[1] In this paper, the essential technique of Grad-Shafranov (GS) reconstruction is reformulated into an inverse boundary value problems (IBVPs) for Laplace's equation on a circle by introducing a Hilbert transform between the normal and tangent component of the boundary gradients. It is proved that the specified IBVPs have unique solution, given the known Dirichlet and Neumann conditions on certain arc. Even when the arc is reduced to only one point on the circle, it can be inferred logically that the unique solution still exists there on the remaining circle. New solution approach for the specified IBVP is suggested with the help of the introduced Hilbert transform. An iterated Tikhonov regularization scheme is applied to deal with the ill-posed linear operators appearing in the discretization of the new approach. Numerical experiments highlight the efficiency and robustness of the proposed method. According to linearity of the elliptic operator in GS equation, its solution can be divided into two parts. One is solved from a semilinear elliptic equation with an homogeneous Dirichlet boundary condition. The other is solved from the IBVP of Laplace's equation. It is concluded that there exists a unique solution for the so-called elliptic Cauchy problem for the essential technique of GS reconstruction.

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Hilbert transforms (HT) have first been used to build the essential technique of Grad‐Shafranov (GS) reconstruction by Li et al. (2013), where the problem of ill posedness in GS reconstruction has been thoroughly investigated. In this study, we present an extended Hilbert transform (EHT) over the plane rectangle. In contrast to previous one (HT over the unit circular region), corner singularities are introduced into these new formulae. It is confronted by problems like the integral with both endpoint singularities, and the semiinfinite integral with one endpoint singularity, as these EHT formulae are used to rebuild the essential technique of GS reconstruction. Two additional mathematic tools are adopted in this study. First, high‐accuracy quadrature schemes are constructed for those improper integrals based on the double exponential (DE) transformations. Benchmark testing with the analytic solutions on a rectangular boundary has shown the efficiency and robustness of the EHT formulae. Second, the data completion or the inverse boundary value problem is solved with the help of a truncated Chebyshev series, which approximates the unknown boundary gradients in very high efficiency under the only assumption that they are Lipschitz continuous on each side of the rectangle. Combining the introduced EHT formulae and the two needed mathematic tools, the essential technique of GS reconstruction is formulated into a linear system of Fredholm equations of the first kind. Then a three‐parameter Tikhonov regularization scheme is developed to deal with the ill‐posed linear operators appearing in the discretized linear system. This new approach for data completion over the plane rectangle is benchmarked with the analytic solutions. Numerical experiments highlight the efficiency and robustness of the proposed method.