A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

Songshu Liu,Lixin Feng

doi:10.1155/2018/1216357

Abstract

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

Highlights

In many industrial applications [1] one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements
We investigate a Cauchy problem of twodimensional (2D) heat conduction equation
We present a numerical example intended to illustrate the behaviour of the proposed method

Summary

Introduction

In many industrial applications [1] one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements This problem leads us to consider a Cauchy problem of heat conduction equation, which can be considered as a data completion problem that means to achieve the remaining information from boundary conditions for both the solution and its normal derivative on the boundary. Its solution does not satisfy the general requirement of existence, uniqueness, and stability under small changes to the input data To overcome such difficulties, a variety of techniques for solving the Cauchy problem of heat equation have been proposed [2,3,4,5,6,7,8,9,10].

Mathematical Formulation of the Cauchy

Revised Tikhonov Regularization Method and Error Estimates

Numerical Aspect

Conclusion