Quasi-reversibility Regularization Method for Solving a Backward Heat Conduction Problem

Non-standard backward heat conduction problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility method to analysis the stability of the problem. Meanwhile, we investigate the roles of regularization parameter in this method. Numerical result shows that our algorithm is effective and stable.


Introduction
In many industrial application one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurement.The backward heat conduction problem is a model of this situation.In general, no solution which satisfies the heat conduction equation with final data and the boundary conditions exists.Even if a solution exists, it will not be continuously dependent on the final data.The BHCP is a typical example of an ill-posed problem which is unstable by numerical methods and requires special regularization methods.In the context of approximation method for this problem, many approaches have been investigated.Such authors as Lattes and Lions [1], Showalter [2], Ames et al. [3], Miller [4] have approximated the BHCP by quasireversibility methods.Schröter and Tautenhahn [5] established an optimal error estimate for a special BHCP.Mera and Jourhmane used many numerical methods with regularization techniques to approximate the problem in [6][7][8], etc.A mollification method has been studied by Haö in [9].Kirkup and Wadsworth used an operatorsplitting method in [10].So far in the literature, most of the authors used the eigenfunctions and eigenvalues to reconstruct the solution of the BHCP by many quasireversibility methods numerically.However, the eigenfunctions and eigenvalues are in general not available and the labor needed to compute these and the corresponding fourier coefficients is very onerous.In this paper, we use a quasi-reversibility regularization method to solve the BHCP in one-dimensional setting numerically, but this method can be generalized to two-dimensional case.
The paper is organized as follows.In the forthcoming section, we will present the mathematical problem on a BHCP; in Section 2, we review a special quasi-reversibility regularization method; some finite difference schemes are constructed for the inverse problem and the numerical stability analysis is provided; in Section 3, numerical example is tested to verify the effect of the numerical schemes.

The Direct Problem
We consider the following heat equation: Solving the equation with given       , , s t l t f x is called a direct problem.From the theory of heat equation, we can see that for      , ,  s t l t f x in some function space there exists a unique solution [11][12][13][14][15][16][17][18][19][20].

The Inverse Problem
Consider the following problem: The inverse problem is to determine the value of for from the data If the solution exists, then the problem has a unique solution [21].
The data   g x are based on physical observations and are not known with complete accuracy, due to the illposedness of the BHCP, a small error in the data   g x can cause an arbitrarily large error in the solution .Now we want to reconstruct the temperature distribution for by quasi-reversibility regularization method.

Quasi-Reversibility Regularization Method
The initial boundary value problem ( 2) is replaced by the following problem: where  is a small positive parameter, for  sufficiently small the solution of (3) approximates the solution (if it exists) of (2) in some sense.This is one of wellknown quasi-reversibility methods.For the above mentioned problem, Ewing [15] has presented a choice rule of regularization parameter  , i.e., where  denotes the noise level of data   g x , and the error estimate between the approximate solution and the exact solution is given in The problem has a unique solution if a solution exists.Now we prove it for two-dimensional case.
Theorem 1.There exists a unique solution(if it exists) for the problem: , , , 0, , ,0 = , . where Proof.We only need to prove the following problem has the zero solution: ,0 = 0, .
x t w x t w x t w x t uwt x t then Due to the Green's second formula, we have since   = 0 w Now we construct the finite difference schemes for solving problem (4), let represent the value of the numerical solution of (4) at the mesh point  

t u x t u x t u x t u x t u x t h u x t u x t u x t u x t h
. 4) is discretized as where Now we discuss the stability of difference sch by verifying the Von Neumann condition.The propagation factor can be found

Numerical Examples
For convenience, we take Example 1.We conside rect problem: ith the initial condition: where 2 = r h  ,and it requires < 1 2 r for nume stability reasons.
The numerical results for is shown in Figure 1, where w we solve the inverse problem by the rical


It is easy to verify the fact that Von Neumann condition π h n, and the time step length = 1 m  , then we solve this problem by an ex scheme in the following form: nce sults are shown in Figure 2, where = 11, = 50 n m .From Example, we conclude that the choice rules of the regularization parameter h  the nu  is very effective.By our numerical experiment, we can see that the accuracy of the numerical results increases with the decreasing T; at