class, but, in general, these problems do not admit analytical solutions and the distributions, while expected to be of near-modified-gamma form, will be somewhat perturbed from it.

It would be highly desirable to use a modified-gamma distribution as the weight function for a set of orthogonal polynomials for reconstructing these distributions so that the role of the higher polynomials is merely to capture the perturbations. In the prior work, in which the polynomials were expanded directly in the usual scaled size variable, it was not possible to derive orthogonal polynomials with a modified-gamma weight function. Instead a normalized gamma distribution weight function was used which resulted in a set of modified-Laguerre polynomials that proved superior to the conventional ones. Examples of successful reconstructions for collisional growth similarity solutions were exhibited over a range of kernel parameters and compared to solutions via sectional models.

While these gamma-weighted polynomials gave better reconstructions of general modified-gamma forms than the regular Laguerre polynomials, they were still not fully satisfactory over the entire range of possible modified-gamma parameters. Furthermore, for similarity solutions to breakage problems that are perturbed from modified-gamma forms, the modified-Laguerre polynomials fail to give good reconstructions.

In this update, a change of variable is employed that converts a normalized modified-gamma distribution into the form of the weight function of the ordinary Laguerre polynomials. With the polynomials expanded in this transformed variable, the reconstructions are now exact for all modified-gamma distributions and the range of applicability of the reconstruction technique is greatly expanded to now include solutions over a broad range of mechanisms and conditions.