An inverse heat problem of finding an unknown parameter
The parameter determination in a parabolic partial differential equation from the overspecified data plays a crucial role in applied mathematics and physics. This technique has been widely used to determine the unknown properties of a region by measuring data only on its boundary or a specified location in the domain. These unknown properties, such as the conductivity medium, are important to the physical process, but they usually cannot be measured directly, or the process of their measurement is very expensive [
In this paper, we solve an inverse problem to a class of reaction-diffusion equation using variational iteration method. The method is capable of reducing the size of calculations and handles both linear and nonlinear equations, homogeneous or inhomogeneous, in a direct manner. The method gives the solution in the form of a rapidly convergent successive approximation that may give the exact solution if such a solution exists. For concrete problems where exact solution is not obtainable, it was found that a small number of approximations can be used for numerical purposes.
Reaction-diffusion systems are mathematical models which explain how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are transformed into each other and diffusion which causes the substances to spread out over a surface in space. This description implies that reaction-diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of nonchemical nature. Examples are found in biology, geology, physics, and ecology [
In this paper, we consider an inverse problem of simultaneously finding unknown coefficients
An additional boundary condition which can be the integral overspecification is given in the following form:
Inverse problems in partial differential equations can be used to model many real problems in engineering and other physical sciences (cf. [
This paper is organized as follows. In Section
The variational iteration method is a powerful tool to search for both analytical and approximate solutions of nonlinear equation without requirement of linearization or perturbation [
Consider the following general nonlinear system:
Assuming
For arbitrary
This section covers the error analysis of the proposed method. Also the sufficient conditions are presented to guarantee the convergence of VIM, when applied to solve the differential equations.
First, we will rewrite (
Let
According to the above theorem, a sufficient condition for the convergence of the variational iteration method is strictly contraction of
In the following theorem, we introduce an estimation of the error of the approximate solution of problem (
Under the conditions of Theorem
Consider
Also, we have
To use the variational iteration method for solving the problem (
By differentiation with respect to the variable
Therefore, the inverse parabolic problem (
In order to solve problem (
Now, the following iteration formula can be obtained as
Having
In this section three examples are presented to demonstrate the applicability and accuracy of the method. These tests are chosen such that their analytical solutions are known. But the method developed in this research can be applied to more complicated problems. The numerical implementation is carried out in Microsoft Maple13.
We consider the following inverse problem:
We consider the inverse problem (
Using (
From the above two examples, it can be seen that the exact solution is obtained by using one iteration step only.
We consider the following inverse problem:
From these results, we conclude that the variational iteration method for this example gives remarkable accuracy in comparison with the exact solution.
Absolute errors of
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In the present work, we have demonstrated the applicability of the VIM for solving a class of parabolic inverse problem in reaction-diffusion equation and introduce an estimation of the absolute error of the approximate solution for the proposed method. The method needs much less computational work compared with traditional methods and does not need discretization. The illustrative examples show the efficiency of the method. By this method, we obtain remarkable accuracy in comparison with the exact solution. Moreover, by using only one iteration step, we may get the exact solution. We expect that, for more general cases, where the inhomogeneous term,
The authors declare that there is no conflict of interests regarding the publication of this paper.