Numerical Analysis of a Fractional Cauchy Problem for the Laplace Equation in an Annular Circular Region
Year: 2025
Type: article
Abstract: 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. 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. Thus, we found a stable method for finding the solution to the problem studied. To illustrate the proposed method, we elaborate synthetic examples and MATLAB programs to implement it. The numerical results show the feasibility of the proposed stable algorithm. In all cases, the solution with regularization gives better results than without regularization.
Source: Fractal and Fractional
Authors: José Julio Conde Mones, Julio Andrés Acevedo Vázquez, Eduardo Hernández Montero, María Monserrat Morín Castillo, Carlos Arturo Hernández Gracidas, José Jacobo Oliveros Oliveros
Institution: Benemérita Universidad Autónoma de Puebla
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,258
OpenAlex ID: W4409892725