ON DETERMINATION OF MINOR COEFFICIENT IN A PARABOLIC EQUATION OF THE SECOND ORDER

An inverse problem of recovering the minor time-dependent coefficient in a parabolic equation of the second order is considere d. The unknown coefficient is the controlling parameter. The inverse problem lies in finding the solution of an initial-boundary value problem for this parabolic equation and this timedependent coefficient using data of the initial-bou ndary value problem and point conditions of overdetermination. Cases of the Diric hlet boundary conditions and oblique derivative conditions are considered. Condi tions under which the theorem of existence and solution uniqueness is applica ble for the given inverse problem is described; the numerical solution method is described, and its justification is given. All the considerations are carried out in Sobolev spaces. Solution of the direct problem is based on the finite element metho d and the finite difference method. The proposed algorithm for the numerical solution consists of three stages: initialization of the massive that describes geomet ry of the area and the boundary vector; implementation of integrative calculation of the desired coefficient using the finite element method; implementation of the fi nite difference method. Results of numerical experiments are presented, and n umerical solution of the model inverse problem is constructed in the case of Ne umann boundary conditions; dependency of an error in calculation of the contro lling parameter on the variation of the equation coefficients and the nois e level of the overdetermination data for domains with different number of nodes that depend on an observation point is described. Results of the calculations sho w a good convergence of the method. In the case when introduced noise level is 10 %, the error between the desired and the obtained solution increases from 8 to 35 times, though the graph of recovered coefficient remains close to the solut ion graph and repeats its outlines.


Introduction
We consider the question of recovering a lower order coefficient in the parabolic equation

t u A x t D u p t u f x t x t Q G T
where G is a bounded domain in n R with boundary 2 C Γ ∈ and A is a second order elliptic operator of the form 1 1 , , ,  (3) Thus, the problem can be stated as follows: given functions ψ , 0 u , g , find a solution u to the equation (1) and the function ( ) p t such that the equalities (1), (2), and (3) hold. The parameter p is actually a control parameter. This inverse problem is a classical problem and numerous examples can be found in [1][2][3][4][5][6]. The existence and uniqueness theorems of solutions to this inverse problem are exposed in [7][8][9][10]. The articles [7,8] contains the conditions of global (in time) solvability of this problem and the local solvability theorems can be found in [9,10] and some other articles. At last, the articles [11][12][13][14][15][16][17][18][19][20] are devoted to numerical calculations of solutions to this problem. The main approach to numerical solving is a reduction of the inverse problem in question to a linear inverse problem by means of the change of variable After the change we arrive at a new inverse problem of recovering the source function of the form ( ) ( ) is an unknown function). The latter problem under the natural conditions on the data is always solvable. However, the inverse change of variables in certain sense is not always possible, since it is not known a priori that the result of recovering, i.e. the function q , is positive (in this case we can determine the function p ). The global existence theorems (the most essential results belongs to Prilepko A.I. [7]) rely on the maximum principle and rather rigid conditions for the data. We do not use this change of variables in contrast to other article and this all allows to treat larger classes of the data. The main numerical methods used in the above-cited articles are the finite difference methods and variational methods. In some cases only some model problems are discussed. In this article we use the theoretical results of the articles [9,10] which are constructive and can serve as the base of a numerical algorithm. The numerical realization relies on the finite element methods. We expose a numerical algorithm and the results of numerical experiments.
In Sect. 1 we present the theoretical justification of the method. Section 2 is devoted to the algorithm of solving the problem. Section 3 contains the description of the numerical realization of the algorithm and the results of numerical experiments are displayed in Section 4.

Basic assumptions and auxiliary results
We use the Lebesgue spaces where 0 1 1/ 2 k p = − in the case of the Dirichlet boundary conditions and 0 1/ 2 1/ 2 Now we can state the existence theorem [10].

Математика
Next, we present some elements of the proof of this theorem. First, we construct an auxiliary function The classical results on solvability of parabolic problems ensure that 0, ; and thus we can assume that and finding a solution w to the problem (11) on the interval where the function w is a solution to the direct problem (11). Note that in view of our conditions

Description of the algorithm
Describe a numerical algorithm. We employ the Neumann boundary condition. In other case the changes in the algorithm are inessential. We take 2 n = and rewrite the equation (1) and the data in the form where n is the unit outward normal to Γ . We also have that ( ) This system for the vector-function ( ) can be written in the form where N is a positive integer, . From (17) and the overdetermination condition we have an approximate equality ( ) where 0 The equality (22) is an analog of the equation (14) and is used in successive approximations.

Numerical realization
Given a triangulation of G , we define the nodes 1 where the symbol 0 ( ) j g denotes the 0 j -coordinate of the vector g , and the quantities . As a result, we obtain the vectors 1 2 , , , N C C C …    and the constants 1 2 , , , N p p p … which define an approximate solution to our problem for a given triangulation and a parameter τ .
To simplify the exposition, we present the results of calculation of the function ( ) p t only. We consider the model problem, where ( ) ( ) x as well as different grids (see the fig. 1). We examine two groups of the data. For the first group, we have ( ) ( ) and different parameters 0 δ > . Table 1 The results of numerical experiments for the first group The error of calculations increases with δ and the proximity of 0 x to the center of the circle (see the table 2).
The input data for the second group are as follows: ( ) Since the error increases when the point 0 x is closer to the center of the circle, we consider the results in both case with the 10 % -noise and without it (Fig. 4). The results are displayed below. We can see on Fig. 4 that the results of calculations of ( ) p t are close to each other for different δ (Fig. 4, a -d). The results are similar to those above (see the table 3).