ИСТИНА

The Grad --- Shafranov equation with affine right-hand side $\Delta u (x) = a u (x) + b$ is considered in plane simply connected domains $G$ with piece-wise $C^{3, \alpha}$--smooth boundary $\Gamma$ at which homogeneous Dirichlet condition and non--local condition $\int_\Gamma \partial_\nu u (x)\, ds = 1$ are prescribed, where $ds$ is length element of arc $\Gamma$, $\partial_\nu$ is normal derivative to $\Gamma$. The non--local condition states an explicit relation between parameters $a$ and $b$ of the equation, $b = b (a)$, and by this fact makes the considering problem, named as a direct one, depending only on parameter $a$. The inverse problem for the equation consists in finding parameter $a$ via information on normal derivative $\partial_\nu u (x)$. Those direct and inverse problems have been studied particularly in [1] in connection to physical applications. In the present work it is stated that parameter $a$ can be found by the value of normal derivative $\partial_\nu u (x)$ in any point $x$ belonging to special subset $\widetilde{\Gamma}$ of boundary $\Gamma$. The necessary and sufficient condition for solvability of the direct problem is that the value $\partial_\nu u (x)$ belongs to half--interval $\mathcal{J}_x$, which depends on $x \in \widetilde{\Gamma}$. A method is elaborated for finding parameter $a$, including an algorithm of constructing subset $\widetilde{\Gamma}$ and half--interval $\mathcal{J}_x$. Those results were obtained with the help of the multipole method that ensures high precision computation of normal derivative $\partial_\nu u (x)$ and by the use of asymptotics [1] as $a \to \infty$ for $\partial_\nu u (x)$ and $\frac{d}{da}\, \partial_\nu u (x)$, $x \in \Gamma$. [1] Demidov A.S., Moussaoui M. An inverse problem originating from magnetohydrodynamics // Inverse Problems. 2004. Vol. 20. P. 137-154. [2] Vlasov V.I. Boundary value problems in domains with curvilinear boundary. (CCAS, Moscow, 1987) [in Russian].