A Note on the Inverse Problem for a Fractional Parabolic Equation

and Applied Analysis 3 Theorem 2.1. Let φ ∈ ◦ C 2α 2 0, π , F ∈ C 0, T , ◦ C 2α 0, π , and ρ′ ∈ C 0, T . Then for the solution of problem 1.1 , the following coercive stability estimates ‖ut‖ C 0,T , ◦ C 2α 0,π ‖u‖ C 0,T , ◦ C 2α 2 0,π ≤ M(x∗, q)∥∥ρ∥∥C 0,T M ( a, δ, σ, α, x∗, q, T ) × ( ∥ ∥φ ∥ ∥ ◦ C 2α 2 0,π ‖F‖ C 0,T , ◦ C 2α 0,π ∥ ∥ρ ∥ ∥ C 0,T ) , ∥ ∥p ∥ ∥ C 0,T ≤ M ( x∗, q )∥ ∥ρ′ ∥ ∥ C 0,T M ( a, δ, σ, α, x∗, q, T ) × [ ∥ ∥φ ∥ ∥ ◦ C 2α 2 0,π ‖F‖ C 0,T , ◦ C 2α 0,π ∥ ∥ρ ∥ ∥ C 0,T ]


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 .

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 ∂u t, x ∂t − a ∂ 2 u t, x ∂x 2 − D 1/2 t u t, x σu t, x p t q x f t, x , 0 < x < π, 0 < t ≤ T, u t, 0 u t, π 0, 0 ≤ t ≤ T, u 0, x ϕ x , 0 ≤ x ≤ π, u t, x * ρ t , 0 < x * < π, 1.1 where u t, x and p t are unknown functions, a x ≥ a > 0, and σ > 0 is a sufficiently large number.Here, D 1/2 t D 1/2 0 is the standard Riemann-Liouville's derivative of order 1/2.Theorems on the stability of problem 1.1 are analyzed by assuming that q x is a sufficiently smooth function, q 0 q π 0 and q x * / 0.

Main Results
In this section, stability estimates for the solution of 1.1 are investigated.For the mathematical substantiation, we introduce the Banach space • C α 0, π , α ∈ 0, 1 , of all continuous functions φ x defined on 0, π with φ 0 φ π 0 satisfying a H ölder condition for which the following norm is finite where C 0, π is the space of all continuous function φ x defined on 0, π with the norm With the help of a positive operator A, we introduce the fractional spaces E α , 0 < α < 1, consisting of all v in a Banach space E for which the following norm is finite: Throughout the paper, positive constants will be indicated by M i α, β, . . . .Here variables are used to focus on the fact that the constant depends only on α, β, . . .and the subindex i is used to indicate a different constant.
Proof.Let us search for the solution of inverse problem 1.1 in the following form see 8 : where η t t 0 p s ds.

2.6
Using the overdetermined condition, we get Using identity 2.8 and the triangle inequality, it follows that for any t, t ∈ 0, T .Here, w t, x is the solution of the following problem:

2.10
For simplicity, we assign

2.12
Note that functions F t, x , Q 1 q, ρ, x, x * , t and Q 2 q, x, x * only contain given functions.Then, we can rewrite problem 2.10 as

2.13
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 where σ is a positive constant, problem 2.10 can be written in the abstract form as an initialvalue problem

2.17
By the Cauchy formula, the solution can be written as w t e −tA ϕ − t 0 e − t−s A F s G s ds.

2.18
Applying the formula we get the following presentation of the solution of abstract problem 2.17 :

2.20
Changing the order of integration, we obtain that where

2.23
Since operators A and exp −tA commute, Applying the definition of norm of the spaces E α and 2.23 and 2.24 , we get for any t, t ∈ 0, T .Estimation of J 2 t is as follows:

2.26
Let us estimate J 3 t :

2.27
It is proven that see 28 Using the definition of norm of the spaces E α , we can obtain that

2.29
Using estimates 2.23 and 2.28 , we get

2.30
Expanding G s , estimation of J 4 t is as follows:

8 Abstract and Applied Analysis
It is known that see 34

2.32
Since Q 1 q, ρ, x, x * , t and Q 2 q, x, x * are known functions, it is easy to see that Estimation of J 5 t can be given similar to the estimation of J 4 t .By 2.23 and 2.32 ,

2.35
Using the Gronwall's inequality, we can write

2.36
From the last estimate, we can obtain the estimate for w t t by using problem 2.17 and wellposedness of the Cauchy problem in C E α see 35 .So the following theorem finishes the proof of Theorem 2.2.

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:

3.1
The exact solution of the given problem is u t, x 1 − t sin x and for the control parameter p t is 1 t.

The First Order of Accuracy Difference Scheme
For the approximate solution of the problem 3.1 , the Rothe difference scheme where x denotes greatest integer less than x is constructed.Throughout the paper, let us denote

3.3
We search the solution of 3.2 in the following form: where Moreover for the interior grid point u k s , we have that From 3.4 , 3.5 , and the condition u k s ρ t k , it follows that where w k n , 0 ≤ k ≤ N, 0 ≤ n ≤ M is the solution of the difference scheme

3.10
First, applying the first order of accuracy difference scheme 3.10 , we obtain M 1 × M 1 system of linear equations and we write them in the matrix form where 12 for any j 1, 2, . . ., k − 2, and

3.13
Here, for any n 1, 2, . . ., M − 1, Abstract and Applied Analysis 13 for any r 0, 1, . . ., k, and D is M 1 × M 1 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 k 1, 2, . . ., N. For the evaluation of w r , r 2, 3, . . ., N, w r−1 is needed.It is obtained in the previous step.Then, the solution pairs u, p are obtained by using the last formulas 3.9 and 3.8 .

The Second Order of Accuracy Difference Scheme
For the approximate solution of the problem 3.1 , the Crank-Nicholson difference scheme Here, 3.17 Moreover, applying the second order of approximation formula for it is obtained see 27

3.19
Here and throughout the paper,

3.20
We search the solution of 3.16 in the following form: where We have that

3.23
Let us denote where 0 ≤ y < 1.Then, one can write 1 − y q s yq s 1 .

3.25
So the values of p t k p t k−1 /2, 1 ≤ k ≤ N can be obtained by the following formula: for r 0, 1, . . ., N.

3.27
For k 1, one can show that w 1 is the solution of the difference scheme Abstract and Applied Analysis

3.28
We have the system of linear equations and we write them in the matrix form where

3.30
Here, for any n 1, 2, . . ., M − 1, and D is M 1 × M 1 identity matrix.Using 3.29 , we can obtain that For k 2, w 2 is the solution of the difference scheme Abstract and Applied Analysis

3.33
The system of linear equations given above can be written in the matrix form where Abstract and Applied Analysis 19

3.36
Using 3.34 , we can obtain that

3.37
For 3 ≤ k ≤ N, we can obtain the following difference scheme: 1 − y q s yq s 1

3.38
Abstract and Applied Analysis

3.42
Applying the last formula step by step, we can reach w k .Then, using 3.21 , 3.25 , and 3.26 , we reach the approximate solutions of u t, x and p t k p t k−1 /2.

Error Analysis
In this part, the results of the numerical analysis is given.The numerical solutions are recorded for different values of N and M and u k n represents the approximate solution of u t, x at grid points t k , x n .Table 1 gives the error analysis between the exact solution and the solutions derived by difference schemes.Table 1 is constructed for N M 15, 45, and 75, respectively.For their comparison, the errors are computed by E max 1≤k≤N 1≤n≤M u t k , x n − u k n .

3.43
Thus, the second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme.

Theorem 2 . 1 .
Let π , and ρ ∈ C 0, T .Then for the solution of problem 1.1 , the following coercive stability estimates

Table 1 :
Comparison of exact solution and approximate solutions.