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This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

Various inverse problems in a parabolic partial differential equation are widely encountered in modeling physical phenomena [

The aim of this paper is to find

When

There are various numerical methods to solve (

When

Although there are many methods for recovering the above inverse problems, those methods only give approximate solution. So it is worth noting that the variational iteration method can give the exact solution.

Professor He proposed variational iteration method (VIM) firstly in 1998 [

The rest of the paper is organized in four sections including Introduction. Section

In this section, we will apply He’s variational iteration method (VIM) to recover time- or space-dependent coefficient problems. The detailed introduction of VIM can be found in [

Using (

Assuming that

Therefore the inverse problem (

From (

Constructing a correction function for the above equation:

In the following, we determine the Lagrange multiplier

Applying

Thus

Now, take

Inserting

From (

Therefore, by (

Using (

Assuming that

Therefore, the inverse problem (

Next, we are concerned with the approximate solutions of (

From (

Constructing a correction function for the above equation,

In the following, we determine the Lagrange multiplier

Applying

Thus

Now, take

If

Now, we prove that

By (

Considering a special case of (

Beginning with

Incorporating the initial condition

Finding

Beginning with

Incorporating the initial condition

We solve the problem (

Beginning with

Incorporating the initial condition

The above three examples are about time-dependent coefficient; in the following we take space-dependent coefficient examples.

Applying the above VIM, we begin with

We take the boundary conditions, initial condition, and additional specification function (

Therefore, the first-order approximation

Finding

From (

We solve the problem (

We can determine

In order to imply the stability of this method, we perturb the additional specification data

Numerical solutions

The VIM has been applied in solving a variety of equations, but it was rarely applied in inverse problems. Here, we develop the new application area of VIM; our contribution is that we apply VIM to solve the inverse problem of time- or space-dependent coefficients in a parabolic partial differential equation and obtain the exact solution. The numerical results fully demonstrate the superiority of VIM for these inverse problems.

To imagine the basic idea behind He’s VIM, we consider the following general differential equation:

To solve (

In summary, we have the following variation iteration formula:

It should be specially pointed out that the more accurate the identification of the multiplier is, the faster the approximations converge to their exact solutions.

We cite an integrate of

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the unknown referees for their careful reading and helpful comments.