Numerical Method for Solving Nonhomogeneous Backward Heat Conduction Problem

Lingde Su,Tongsong Jiang

doi:10.1155/2018/1868921

Abstract

In this paper we consider a numerical method for solving nonhomogeneous backward heat conduction problem. Coupled with the likewise Crank Nicolson scheme and an intermediate variable, the backward problem is transformed to a nonhomogeneous Helmholtz type problem; the unknown initial temperature can be obtained by solving this Helmholtz type problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several problems in both two and three dimensions. The results show that this numerical method can solve nonhomogeneous backward heat conduction problem effectively and precisely, even though the final temperature is disturbed by significant noise.

Highlights

The results show that this numerical method can solve nonhomogeneous backward heat conduction problem effectively and precisely, even though the final temperature is disturbed by significant noise
The heat conduction equation is a kind of very important time-dependent parabolic partial differential equation; it describes the distribution of heat or temperature in a given region over time and is widely used in diverse scientific fields, such as the study of Brownian motion [1], to solve the BlackScholes partial differential equation [2] and the research of chemical diffusion
We proposed a new numerical scheme to solve nonhomogeneous backward heat conduction problem

Summary

Introduction

The heat conduction equation is a kind of very important time-dependent parabolic partial differential equation; it describes the distribution of heat or temperature in a given region over time and is widely used in diverse scientific fields, such as the study of Brownian motion [1], to solve the BlackScholes partial differential equation [2] and the research of chemical diffusion. Many works have been done to study the heat conduction problem [3,4,5]. The purpose of this article is to numerically solve nonhomogeneous backward heat conduction problem. This problem is one kind of inverse problem, called final value problem or time inverse problem. In many engineering and application areas we need to reconstruct the unknown initial heat energy or temperature from the final measured one; it is a typical inverse problem which is related to the initial boundary value problems in heat conduction. The backward heat conduction problem is a time inverse problem, in which the initial conditions are unknown; instead the final data are observable

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Conclusion