Existence and Uniqueness of the Solution for an Inverse Problem of a Fractional Diffusion Equation with Integral Condition

The solvability of the fractional partial di ﬀ erential equation with integral overdetermination condition for an inverse problem is investigated in this We analyze the direct problem solution by using the “ energy inequality ” method. Using the ﬁ xed point technique, the existence and uniqueness of the solution of the inverse problem on the data are established.


Introduction
This work devoted to study the solvability of a pair of functions fu, f g satisfying the following fractional parabolic problem: with the initial condition the boundary condition and the nonlocal condition ð Here, Ω is a bounded domain in ℝ n with smooth boundary ∂Ω:. The functions g, φ, and θ are known functions, and β is a positive constant. And Γ ð·Þ denotes the gamma function. For any positive integer 0 < α < 1, the left Caputo derivative is defined as Inverse parabolic equation problems occur naturally in many fields, and there is extensive literature on inverse heat equation problems (see [1][2][3][4], and references therein). The form (4) is an additional information of problem.
In engineering and physics, the parameter recognition in a partial differential equation from the data of the integral overdetermination condition plays an important role [5][6][7][8][9][10]. From a physical point of view, these conditions can be interpreted by a system averaging the domain of spatial variables as measurements of the temperature uðx, tÞ.
Note that nonlocal problems related with integral overdetermination [11,12]. Studies have shown that when we deal with these kinds of nonclassical problems, classical approaches sometimes do not work [13,14]. To date, different methods for addressing problems resulting from nonlocal problem have been suggested. The choice of approach depends on the form of nonlocal boundary value that are involved.
We note that several authors have studied the inverse parabolic problem with condition of type (4) and its special solubility (see, for example, [3,4,[15][16][17][18][19][20]). There are also several articles dedicated to the study of the existence and uniqueness of inverse problem solutions for different parabolic equations with unknown source functions. Inverse problems related by determining unknown function in source term of a parabolic equation with overdetermination condition [21,22].
In recent years, fractional differential equations have created growing interest from engineers and scientists and have great importance in modeling complex phenomena. Because FDEs have memory, nonlocal space, and time relationships, using these equations, complex phenomena can be modelled [23][24][25][26][27][28].
Namely, in the present paper, a new research on the inverse problem of a fractional parabolic equation is discussed, for which the solvability of the problem (1)-(4) is reduced to the concept of a fixed point technique. This work is divided into four sections; we start with an introduction then we give some definitions of function space and important lemmas. The third section is devoted to studying the solvability of the direct fractional parabolic problem. Finally, in the last section, we prove the existence and uniqueness of the solution to the main problem.

Functional Space
Definition 1. Let us introduce certain notations used below, we set We denote by Cðð0, TÞ, L 2 ðΩÞÞ the space is composed of all continuous functions on ð0, TÞ with values in L 2 ðΩÞ. For any 0 < α < 1, the Caputo and Riemann-Liouville derivatives are defined, respectively, as follows: (i) The left Caputo derivatives: (ii) The left Riemann-Liouville derivatives: (iii) The right Riemann-Liouville derivatives: Many authors believe that the Caputo version is more natural because it makes it easier to manage inhomogeneous initial conditions. Then, the following relationship is related to the two concepts (7) and (8), which can be checked by a direct calculation: Definition 2 (see [29]). For any real σ > 0, we define the space l H σ 0 ðIÞ as the closure of C ∞ 0 ðIÞ with respect to the following norm k·kl H σ 0 ðIÞ : where u j j 2 Definition 3. For any real σ > 0, we define the space r H σ 0 ðIÞ as the closure of C ∞ 0 ðIÞ with respect to the following norm k·kr H σ 0 ðIÞ : where u j j 2 Lemma 4 (see [29,30]). For any real σ ∈ ℝ + , if u ∈ l H α ðIÞ and v ∈ C ∞ 0 ðIÞ, then Lemma 5 (see [29,30]). For 0 < σ < 2, σ ≠ 1, u ∈ H σ/2 0 ðIÞ, on a Journal of Function Spaces Lemma 6 (see [29,30] Lemma 7 (see [29]). For any real σ > 0, the space l H σ 0 ðIÞ with respect to the norm (11) is complete. Definition 8. We denote by L 2 ð0, T, L 2 ð0, dÞÞ ≔ L 2 ðQÞ the space of square functions, integrated with the scalar product in the Bochner sense, Since the space L 2 ð0, dÞ is a Hilbert space, it can be shown that L 2 ð0, T, L 2 ð0, dÞÞ is a Hilbert space as well. Let C ∞ ð0, TÞ denote the space of infinitely differentiable functions on ð0, TÞ and C ∞ 0 ð0, TÞ denote the space of infinitely differentiable functions with compact support in ð0, TÞ.

Solvability of the Direct Fractional
Parabolic Problem

Position of Problem.
In the rectangular domain Q = ð0, dÞ × ð0, TÞ = Ω × I, with d, T < ∞ and 0 < α < 1, we shall study the existence and uniqueness of solutions u = uðx, tÞ to the following fractional parabolic problem: We consider the following fractional parabolic equation of the type with the initial condition and Dirichlet condition where b ∈ ℝ + * ;f and φ are known functions. We shall assume that the function φ satisfies a compatibility conditions, i.e., where So, we get Such that with the initial condition the boundary condition of Dirichlet type where 3 Journal of Function Spaces

A Priori Estimate.
In this section, we illustrate the existence and uniqueness of the problem's solution (27)-(29) as a solution of the operator equation where L = ðL, ℓÞ, with domain of definition B consisting of functions v ∈ L 2 ðQÞ, such that v, C D α t v, ð∂v/∂xÞ ∈ L 2 ðQÞ, and v verify (29).
The operator L is considered from B to F, where B is the Banach space consisting of all functions vðx, tÞ having a finite norm v k k 2 and F is the Hilbert space consisting of all elements Fourier = ð f , 0Þ for which the norm L 2 ðQÞ is finite.
where k is a positive constant independent of v.
Proof. Multiplying equation (27) by the following function: and integrating over Q = ð0, dÞ × ð0, TÞ, we get As vðx, 0Þ = 0, so by applying Lemmas 4, 5, and 6 becomes So, we obtain So, we get which give So, we have On the other hand, we have ∂v ∂x By combining (41), (42), and (43), for ε < b/2, we get Journal of Function Spaces Finally, it follows that with Therefore, we obtain that Hence, the uniqueness of the solution.
Remark 10. This inequality kvk B ≤ kkLvk F gives the uniqueness of the solution, indeed: Let v 1 and v 2 two solutions, so then which gives the uniqueness of the solution.
Proposition 11. The operator L from B to F admits a closure.
Proof. Let ðv n Þ n∈ℕ ⊂ DðLÞ be a sequence such that: it must be shown that As the continuity of the fractional derivation(2) and the derivation of the first order (as a particular case of the fractional derivative) of ðC ∞ 0 ðQ T ÞÞ ′ in ðC ∞ 0 ðQ T ÞÞ ′ , then (52) implies On the other hand, the convergence of Lv n to f in F = L 2 ðQ T Þ implies that By virtue of the uniqueness of the limit in ðC ∞ 0 ðQ T ÞÞ ′ , we conclude between (53) and (54) that Hence, the operator L is closable. Theorem 9 is valid for a generalized strong solution, i.e., we have the following inequality: Consequently, this last inequality entails the following corollaries: So there is a corresponding sequence ðv n Þ n ⊂ DðLÞ such that From the estimate (41), we obtain We can deduce that ðv n Þ n is a Cauchy sequence in B, so So we conclude here that Rð LÞ is closed because it is complete (any complete subspace of a metric space (not necessarily complete) is closed).
It remains to show the opposite inclusion. Let z ∈ Rð LÞ, then there is a sequence of ðz n Þ n in F consists of the elements of the set Rð LÞ such that where z ∈ Rð LÞ, because Rð LÞ is closed subset of a complete space F; then, Rð LÞ is complete. So there is a corresponding sequence ðv n Þ n ⊂ Dð LÞ such that Lv n = z n : ð65Þ From the estimate (57), we obtain We can deduce that ðv n Þ n is a Cauchy sequence in B, so Once more, there is a corresponding sequence ðLðv n ÞÞ n ∈ RðLÞ such that Lv n = Lv n over R L ð Þ,∀n: ð68Þ Then lim n⟶+∞ Lv n = z: Consequently, z ∈ RðLÞ, and then, we conclude that Proof. The scalar product of F is defined by If we put w ∈ RðLÞ ⊥ , we have where C D α t v, ∂v/∂x, v ∈ L 2 ðQ T Þ, with v satisfies the boundary conditions of (27)- (29). From (72), we get the equality And from the equality (73), we give the function w in terms of v as follows: then w ∈ L 2 ðQ T Þ.
Replacing w in (73) by its representation (74) and integrating by parts each term of (73) and by taking the condition of v, we obtain ð Hence And thus, v = 0 in Q T which gives w = 0 in Q T . This proves Theorem 15. So RðLÞ = F.

Existence and Uniqueness of the Solution of Main Problem
We are finding a solution in the form of the original inverse problem. fu, f g = fz, f g + fy, 0g where y is the solution of the direct problem Journal of Function Spaces while the pair fz, f g is the solution of the inverse problem where We will assume that the functions that appear in the problem data are measurable and fulfill the following conditions: The correspondence between f and z can be seen as one way of defining the linear operator.
with the values In this view, the linear equation of the second form for the function is rational to refer to f over the space L 2 ð0, TÞ : where Remark 16.
As fu, f g = fz, f g + fy, 0g where y is the solution of the direct problem (78)-(80). Obviously, the previous section implies that y exists and is unique, but instead of demonstrating the solvability of the initial problem (1)-(4), we demonstrate the existence and uniqueness of the inverse problem (81)-(84) solution.
holds, then a solution to the inverse problem exists. Proof.
This gives that f solves equation (89).
(ii) Equation (89) has a solution in space, according to the assumption, L 2 ð0, TÞ, say f . The resulting relationship (81)-(83) can be viewed as a direct problem with a unique solution z∈W 1 2 ðQ T Þ when inserting this function in (81). Let us show that the z function also satisfies the condition of integral overdetermination (84). By equation (92), the function z is subject to the following relation Subtracting equation (92) from equation (93), we get Integrating the preceding differential equation and taking into account the compatibility condition (89), we find that the overdetermination condition (84) is satisfied by z and the function pair fz, f g is a solution to the inverse problem (81)-(84).
This completes the theorem's proof.
Now, we are touching on some properties of operator A.
Lemma 18. Let the condition ðHÞ hold. Then, there exists a positive ε for which A is a contracting operator in L 2 ð0, TÞ.

Journal of Function Spaces
Multiplying both sides of (81) by z scalarly in L 2 ðQ T Þ and integrating the resulting by parts with use of (82), we get Thus, by using the Cauchy's ε-inequality, we obtain Choosing 0 < ε < 2β, we get Omitting some terms on the left-hand side (99) leads to ▽z k k 2 According to (95) and (100), we can obtain the following estimate: where So, we obtain It is obvious from the above that there is positive ε such that Inequality (103) shows that the operator A is a contracting mapping on L 2 ð0, TÞ.
The following result may be useful with respect to the particular solvability of the inverse problem concerned. Theorem 19. Let the compatibility condition (91) and the condition ðHÞ hold. Then, the inverse problem (81)-(84) has a unique solution fz, f g.
Proof. This means that the equation (89) has a unique solution f in L 2 ð0, TÞ.
The existence of a solution to the inverse problem (81)-(84) is verified, according to Lemma 6.
The uniqueness of this solution has yet to be proven. Suppose the contrary that there are two distinct solutions fz 1 , f 1 g and fz 2 , f 2 g of the under consideration inverse problem.
Also, the linear operator A is contracting on L 2 ð0, TÞ from Lemma 18, which gives that f 1 = f 2 ; then, by the theorem of the uniqueness of the solution of main direct problem (78)-(80), we will just have z 1 = z 2 .
Corollary 20. The solution f to equation (91) depends continuously, under the conditions of Theorem 19, on the data W.
Proof. Let V 1 and V 2 twosets of data that satisfy Theorem 19's assumptions.
Let f and g be solutions of the equation (89) corresponding to the data V 1 and V 2 , respectively. According to (103), we have Let us estimate the difference first, f − g. It is easy to see with the use of (103) that so, we get

Conclusion and Perspectives
This work contains a new inverse problem by investigating the fractional derivatives where we develop the method of fixed point and energy inequality method for proving the solvability of an inverse fractional problem. We note that our work extends to the existence of open problems as a study of the nonlinear case of this problem and the numerical part.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare no conflict of interest.