Examining the Mathematica algorithm for general Heun function calculation: a comparative analysis

Droghei, Riccardo; Cesarano, Clemente; Ishkhanyan, Artur

doi:10.2478/caim-2024-0013

Abstract

We investigate the numerical calculation of the general Heun equation using Wolfram Mathematica’s functions, comparing the numerical solutions with hypergeometric and explicit solutions. This exploration sheds light on the efficacy and accuracy of the numerical algorithm implemented in Mathematica for computing Heun functions.

References

A. Ronveaux, Heun’s Differential Equations. Oxford University Press, 1995.
Search in Google Scholar Back to article
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions. Cambridge University Press, 2010.
Search in Google Scholar Back to article
S. Y. Slavyanov and W. Lay, Special Functions. Oxford University Press, 2000.
Search in Google Scholar Back to article
A. M. Ishkhanyan, Schrödinger potentials solvable in terms of the general heun functions, Annals of Physics, vol. 388, pp. 456–471, 2018.
Search in Google Scholar Back to article
H. Grad and H. Rubin, Hydromagnetic equilibria and force free fields, J. Nucl. Energy, vol. 7, p. 284, 1958.
Search in Google Scholar Back to article
V. D. Shafranov, Equilibrium of a plasma toroid in a magnetic field, Sov. Phys. JETP, vol. 37, p. 775, 1960.
Search in Google Scholar Back to article
V. S. Mukhovatov and V. D. Shafranov, Plasma equilibrium in a tokamak, Nucl. Fusion, vol. 11, p. 605, 1971.
Search in Google Scholar Back to article
A. Lupica, C. Cesarano, F. Crisanti, and A. Ishkhanian, Analytical solution of the three-dimensional laplace equation in terms of linear combinations of hypergeometric functions, Mathematics, vol. 9, p. 3316, 2021.
Search in Google Scholar Back to article
F. Crisanti, C. Cesarano, and A. Ishkhanyan, Grad-shafranov equation in cap-cyclide coordinates: the heun function solution, Mathematics, vol. 11(9), p. 3316, 2023.
Search in Google Scholar Back to article
A. M. Ishkhanyan, New solutions of heun’s general equation, J. Phys. A, vol. 36, p. L81, 2003.
Search in Google Scholar Back to article
Maple Online Help, The five second order linear Heun equations and the corresponding Heun function solutions. Accessed on 30.03.2024.
Search in Google Scholar Back to article
T. A. Ishkhanyan, From sine to Heun: 5 new functions for mathematics and physics in the Wolfram language. Accessed on 30.03.2024.
Search in Google Scholar Back to article
Wolfram Research, HeunG, Wolfram language function.
Search in Google Scholar Back to article
T. A. Ishkhanyan, T. A. Shahverdyan, and A. M. Ishkhanyan, Expansions of the solutions of the general heun equation governed by two-term recurrence relations for coefficients, Advances in High Energy Physics, vol. 2018, p. 4263678, 2018.
Search in Google Scholar Back to article
A. M. Ishkhanian, Generalized hypergeometric solutions of the heun equation, Theor. Math. Phys., vol. 202, p. 1, 2020.
Search in Google Scholar Back to article
L. J. Slater, Generalized Hypergeometric Functions. Cambridge University Press, 1966.
Search in Google Scholar Back to article
R. S. Maier, The 192 solutions of the heun equation, Mathematics of Computation, vol. 76, pp. 811–843, 2007.
Search in Google Scholar Back to article

Examining the Mathematica algorithm for general Heun function calculation: a comparative analysis

Abstract

Paradigm

My account