We consider the problem of identification of the unknown source in a heat equation. The problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. Meyer wavelets have the property that their Fourier transform has compact support. Therefore, by expanding the data and the solution in the basis of the Meyer wavelets, high-frequency components can be filtered away. Under the additional assumptions concerning the smoothness of the solution, we discuss the stability and convergence of a wavelet-Galerkin method for the source identification problem. Numerical examples are presented to verify the efficiency and accuracy of the method.

Inverse source identification problems are important in many branches of engineering sciences; for example, an accurate estimation of pollutant source is crucial to environmental safeguard in cities with high population. This inverse problem has been investigated in some theoretical papers concerned with the conditional stability and the data compatibility of the solution, notably in [

We consider the following initial value problem for the nonhomogeneous heat equation:

The problem (

In this paper we consider the inverse problem of determining the function

It is easy to know that the function

Since the convergence rates can only be given under a priori assumptions on the exact solution [

Meyer’s wavelets are special because, unlike most other wavelets, they have compact support in the frequency domain but not in the time domain (however, they decay very fast). The wavelet-Galerkin method for approximation solutions of the sideways heat equation has been used in [

The purpose of this paper is to demonstrate that, using a wavelet-Galerkin approach, we can solve problem (

The outline of the paper is as follows. First in Section

Let

It is easy to see from (

From (

We now introduce the Galerkin method for approximation of the solutions of problem (

Consider the weak formulation of the differential equation with test functions from

The infinite matrix

The proof is similar to [

Let us denote

We are interested in the norm estimation of the distance between the Galerkin solution

First, let us consider the problem of stability of the Galerkin solution with respect to perturbations of the data function

Let

Let

Before investigating the relation between the exact solution and the corresponding Galerkin solution with the exact data

If the exact solution

Suppose condition of Lemma

If (

Due to (

It remains to reckon with the second term in the right-hand side of (

Let

Since the Galerkin equation (

Theorems

Let

In this section, we will describe a numerical complement of the proposed method.

Note that the problem (

We select

In the numerical solution of (

We will use DMT as a short form of the “discrete Meyer (wavelet) transform.” Algorithms for discretely implementing the Meyer wavelet transform are described in [

In the solution of problem (

In this section some numerical examples are presented to demonstrate the usefulness of the approach. The tests were performed using Matlab and the wavelet package WaveLab 850.

Suppose that the sequence

The numerical examples were constructed in the following way. First we selected function

It is easy to verify that the function

We examine the reconstruction of a Gaussian normal distribution

Consider a continuous piecewise smooth heat source; namely,

This example involves reconstructing a discontinuous heat source given by

The results from these examples are given in Figures

Exact solution (solid) and its approximation (dashed) for Example

Exact solution (solid) and its approximation (dashed) for Example

Exact solution (solid) and its approximation (dashed) for Example

Exact solution (solid) and its approximation (dashed) for Example

Figures

For the proof we use the following two lemmas.

The matrix

It can be easily shown by integration by parts that

For

The Fourier coefficients are given by

We can now prove Proposition

First, due to

Since

Since

The work described in this paper was partially supported by a grant from the Fundamental Research Funds for the Central Universities of China (Project no. ZYGX2009J099) and the National Natural Science Funds of China (Project no. 11171054).