Solving the Inverse Problems of Laplace Equation to Determine the Robin Coefficient/Cracks' Position Inside a Disk

Abstract

We consider an inverse problem of Laplace equation by recoverning boundary value on the inner circle of a two-dimensional annulus from the overde- termined data on the outer circle. The numerical results can be used to determine the Robin coefficient or crack's position inside a disk from the measurements of Cauchy data on the outer boundary. The Fourier series is used to formulate the first kind Fredholm integral equation for the unknown data f(q) on the inner circle. Then we consider a Lavrentiev regularization, by adding an extra term a f(q) to obtain the second kind Fredholm integral equation. The termwise separable prop- erty of kernel function allows us to obtain a closed-form regularized solution, of which the uniform convergence and error estimation are proved. Then we apply this method to the inverse Cauchy problem, the unknown shape of zero-potential problem, the problem of detecting crack position, as well as the problem of un- known Robin coefficient. These numerical examples show the effectiveness of the new method in providing excellent estimates of the unknown data.