Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

TL;DR

This paper introduces a novel numerical method for solving an ill-posed Cauchy problem for a quasilinear parabolic PDE using a Carleman weight function, marking the first such approach for nonlinear equations.

Contribution

It presents the first numerical solution for a quasilinear PDE Cauchy problem employing a Carleman weight, extending methods previously limited to linear equations.

Findings

Successfully minimized a convex cost functional with Carleman weight

Demonstrated numerical stability and convergence for the quasilinear problem

Extended the applicability of Carleman-based methods to nonlinear PDEs

Abstract

This is the first publication in which an ill-posed Cauchy problem for a quasi- linear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is to minimize a strictly convex cost functional with the Carleman Weight Function in it. Previous publications about numerical solutions of ill-posed Cauchy problems were considering only linear equations.