Heat and Mass Transfer Research Journal (HMTRJ)

Research Article

A Literature Survey on the Methods for Finding Zeros of the Kummer Function

Paul Koziel, William S. Janna and Jeffry G. Marchetta

DDepartment of Mechanical Engineering, The University of Memphis, Memphis, TN 38152, USA.

Submitted: November 14, 2020; Accepted: December 15, 2020

Abstract

Algorithms for determining the real eigenvalues of the confluent hypergeometric function known as the Kummer function are reviewed. There is a need for a large number of eigenvalues in order to describe thermally developing flow in the classic Graetz problem. Numerical approaches using the power series definition for the real portion of the Kummer function M(a; b; z) may be implemented through userfriendly MATLAB functions to compute 150 eigenvalues for ducts of circular, triangular, and square cross-section. Methods of iterative root calculation using bisection, secant method, Newton’s method, and Brent’s method are also described. Comparison may be made with Graetz problem eigenvalues published in the literature using a finite number of terms for the power series, hypergeometric function calculators, as well as asymptotic approximations is provided to predict accuracy. The length of time needed to calculate a finite number of eigenvalues is discussed for the algorithms described..

Keywords

Kummer function; Graetz problem; MATLAB; Eigenvalues; Hypergeometric.

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