Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

Antonella Lupica,Artur Ishkhanyan,Clemente Cesarano,Flavio Crisanti

doi:10.3390/math9243316

Antonella Lupica, Artur Ishkhanyan + Show 2 more

Open Access

https://doi.org/10.3390/math9243316

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Abstract

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry.

Highlights

In 2019, the European Union (27 countries) produced a total of around 617.52 Mtoe of electricity for its energy consumption needs, of which 100.63 Mtoe concern solid fossil fuels [1]
Cristanti faces the problem of finding the analytical solution for the Grad–Shafranov equation in vacuum when the reference system is written in toroidal prolate elliptic cap-cyclide coordinates
Since the Laplace equation and the Grad–Shafranov equation differ by one sign, the procedure presented in this manuscript can be repeated to find the analytical solution of the Grad–Shafranov equation

Summary

Introduction

In 2019, the European Union (27 countries) produced a total of around 617.52 Mtoe of electricity for its energy consumption needs, of which 100.63 Mtoe concern solid fossil fuels [1]. The exact analytical solution of the Grad–Shafranov equation in vacuum has been found only if the reference system has a standard circular shape or has an oblate elliptical toroidal geometry [4,5]. Both of these geometries are unsuitable for current tokamak experiments, which are all based on prolate elliptical geometry. Cristanti faces the problem of finding the analytical solution for the Grad–Shafranov equation in vacuum (and of the Laplace equation) when the reference system is written in toroidal prolate elliptic cap-cyclide coordinates.

Hypergeometric Functions

Heun Functions

From Grad–Shafranov Equation to Heun Equation

Standard Toroidal Geometry as a Particular Case of Cap-Cyclide Geometry

Positive Integer γ

Conclusions and Discussion