We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.
- alternating method
- Cauchy problem
- Green's functions
- Laplace equation
- trigonometric- and sinc-quadrature rules
- unbounded domain
Chapko, R., & Johansson, B. T. (2009). An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant. Inverse Problems in Science and Engineering, 17(7), 871-883. https://doi.org/10.1080/17415970902899464