Simultaneous Identification of Coefficient and Initial State for One-Dimensional Heat Equation from Boundary Control and Measurement

TL;DR

This paper develops a method to simultaneously identify the diffusion coefficient and initial state of a 1D heat equation using boundary measurements, transforming the problem into spectral data reconstruction and applying a matrix pencil algorithm.

Contribution

It introduces a novel spectral reconstruction approach for inverse heat problems, combining Dirichlet series representation with matrix pencil method for simultaneous parameter estimation.

Findings

Successful reconstruction of spectral data with measurement error

Algorithm verified through numerical simulations

Error analysis confirms method robustness

Abstract

In this paper, we consider simultaneous reconstruction of the diffusion coefficient and initial state for a one-dimensional heat equation through boundary control and measurement. The boundary measurement is known to make the system exactly observable, and both coefficient and initial state are shown to be identifiable by this measurement. By a Dirichlet series representation for observation, we can transform the problem into an inverse process of reconstruction of the spectrum and coefficients for Dirichlet series in terms of observation. This happens to be the reconstruction of spectral data for an exponential sequence with measurement error. This enables us to develop an algorithm based on the matrix pencil method in signal analysis. An error analysis is made for the proposed method. The numerical simulations are presented to verify the proposed algorithm.