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

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.

Stationary rotating and axially symmetric dust systems as peculiar General Relativistic objects

“…The fact that the Grad-Shafranov equation in vacuum coincides with the Laplace equation for the toroidal component of the vector potential [19,20] suggests that the solutions can be found in analogy with electromagnetism. For instance, since A = m ∧ x |x| 3 is the solution of the Laplace equation describing the vector potential of a magnetic-dipole m, we see that…”

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“…The hypergeoemtric series plays an important role in mathematics and physics (cf. [14,15]). There exist numerous summation and transformation formulae of classical hypergeometric series (see [3,Chapter 8] and [6][7][8][9][10]12,13,16,17,20]).…”

The Grad–Shafranov Equation in Cap-Cyclide Coordinates: The Heun Function Solution

“…Recently, Crisanti [13] came to the solution of the Grad-Shafranov equation in the case of the elliptic prolate geometry, writing Equation (1) in the cap-cyclide coordinates; but in this case, too, the solution was written in terms of the Wangerin functions and hence its actual representation was not illustrated and/or suggested. Eventually, it was found [14] that the three-dimensional Laplace equation can be transformed into the general Heun equation [15,16] by an appropriate coordinate transformation. Further, for some particular cases of the parameters, the solution can be expressed as a linear combination of the generalized [17,18] or ordinary [18][19][20] hypergeometric functions.…”