Numerical Analysis of a Fractional Cauchy Problem for the Laplace Equation in an Annular Circular Region

The Cauchy problem for the Laplace equation in an annular bounded region consists of finding a harmonic function from the Dirichlet and Neumann data known on the exterior boundary. This work considers a fractional boundary condition instead of the Dirichlet condition in a circular annular region. We found the solution to the fractional boundary problem using circular harmonics. Then, the Tikhonov regularization is used to handle the numerical instability of the fractional Cauchy problem. The regularization parameter was chosen using the L-curve method, Morozov’s discrepancy principle, and the Tikhonov criterion. From numerical tests, we found that the series expansion of the solution to the Cauchy problem can be truncated in N=20, N=25, or N=30 for smooth functions. For other functions, such as absolute value and the jump function, we have to choose other values of N. Thus, we found a stable method for finding the solution to the problem studied. To illustrate the proposed method, we elaborate on synthetic examples and MATLAB 2021 programs to implement it. The numerical results show the feasibility of the proposed stable algorithm. In almost all cases, the L-curve method gives better results than the Tikhonov Criterion and Morozov’s discrepancy principle. In all cases, the regularization using the L-curve method gives better results than without regularization.