A numerical technique for backward inverse heat conduction problems in one-dimensional space

Abstract

In this article, we consider an inverse heat conduction problem in one-dimensional space with two unknown terms. In this problem, the initial temperature and histories of temperature in the finite interval of time, may be determined. Then, by using a semi-implicit finite difference method, an estimation of these unknown terms will be found. Obviously, this problem is ill-posed. But by applying a suitable parameter, we can obtain the best of these estimations. Finally, some numerical experiments are given in the final section of this paper.

Introduction

Inverse problems occur in many branches of engineering natural science. This is because such problems are connected directly with experimental data. Such problems are much more difficult to solve than direct once, because inverse problems are usually ill-posed. On the other hand, methods of solving inverse problems are very important for a wide range of problems that cannot be analyzed by simple methods.

These problems are of two types: backward inverse heat conduction problems (BIHCP) and determining unknown temperature histories and heat flux on a part of boundary, from known values in the body is the idea of many mathematicians. Physically, let us consider a heated bar with unit length in one dimensional space. A direct measurements of the heat flux or temperature in a boundary and initial time of hot body is almost impossible. Consequently, this problem is an inverse heat conduction problem (IHCP). Some numerical and theoretical approaches for solving IHCPs when the histories of temperature at the initial time or in a boundary, not both, are unknown, have been summarized in [2], [3], [4], [5], [8]. Beck in [2] has shown that, if an error is made in a known boundary condition, then there will be errors in the unknown heat flux on the other boundary. In [2], [3], [9], an estimated of the solution an BIHCP, by using the regularization method, is derived. These results are consistent with earlier observations that small values of time can produce large errors in surface flux.

In this work, an IHCP to the above form, is considered. By using a semi-implicit type in a finite difference method, the problem converts to a system of ordinary differential equations with initial conditions. Then by choosing appropriate regularization of parameters, we obtain a well-posed solution for this problem.

Now, suppose that for any given t, 0 ⩽ t ⩽ T, u(x, t) ∈ H¹[0, 1] and satisfying

$$\rho c\frac{\mathrm{\partial }u(x\text{,}t)}{\mathrm{\partial }t}=\lambda (t)\frac{{\mathrm{\partial }}^{2}u(x\text{,}t)}{\mathrm{\partial }{x}^2}\text{,}\mathrm{in}D=\{(x\text{,}t)|0<x<1\text{,}0<t<T\}\text{,}$$

$$u(x\text{,}T)=f(x)\text{,}0⩽x⩽1\text{,}$$

$$u(0\text{,}t)=g(t)\text{,}0⩽t⩽T\text{,}$$

$$\frac{\mathrm{\partial }u(0\text{,}t)}{\mathrm{\partial }x}=0\text{,}0⩽t⩽T\text{,}$$

$$u(1\text{,}t)=h(t)\text{,}0⩽t⩽T\text{,}$$

$$u(x\text{,}0)=\varphi (x)\text{,}0⩽x⩽1\text{,}$$

where g(t) ∈ H¹([0, T]), f(x) is a known function, λ(t) ∈ L^∞((0, T]) is the known thermal conductivity, T, ρ and c are final time, density and specific heat of the material, respectively, and they are constant numbers. Consequently h(t), ϕ(x), and u(x, t) are unknown functions, which remain to be determined. Then by using [5], [7], we note that, if for any t ∈ [0, 1], λ(t) ⩾ K > 0, where K is a positive constant number, the problems (1), (2), (3), (4), (5), (6) have a unique solution in L²(0, T;H¹[0, 1]). It can be shown that, no solution of this inverse problem exists, unless f is analytic ([6], [7]).

In the next section, we discrete the variable t and reduce (1), (2), (3), (4) to a system of linear, nonhomogeneous second order differential equations. Then, we obtain the solutions of this system. This solution is not well-posed. Then, we express a relation of a parameter and increment time, and by using this relation we modify these solutions. Some numerical results and discussion are given in Section 3.