ИСТИНА

The well--known model [1] of magnetic field in tokamak reduces to solving the Grad - Shafranov equation $\Delta u (x) = a u (x) + b$ in the cross--section $G$ of plasma coil with homogeneous Dirichlet condition on its boundary $\Gamma$, which is supposed to be piece-wise $C^{3, \alpha}$--smooth; constants $a$, $b$ are unknown. In [2] this statement was supplemented with the non--local condition $\int_\Gamma \partial_\nu u (x)\, ds = 1$, which physically means prescribing the value of full current; here $ds$ is length element of arc $\Gamma$, \partial_\nu$ is normal derivative to $\Gamma$. The non--local condition states an explicit relation $b = b (a)$ between parameters $a$ and $b$ of the equation, and so makes the problem stated above depending only on parameter $a$. The inverse problem for the Grad - Shafranov equation with non-local condition consists in finding parameter $a$ via the value of normal derivative $\partial_\nu u (x)$ in any point $x$ belonging to special subset $\widetilde{\Gamma}$ of boundary $\Gamma$. In the present work the necessary and sufficient conditions of unque solvability of the inverse problem are stated. An effective analytic--numerical method is elaborated for finding parameter $a$, including an algorithm of constructing subset $\widetilde{\Gamma}$. Those results were obtained by the use of the multipole method [3] that ensures high precision computation of normal derivative $\partial_\nu u (x)$ and by the use of asymptotics [4] as $a \to \infty$ for $\partial_\nu u (x)$ and $\frac{d}{da}\, \partial_\nu u (x)$, $x \in \Gamma$. [1] A.Demidov, 2nd Conference on Inverse Proplems, Control and Shape Optimization. Carthage, Tunisie, April 10-12, 2002, Abstracts, 93--94. [2] V.I.Vlasov, Boundary value problems in domains with curvilinear boundary, Computing Centre AS USSR, Moscow, (1987) [in Russian]. [3] A.S.Demidov, M.Moussaoui, Inverse Problems, 20 (2004), 137-154.