For a fractional inverse problem with an unknown time-dependent source term, stability estimates are obtained by using operator theory approach. For the approximate solutions of the problem, the stable difference schemes which have first and second orders of accuracy are presented. The algorithm is tested in a one-dimensional fractional inverse problem.
1. Introduction
Inverse problems arise in many fields of science and engineering such as ion transport problems, chromatography, and heat determination problems with an unknown internal energy source. Different typed of inverse problems have been investigated, and the main results obtained in this field of research were given by many researchers (see [1–10]). More than three centuries the theory of fractional derivatives developed mainly as a pure theoretical field of mathematics. Fractional integrals and derivatives appear in the theory of control of dynamical systems, when the controlled system or/and the controller is described by a fractional differential equation (see [11]). Recently, many application areas such as bioengineering applications, image and signal processing are also related to fractional calculus. Methods of solutions of problems and theory of fractional calculus have been studied by many researchers [11–28]. Among them finite difference method is used for solving several fractional differential equations (see [20, 22, 23, 27] and the references therein).
1.1. Statement of the Problem
Many scientists and researchers are trying to enhance mathematical models of real-life cases for investigating and understanding the behavior of them. Therefore, some phenomena have been modeled and investigated as fractional inverse problems (see [29–33] and the references therein). In this paper, we consider the fractional parabolic inverse problem with the Dirichlet condition
where and are unknown functions, , and is a sufficiently large number. Here, is the standard Riemann-Liouville’s derivative of order .
Theorems on the stability of problem (1.1) are analyzed by assuming that is a sufficiently smooth function, and .
2. Main Results
In this section, stability estimates for the solution of (1.1) are investigated. For the mathematical substantiation, we introduce the Banach space , of all continuous functions defined on with satisfying a Hölder condition for which the following norm is finite
where is the space of all continuous function defined on with the norm
With the help of a positive operator , we introduce the fractional spaces , consisting of all in a Banach space for which the following norm is finite:
Throughout the paper, positive constants will be indicated by . Here variables are used to focus on the fact that the constant depends only on and the subindex is used to indicate a different constant.
Theorem 2.1. Let , and . Then for the solution of problem (1.1), the following coercive stability estimates
hold.
Proof. Let us search for the solution of inverse problem (1.1) in the following form (see [8]):
where
Using the overdetermined condition, we get
Using identity (2.8) and the triangle inequality, it follows that
for any . Here, is the solution of the following problem:
For simplicity, we assign
where
Note that functions and only contain given functions. Then, we can rewrite problem (2.10) as
So, the end of proof of Theorem 2.1 is based on estimate (2.9) and the following theorem.
Theorem 2.2. For the solution of problem (2.10), the following coercive stability estimate
holds.
Proof. In a Banach space , with the help of the positive operator defined by
with
where is a positive constant, problem (2.10) can be written in the abstract form as an initial-value problem
By the Cauchy formula, the solution can be written as
Applying the formula
we get the following presentation of the solution of abstract problem (2.17):
Changing the order of integration, we obtain that
where
Now, we estimate separately. It is known that [13]
Since operators and commute,
Applying the definition of norm of the spaces and (2.23) and (2.24), we get
for any . Estimation of is as follows:
Let us estimate :
It is proven that (see [28])
Using the definition of norm of the spaces , we can obtain that
Using estimates (2.23) and (2.28), we get
Expanding , estimation of is as follows:
It is known that (see [34])
Since and are known functions, it is easy to see that
Estimation of can be given similar to the estimation of . By (2.23) and (2.32),
Finally combining estimates (2.25), (2.26), (2.30), (2.33), and (2.34), we get
Using the Gronwall’s inequality, we can write
From the last estimate, we can obtain the estimate for by using problem (2.17) and well-posedness of the Cauchy problem in (see [35]). So the following theorem finishes the proof of Theorem 2.2.
Theorem 2.3 (see, [36]). For , the norms of the spaces and are equivalent.
3. Numerical Results
We have not been able to obtain a sharp estimate for the constants figuring in the stability inequalities. So we will provide the following results of numerical experiments of the following problem:
The exact solution of the given problem is and for the control parameter is .
3.1. The First Order of Accuracy Difference Scheme
For the approximate solution of the problem (3.1), the Rothe difference scheme
where denotes greatest integer less than is constructed. Throughout the paper, let us denote
We search the solution of (3.2) in the following form:
where
Moreover for the interior grid point , we have that
From (3.4), (3.5), and the condition , it follows that
where is the solution of the difference scheme
First, applying the first order of accuracy difference scheme (3.10), we obtain system of linear equations and we write them in the matrix form
where
for any , and
Here, for any ,
and is identity matrix. Using (3.11), we can obtain that
To solve the resulting difference equations, we apply the method given in (3.15) step by step for . For the evaluation of is needed. It is obtained in the previous step. Then, the solution pairs are obtained by using the last formulas (3.9) and (3.8).
3.2. The Second Order of Accuracy Difference Scheme
For the approximate solution of the problem (3.1), the Crank-Nicholson difference scheme
is constructed.
Here,
Moreover, applying the second order of approximation formula for
it is obtained (see [27])
Here and throughout the paper,
We search the solution of (3.16) in the following form:
where
We have that
Let us denote
where . Then, one can write
So the values of can be obtained by the following formula:
Let denote
For , one can show that is the solution of the difference scheme
We have the system of linear equations and we write them in the matrix form
where
Here, for any ,
and is identity matrix. Using (3.29), we can obtain that
For , is the solution of the difference scheme
The system of linear equations given above can be written in the matrix form
where
Here, for any ,
Using (3.34), we can obtain that
For , we can obtain the following difference scheme:
