Consider the inverse problem of determining the explicit form of the unknown coefficient via the semigroup approach. Let the parabolic equation be arranged as follows:
Here we treat the term as a right-hand side function, i.e., . In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, a new function needs to be defined to make the boundary values equal to zero:
(3)
which satisfies the following parabolic problem:
(4)
We can represent the solution of this problem in terms of a semigroup of linear operators. Here, is a second-order differential operator and its domain is , where and are Sobolev spaces. Obviously, by completion since the initial value function belongs to . Hence is dense in , which is a necessary condition for being an infinitesimal generator.
In the following, despite doing the calculations in the smooth function space, by completion they are valid in the Sobolev space. Denote by the semigroup of linear operators generated by the operator −A [5, 6]. Since the initial boundary value problem (4) possesses the property of uniqueness of its solutions, then the solution of its homogeneous part can be written as follows:
Furthermore, by using the variations of parameters formula, we can write the solution of the initial boundary value problem (3) as follows:
In order to construct the semigroup , we need to identify the eigenvalues and eigenfunctions of the infinitesimal operator A. Hence, the following eigenvalue problem must first be considered:
(5)
This problem is called the Sturm-Liouville problem. The eigenvalues are determined with for all the corresponding eigenfunctions as . In this case, the semigroup can be represented in the following way:
(6)
where . It is well known that the Sturm-Liouville problem (5) generates a complete orthogonal family of eigenfunctions so that the null space of the semigroup is trivial, i.e., . The null space of the semigroup of the linear operators can be defined as follows:
The unique solution of the initial-boundary value problem (4) in terms of the semigroup can be represented in the following form:
Hence, by using identity (3) and taking the initial value into account, the solution of the parabolic problem (1) in terms of a semigroup can be written in the following form:
(7)
Now, differentiating both sides of identity (7) with respect to t yields
Using the semigroup property ,
is obtained. Taking in the above identity, we get
(8)
Taking in the above identity, we obtain
Since , . Taking this into account yields
By solving the equation for , the following explicit formula for the value of the unknown coefficient is obtained:
(9)
Now, differentiating both sides of identity (8) with respect to x yields
Taking in the above identity, we obtain
Since , . Taking this into account yields
By solving the equation for , the following explicit formula for the value of the unknown coefficient is obtained:
(10)
By making use of the values and , we can write a linear approximation for near as follows:
(11)
Substituting the linear approximation (11) into equation (8) yields
(12)
which enables us to obtain the approximate solution of the parabolic problem (1). For this, we first need to show that
by using the well-known property of the semigroup of the linear operators
which leads us to the determination of the unknown coefficient analytically.
Lemma Let be the semigroup of the linear operators, and let be the solution of the equation , where A is the infinitesimal generator of . Then the following identity holds:
(13)
for each .
Proof First of all, observe that since eigenfunctions of the infinitesimal generator A generate a complete orthogonal family, we can write in terms of eigenfunctions in the following form:
Taking derivative of both sides with respect to the variable t produces
Notice that this identity implies that is in the domain of the semigroup of linear operators . Hence we can apply to :
(14)
Now we know that
Taking derivative of both sides with respect to the variable t produces
which is exactly the same as (11). This is exactly what is desired. However, notice that the identity
does not hold. □
This lemma leads us to the solution of the parabolic problem (1) without knowing the unknown coefficient , but knowing and . Then the solution enables us to determine the unknown coefficient approximately.
Now, applying the identity (13) to (12) and using the linearity, we get
Hence we can obtain the solution from the above identity by integrating with respect to t as follows:
(15)
where denotes the integration constant. We can determine it by using the initial condition
which yields
Substituting the integration constant into the identity (15), we obtain the approximate solution of the parabolic problem (1). Hence, by using this solution and the parabolic equation, we are able to determine the unknown coefficient approximately.