The Fourier Regularization for Solving a Cauchy Problem for the Laplace Equation with Uncertainty

Abstract

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in the data can lead to significant errors in the solution and lack stability. Secondly, the determination of the parameters of the classical Laplace equation is difficult to adapt to the requirements of complex applications. For this purpose, in this paper, the Laplace equation with uncertain parameters is defined, and the uncertainty is represented by fuzzy numbers. In the case of granular differentiability, it is transformed into a granular differential equation, proving its serious ill-posedness. To overcome the ill-posedness, the Fourier regularization method is used to stabilize the numerical solution, and the stability estimation and error analysis between the regularization solution and the exact solution are given. Finally, numerical examples are given to illustrate the effectiveness and practicability of this method.