Consider now the inverse problem with one measured output data at . In order to formulate the solution of parabolic problem (1) by using the Fourier method of the separation of variables, let us first introduce an auxiliary function as follows:
by which we transform problem (1) into a problem with homogeneous boundary conditions. Hence the initial boundary value problem (1) can be rewritten in terms of in the following form:
The unique solution of the initial-boundary value problem can be represented in the following form :
Moreover, , being the generalized Mittag-Leffler function defined by
Assume that is the solution of the following Sturm-Liouville problem:
The Neumann-type measured output data at the boundary in terms of can be written in the following form:
In order to arrange the above solution, let us define the following:
The solution in terms of and can then be rewritten in the following form:
Differentiating both sides of the above identity with respect to x and substituting yields
Taking into account the over-measured data ,
is obtained, which implies that can be determined analytically. Substituting into this yields
Hence we obtain the following explicit formula for the value of the unknown coefficient
Under the determined value , the set of admissible coefficients can be defined as follows:
The right-hand side of identity (4) defines the input-output mapping on the set of admissible source functions
The following lemma implies the relation between the parameters at and the corresponding outputs , .
Lemma 1 Let and be the solutions of direct problem (2), corresponding to the admissible parameters . If , , are the corresponding outputs. If the condition , then the outputs , , satisfy the following integral identity:
for each , where , .
Proof By using identity (4), the measured output data , , can be written as follows:
respectively. Note that the definition of implies that . Hence, the difference of these formulas implies the desired result. □
The lemma and the definitions of and given above enable us to reach the following conclusion.
Corollary 1 Let the conditions of Lemma 1 hold. If in addition
then , .
Proof If , , then . If , then . Since depends on , then from the uniqueness of solution .
Since , form a basis for the space and , , then implies that at least for some . Hence by Lemma 1 we conclude that , which leads us to the following consequence: implies that . □
Theorem 1 Let conditions (C1), (C2) hold. Assume that is the input-output mapping defined by (4) and corresponding to the measured output . In this case the mapping has the distinguishability property in the class of admissible parameters , i.e.,