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

The Grad–Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of first Tokamaks. poloidal surface Tokamak has evolved over years from a circular to D-shaped ellipse. natural geometry that describes such is prolate elliptical one, i.e., cap-cyclide coordinate system. When written this can be terms general Heun function. In paper, we obtain complete analytical solution functions and compare result with limiting case standard Fock functions.