The present paper reveals an analytically computational method for the inverse Cauchy problem of Laplace equation. For the sake of analyticity, and also for the frequent use of rectangular plate in engineering structure, we only consider the analytical solution in a two-dimensional rectangular domain, wherein a missing boundary condition is recovered from a partial measurement of the Neumann data on an accessible boundary. The Fourier series is used to formulate a first-kind Fredholm integral equation for the unknown function of data. Then, we consider a Lavrentiev regularization amended to a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us to obtain a closed-form solution of the regularization type. The uniform convergence and error estimation of the regularization solution are proven. The numerical examples show the effectiveness and robustness of the new method.
