Regularising the Cauchy problem for Laplace's equation by fractional operators

TL;DR

This paper explores regularising the ill-posed Cauchy problem for Laplace's equation using fractional derivatives within a cylindrical geometry, demonstrating improved stability and convergence through numerical methods.

Contribution

It introduces fractional operators as a regularisation technique for classical ill-posed PDE problems, with new methods and convergence analysis in a cylindrical setting.

Findings

Fractional derivatives improve stability in the Cauchy problem

Numerical reconstructions show successful regularisation

Convergence results support the effectiveness of the methods

Abstract

In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional ones. We do so in the spirit of quasi reversibility, replacing a classically severely ill-posed PDE problem by a nearby well-posed or only mildly ill-posed one. In order to be able to make use of the known stabilising effect of one-dimensional fractional derivatives of Abel type we work in a particular rectangular (in higher space dimensions cylindrical) geometry. We start with the plain Cauchy problem of reconstructing the values of a harmonic function inside this domain from its Dirichlet and Neumann trace on part of the boundary (the cylinder base) and explore three options for doing this with fractional operators. The two other related problems are the recovery of a free boundary and then this together with simultaneous recovery of the impedance function in the boundary condition. Our main technique here will be Newton's method. The paper contains numerical reconstructions and convergence results for the devised methods.