Resolving the Problem of Multiple Control Parameters in Optimized Borel-Type Summation

TL;DR

This paper introduces a novel method to resolve the issue of multiple solutions in optimized Borel-type summation by using cost functional minimization, enhancing the accuracy of summing divergent series in physics.

Contribution

It proposes a new approach employing cost functionals and fractional transforms to uniquely determine control parameters in Borel-type summation.

Findings

Method effectively resolves multiple control parameter solutions.

New transformations improve convergence and accuracy.

Performance demonstrated on various models.

Abstract

One of the most often used methods of summing divergent series in physics is the Borel-type summation with control parameters improving convergence, which are defined by some optimization conditions. The well known annoying problem in this procedure is the occurrence of multiple solutions for control parameters. We suggest a method for resolving this problem, based on the minimization of cost functional. Control parameters can be introduced by employing the Borel-Leroy or Mittag-Leffler transforms. Also, two novel transformations are proposed using fractional integrals and fractional derivatives. New cost functionals are advanced, based on lasso and ridge selection criteria, and their performance is studied for a number of models. The developed method is shown to provide good accuracy for the calculated quantities.