Identification of a Time-Dependent Source Term in a Nonlocal Problem for Time Fractional Diffusion Equation

. This paper is concerned with the inverse problem of recovering the time dependent source term in a time fractional diffusion equation, in the case of non-local boundary condition and integral overdetermination condition. The boundary conditions of this problem are regular but not strongly regular. The existence and uniqueness of the solution are established by applying generalized Fourier method based on the expansion in terms of root functions of a spectral problem, weakly singular Volterra integral equation and fractional type Gronwall’s inequality. Moreover, we show its continuous dependence on the data


Introduction
Differential equations involving fractional time derivatives of order less than 1, namely time fractional diffusion equation (TFDe) have become an essential tool in modeling slow diffusion (subdiffusion) processes in chemistry, physics, viscoelasticity, biology, nuclear power engineering [7,26,27].The problems with TFDe should be more difficult since some standard methods cannot be applied for a non-classical derivative.The difficulty comes from the definition of the fractional-order derivatives, which is essentially an integral with the kernel of weak singularity.
It is important to emphasize that the first theoretical results for the inverse problem of determining the coefficients in TFDe are obtained in [10,20,22,33,39].Inverse source problems for TFDe have been intensively investigated by many researchers under various initial, boundary and overdetermination conditions.The inverse problem of finding of a space-dependent source term from final temperature distribution was considered in [4,6,14,15,21,22,30,37] and recovering a space-dependent source term from total energy measurement has been discussed in [8,24,25].For a TFDe inverse problem of identification a time-dependent source term from temperature measurement at the selected point in the spatial domain was considered in [5,19] and determining a time-dependent source term from integral type overdetermination condition was studied in [2,3,16,18].The initial and boundary data identification from final measurements in the initial boundary value problem for time-fractional heat equation was studied in [1,21] and [9], respectively.
In studying inverse source problem for TFDe there are several approaches in the literature, one of which is the generalized Fourier method.The papers [1,2,3,4,18,22,30] consider the inverse source problems in point of view of spectral analysis that the temperature distribution is expanded in terms of root functions (the system of eigenfunction and associated function) of a spectral problem with the boundary conditions which are used in related problem.If the boundary conditions includes some nonlocal characters then the classical self-adjoint eigenfunction expansion results can not be applied in auxiliary spectral problem and it neeeds additional eigenfunction expansion investigation [28].Non-self-adjoint operators appear e.g. in the modeling of processes with dissipation [17,36].In many cases a nonlocal condition is more realistic in handling physical problems than the classical local conditions, which motivates the investigation of nonlocal boundary-value problems.
In this paper, we are concerned with an inverse source problem of recovering a time-dependent source term a(t) and u(x, t) for the following TFDe, supplemented with the initial condition and the nonlocal family of boundary conditions where 0 < q < 1, Ω T = {(x, t) : 0 < x < 1, 0 < t ≤ T } , f (x, t), φ(x) are given functions and α is real constant.The direct problem is to determine the solution u(x, t) that satisfies (1.1)-(1.3),when the function a(t) is known.The structure of the source term a(t)f (x, t) in (1.1) appears in microwave heating process, in which a(t) is proportional to power of external energy source and f (x, t) is local conversion rate of microwave energy.The external energy is supplied to a target at a controlled level by the microwave generating equipment.Inverse source problem for such a model gives an idea of how total energy content might be externally controlled.However, we are interested in finding the pair of solution {a(t), u(x, t)} from (1.1)-(1.3)with integral overdetermination condition where E(t) are given functions.The integral condition (1.5) arises naturally and can be used as supplementary information in the identification of the source term.Such type of condition can describe several physical phenomena in context of chemical engineering, thermoelasticity, heat conduction and diffusion process, fluid flow in porous media [18].
Compared with the inverse source problem for TFDe with local boundary conditions, many less results are available for the nonlocal problems.
The paper [22] investigated first an inverse problem of finding the temperature distribution and the space dependent heat source from final data.The obtained results were extended to the two-dimensional problem in [23].For the case α = 0 in (1.3), the authors in [2] determined a source term independent of the space variable and solution for TFDe from integral type extra condition.The well-posedness of the inverse problem in more general case was proved by using eigenfunction expansion of a non-self adjoint spectral problem along the generalized Fourier method in [18].For a two parameter TFDe, the authors in [15] constructed the series representations of the solution and space dependent source term by utilizing solvability of a 2 × 2 linear system with a coefficient matrix involving Mittag-Leffler functions.Recovering time dependent coefficient of a generalized TFDe from energy measurement and space dependent source term from final data was performed in [13] and [14], respectively.In all the previous works cited above, the problems considered involve the special case of (1.3): α = 0 in [2] and α = 1 in [13,14,15,22,23].The general case α ̸ = 0 for boundary condition and weakly singular Volterra integral equation approach for unique determination of solely time dependent source term is considered for the first time in our work.
The rest of the manuscript is organized as follows: in Section 2, we present the mathematical formulation of auxiliary spectral problem and construction of the bi-orthogonal system.In Section 3, existence and uniqueness of the solution of the inverse time-dependent source problem is proved by generalized Fourier method and Volterra type integral equation with the kernel may have a diagonal singularity.Moreover, the continuous dependence upon the data of the solution of the inverse problem is also shown by utilizing the fractional type Gronwall's inequality in this section.Finally, in Section 4, we recall some basic definitions and facts on fractional calculus and present some necessary lemmas needed for further investigations.

Auxiliary spectral problem and construction of the bi-orthogonal system
The spectral problem for the initial boundary value problem (1.1)-(1.3) is given by If α ̸ = −1, then the boundary conditions in (2.1) are regular but not strengthened regular [28].In this case the problem (2.1) has double eigenvalues λ n = (2πn) 2 (except for the first λ 0 = 0).The set of eigenfunctions and eigenvalues of the problem (2.1) are the following: It is well-known that the main difficulty in applying the Fourier method is its basisness, i.e., the expansion in terms of eigenfunctions of the auxiliary spectral problem.The set of eigenfunctions {X n (x)} (n = 0, 1, 3, 5, ...) of the spectral problem (2.1) is not complete in the space L 2 (0, 1).The set of eigenfunctions with the associated eigenfunctions is supplemented to make the set complete on L 2 (0, 1).We use the equation to avoid the problem of the choice of associated functions for their construction.
Then, associated functions are: (2.3) In the rest of the paper, we will consider the following class of functions to investigate well-posedness of the inverse source problem (1.1)-(1.4): then we have: where c 1 and c 2 are some positive constants.

Well-posedness of the inverse problem
A solution of the inverse problem (1.1)-(1.4),which we called the classical solution, is a set of functions {a(t), u(x, t)} satisfying a(t The choice of the term D q 0+ (u(x, t) − u(x, 0)) instead of the usual term D q 0+ u(x, t) is not only to avoid the singularity at zero, but also to include certain initial conditions [31].
In this part, we will examine the existence, uniqueness and stability of the solution of the inverse initial-boundary value problem for the TFDe (1.1) with time-dependent source term.For the proof, we will need to use the eigenfunctions expansion method, some properties of the Volterra integral equation with weak singular kernel and fractional type Gronwall inequality.
where e q (t, λ) := E q (−λt q ) and e q,q (t, λ) := t q−1 E q,q (−λt q ).According to (4.1), (3.5), the solution v 1n (t), n=1, 2, ... of (3.4) is given by where ′ * ′ is the convolution integral, The formulas (3.1) and (1.4) yield a following Volterra integral equation of the first kind with respect to a(t) : where Further, the Equation (3.6) yields the following Volterra integral equation of the second kind by taking fractional derivative D q 0+ : By using the properties (iii) in Proposition 1, it is easy to show that and a(t) lim τ →t−0 According to (3.7)-(3.10),we obtain the Volterra integral equation of the second kind with respect to a(t) in the form with the free term P and kernel Q According to the Lemmas 2 and 3 we estimate the kernel of (3.11) in the following form: where Because the kernel Q(t, τ ) belongs to the class S ν with 0 < ν < 1, Volterra integral equation (3.11) is weakly singular.Then, it has a unique solution a ∈ C[0, T ] according by Lemmas 5 and 6.First, let us show that the solution of inverse problem (1.1)-(1.4) is unique.To prove the uniqueness of the solution, we suppose that {a(t), u(x, t)} and {r(t), v(x, t)} are two solution sets of inverse problem (1.1)- (1.4).According to form of solutions (3.1) and (3.11), we have Now, we denote R(t) = |a(t) − r(t)|, thus (3.12) and (3.14) imply the inequality In view of the Lemma 7 we have R(t) ≤ 0 that yields a(t) = r(t).Consequently, we have u = v after inserting a = r in (3.13).
So far, we have proved the uniqueness of the solution of the inverse problem.Because the solution u(x, t) is formally given by the series form (3.1), we need to show that the series corresponding to u(x, t), u x (x, t), u xx (x, t) and D q 0+ (u(x, .)− u(x, 0)) represent continuous functions.Under the assumptions (A 1 )-(A 3 ) and Lemma 1, for all (x, t) ∈ Ω T , the series corresponding to u(x, t) is bounded above by the series where max The majorizing series (3.15) is convergent by using Lemma 1, Schwarz inequality and p-series test.This implies that by the Weierstrass M-test, the series (3.1) is uniformly convergent in the rectangle Ω T and therefore, the solution u(x, t) is continuous in the rectangle Ω T .
The majorizing series for x-partial derivative is 1n . (3.16) The majorizing series (3.16) is convergent by employing Lemma 1, Schwarz inequality and p-series test.Hence, by the Weierstrass M-test, the series obtained for x-partial derivatives of (3.1) is uniformly convergent in the rectangle Ω T .Therefore, their sum u x (x, t) is continuous in Ω T .Similarly, we can show that the series corresponding to xx-partial derivatives of (3.1) is uniformly convergent and u xx (x, t) represents continuous function.Now, it remains to show that q-fractional derivative of the series u(x, t) − u(x, 0) represents continuous function on Ω T .We will show that for any ϵ > 0 and t ∈ [ϵ, T ], the following series is generated according to Lemma 4 corresponding to q-fractional derivative of the function u(x, t) − u(x, 0) is uniformly convergent.Now, we can see that Equations (3.2)-(3.4)yield We have the following estimates , and we obtain a majorant series as follows Consequently, It is important to note the phenomenon that the smoothness of coefficient of (1.1) does not imply smoothness of the solution u in the closed domain.It is discussed in more detail in [38] that ∂ ∂t u(x, t) blows up as t → 0 + which forces, for example, the overdetermination data E(t) to belong to the class (A 2 ).
The following result on continuously dependence on the data of solution of the inverse problem (1.1)-(1.4)holds.
Theorem 2. (Lipschitz stability) Let ℑ be the class of triples in the form of {f, φ, E}, which satisfies the assumptions (A 1 )-(A 3 ) of Theorem 1 and for some positive constants N i , i = 0, 1, 2, 3.
Then, the solution pair {a(t), u(x, t)} of the inverse problem (1.1)-(1.4)depends continuously upon the data in ℑ.
Finally, using (3.11) and (3.18), we obtain , (3.19) where ϵ 3 = E q (ϵ 2 T q ).By virtue of the expression for P (t) − P (t), one can estimate that . By using the inequality (3.20), from (3.19) we get ).This shows that a(t) depends continuously upon the input data.From (3.13), a similar estimate is also obtained for the difference u − u as This completes the proof of Theorem 2. ⊓ ⊔

Conclusions
The paper considers an inverse source problem of identification of the timedependent source term from the energy measurement for TFDe with a general form of nonlocal boundary conditions which is regular but not strongly regular.The well-posedness of the inverse problem is proved by means of Fourier expansion method, some properties of Volterra integral equation with weak singular kernel and fractional type Gronwall's inequality.In all the previous works cited in introduction, the problems studied involve the special case of (1.3): α = 0 in [2] and α = 1 in [13,14,15,22,23].The general case α ̸ = 0 for boundary condition and weakly singular Volterra integral equation approach for unique determination of solely time dependent source term is considered for the first time in this work.This approach can be extended by considering different type of fractional derivative, boundary and overdetermination conditions, which are the line of future investigations.Another future line of investigation is to extend the fractional derivative to multi-dimensional and nonlinear evolutional equations, see [30] and references therein.The paper [30] demonstrates the importance of addressing the given inverse problem for establishing the existence of a solution to the nonstationary Navier-Stokes equations with a prescribed flow rate within an infinite cylinder.

⊓ ⊔
We will also need to recall the following result.
corresponding to the data  and , respectively.