Abstract
This study is motivated by solving the inverse boundary problem of the static Klein-Gordon equation (SKGE), which usually occurs in data assimilation problems. For the purpose of obtaining boundary conditions, the pro forma solution of the problem is provided by using Green’s function. The representation in double series of Green’s function for the SKGE on a rectangular region is obtained by means of the method of images. Convergence analysis shows that the representation is uniformly convergent, which is computer friendly, and can be applied to approximate computations.
Highlights
1 Introduction This research is motivated by solving the inverse boundary value problem of the static Klein-Gordon equation (SKGE), which is formulated from data assimilation
3 Green’s function for the SKGE stated on a rectangular region In the following paragraph of this paper, we address the elliptic two-dimensional static Klein-Gordon equation (SKGE)
We focus on dealing with the SKGE on a rectangular region, whose boundary is = ∪
Summary
This research is motivated by solving the inverse boundary value problem of the static Klein-Gordon equation (SKGE), which is formulated from data assimilation. Data assimilation can be transferred into a variational problem, whose corresponding Euler equation is an elliptic PDE, such as a SKGE These data assimilation problems have some specific features: the boundary conditions are unknown, and some parameters, such as weighted coefficients, are unknown. The series of Green’s functions for the rectangle is not computer friendly, as it is not uniformly convergent To address this issue, in [ , ], the author provided explicit formulas for the Green’s function of an elliptic PDE in the half-plane and the infinite strip, which were expressed in elementary and special function forms by means of Fourier transformations.
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