An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time-and Space-Dependent Sources in Fractional Diffusion and Wave Equations

: In this article, we consider two inverse problems with a generalized fractional derivative. The ﬁrst problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the ﬁnal time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of ﬁnal time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.


Introduction
Fractional derivatives are increasingly used in modeling various processes in physics, biology, economics, engineering sciences, etc. [1].In addition to classical fractional derivatives, several generalizations have been introduced to better match the models to the reality in different situations.In this paper, we work with generalized fractional derivatives of Riemann-Liouville and Caputo type where the power-type kernel (fractional derivative case) is replaced by an arbitrary function k.Such a generalization was previously used in [2][3][4][5] and covers many specific cases that are important in applications (see Section 2.1).
Fractional derivatives of Riemann-Liouville and Caputo type are non-local: the derivative of a function u(t) at t = T depends on values of u at t < T. We consider an inverse problem (IP1) to recover a history of a function u at 0 < t < T by means of measurements of u(t) and its generalized fractional derivative in a left neighborhood of T. To the authors' knowledge, such a problem has not yet been considered in the literature.
We use the results obtained for IP1 in order to investigate an inverse problem of reconstruction of a history of a source in a general PDE that includes as particular cases fractional diffusion and wave equations from the measurements in a left neighborhood of final time T (IP2).
Quite often in the inverse source problem, the goal is to determine a source that is either a spaceor time-dependent function.The space-dependent source term is usually reconstructed based on the final time overdetermination condition [6][7][8][9][10][11].The time-dependent source term can be recovered from additional boundary measurements [7] or from integral conditions [12,13].In this paper [14], the source term dependent on time and part of the space variables has been determined.In this paper, we assume that the overdetermination condition is given not only at the final moment of time T, but in its neighborhood.This enables us to reconstruct the source term that depends on both time and all space variables.
In Section 2, we explain the concept of generalized fractional derivative with examples.Next, we formulate the inverse problems and give hints to their physical applications.In Section 3, we prove the uniqueness for a general class of kernels k and reduce IP1 to an integral equation that is further used to derive the solution formulas.Finally, in Section 4, we derive the solution formulas in some particular cases of k based on the expansion with the Legendre polynomials.

Generalized Fractional Derivatives
In this paper, L p (0, T) and W n p (0, T) stand for real Lebesgue and Sobolev spaces.We are solving problems with a generalized fractional derivative.This concept has been used in [2][3][4][5].We utilize D {k},n a as a unified notation that stands for the generalized fractional derivatives in The notation of generalized fractional derivative incorporates the following possibilities.The basic case is is the Riemann-Liouville fractional integral of the order Often a memory is not of power-type.A direct generalization of (k1) leads to multiterm and distributed order fractional derivatives [15][16][17].These derivatives have the following kernels: , respectively.Distributed order and multiterm derivatives enable to model accelerating and retarding sub(super) diffusion, since different powers of t dominate as t → 0 + and t → ∞ in the kernel.A proper choice of p in (k3) allows modelling ultraslow diffusion [16].
The cases (k2) and (k3) can be unified to a form of Lebesque-Stiltjes integral k(t) = 1 0 , but we will treat them separately.
Tempered fractional derivatives are used to describe slow transition of anomalous diffusion to a normal one.There are two models of this type in the literature that differ in their mathematical derivations.The corresponding kernels are: [19,20].
We will call derivatives with kernels (k4) and (k5) tempered fractional derivatives of type I and II, respectively.

Formulation of Inverse Problems
Let 0 < t 0 < T < ∞.Our basic inverse problem consists in a reconstruction of a function in (0, t 0 ) provided that this function and its derivative are given in (t 0 , T).
An example of IP1 is the reconstruction of physical quantities in constitutive relations involving fractional derivatives.In the Scott-Blair model of viscoelasticity, the stress is proportional to a time fractional derivative of the strain [25].In this context, IP1 means the reconstruction of a history of the strain of a body by means of the measurement of strain and stress in a left neighborhood of a time value T. A similar meaning for IP1 can be given in the subdiffusion where the flux is proportional to a time fractional derivative of the concentration (temperature) gradient [26].
Next, we formulate IP2 that is an inverse source problem that can be reduced to IP1: is fulfilled and Here, Ω ⊆ R N with some N ∈ N, D l = l ∑ j=1 q j ∂ j ∂t j with some l ∈ N, q j ∈ R, and A and B are operators that act on functions depending on x.Throughout the paper, assume that A and B with their domains D(A) and D(B) are such that A : D(A) ⊆ C(Ω) → C(Ω), B : D(B) ⊆ C(Ω) → C(Ω).We also assume that B is invertible.[13,27,28], the attenuated wave equation
We point out that the operators A and B in (2) are not necessarily linear.
In case if Φ = 0, IP2 means a reconstruction of a source that was active in the past using a measurement of the state of u in a left neighborhood of T. Such an inverse problem may occur in seismology, ground water pollution, etc.Now, we reduce IP2 to IP1.Let (u, F) solve IP2.Then, Equation (2) restricted to Ω × (t 0 , T) has the form (D {k},n 0 Bu)(x, t) + D l ϕ(x, t) − Aϕ(x, t) = Φ(x, t).Therefore, Bu is a solution of the following family of IP1: where The solution of IP2 is expressed by means of Bu explicitly: Bu + D l u − Au.

Results in Case of
Using the analyticity of k C , it is possible to show that functions u and v involved in the formula w C (t + is) = u(t, s) + iv(t, s), are continuously differentiable and satisfy Cauchy-Riemann equations in {(t, s) : t + is ∈ D t 0 }.This implies that w C is complex analytic in D t 0 .On the other hand, its restriction to the subset {z = t + i0 : t ∈ (t 0 , ∞)} is the function w.Therefore, w is real analytic in (t 0 , ∞).

We will denote the Laplace transform of a function
The symbol * will stand for the time convolution, i.e., ( We prove a uniqueness theorem for IP1.Theorem 1. Assume that k satisfies the following conditions: k(s) cannot be meromorphically extended to the whole complex plane C.
Then, the following assertions hold.
Proof.(i) Let us extend u(t) by zero for t > T and define the function f : (0, ∞) → R: Since u(t) = 0, t > t 0 , it holds that The function k is real analytic, therefore, k (n) is also real analytic.Hence, Lemma 1 implies that f is real analytic in (t 0 , ∞).Since f (t) = 0, t ∈ (t 0 , T), and f is real analytic, we obtain that f (t) = 0, t > t 0 .
Due to (5) the k(s) exists and is holomorphic for Res > µ.Moreover, in view the properties of f , the f (s) also exists and is expressed by the formula Therefore, for any s such that Res > µ and s n u(s) = 0.
Since the values f (t) and u(t) vanish for t > t 0 , f and u are entire functions.Thus, the function f (s) + p 0 s n−1 + ... + p n−1 is also entire.Assume that u does not vanish on C.Then, by Identity theorem and the fact that u is entire the set of zeros of u does not contain accumulation points.This implies that the extension of k is meromorphic on C.This contradicts to the assumption (7) of the theorem.Therefore, the assumption u ≡ 0 is invalid, which implies u = 0 in L 1 (0, T).
Let us compute the Laplace transform for the kernels from Section 1 to see if they satisfy the conditions of Theorem 1.
The kernel of Caputo-Fabrizio fractional derivative (k6) does not satisfy (7) because it has the meromorphic in C Laplace transform.IP1 with this kernel has infinitely many solutions.Any function such that and any function such that . Now, we proceed to IP2.We define the following set related to operators A, B and D l : From Theorem 1, we can immediately deduce a uniqueness statement for IP2.

Reduction to Integral Equations
In this subsection, we reduce IP1 to integral equations.Let us assume that k satisfies (6).
Firstly, we consider the case D for t ∈ (t 0 , T), where the left hand side belongs to W n 1 (t 0 , T) and the first addend in the right-hand side belongs to C ∞ (t 0 , T].Thus, the data ϕ necessarily satisfies ∀δ ∈ (t 0 , T). Applying d n dt n to (8), using the second condition in (1) and rearranging the terms, we obtain the following integral equation of the first kind for u| (0,t 0 ) : Secondly, let us consider the case D {k},n 0 Since lim the integration: Due to Lemma 1, the integral operators involved in ( 9),( 10) map L 1 (0, t 0 ) into the space of functions that are real analytic in t > t 0 .This means that IP1 is severely ill-posed and necessarily, f is real analytic in (t 0 , T).In the next section, we will derive solution formulas for IP1 that contain the quantities where t 1 is an arbitrary point in (t 0 , T).

Solution Formulas in Particular Cases of k 4.1. A Basic Theorem
Theorem 2. Let α ∈ R \ Z, t 1 > t 0 > 0 and f ∈ C ∞ (t 0 , ∞).Let us introduce the following family of sums that depend on a variable t ∈ (0, t 0 ) and parameters α, f , t 1 , t 0 : Here, N ∈ {0} ∪ N ∪ {∞}, P n are normalized in L 2 (−1, 1) Legendre polynomials and Assume that v ∈ L 2 (0, t 0 ) and f is given by f (t) = t 0 0 converges pointwise in (0, t 0 ) and the estimate is valid: where c(t) is a positive constant depending on t.
Proof.For t 1 > t 0 we have The substitution s = 1 t 1 −τ under the integral takes (12) to the form where w(s) = s −α−2 v t 1 − 1 s .We would like to expand our function into series by means of orthonormal Legendre polynomials; thus, we apply a linear substitution that takes us from , where such an expansion can be applied: We also denote w( s) = w(s).Since the performed changes of variables under the integrals are diffeomorphic, v ∈ L 2 (0, Since w ∈ L 2 (−1, 1), it can be expanded into the Fourier-Legendre series.It follows from ( 13) that for n ∈ {0} ∪ N and, therefore, It implies that for the normalized Legendre polynomials 1 Then, w( s) = ∑ ∞ n=0 A n P n (s).This series converges in L 2 (−1, 1) and for almost every s ∈ (−1, 1) [34].
Since the change of variables s = a t 1 −t + b, t ∈ [0, t 0 ], is diffeomorphic and v(t) = (t 1 − t) −α−2 w( a t 1 −t + b), all assertions of the theorem follow from the proved properties of the series w( s) = ∑ ∞ n=0 A n P n (s).
Remark 1.It follows from (11) that for f of form f (t) = t 0 0

Solution Formulas in Case of Usual Fractional Derivatives
In this subsection, we consider the case are the Riemann-Liouville and Caputo fractional derivatives of the order n + β − 1, respectively.
Then, the following assertions hold.
where the operator F β,n R,t 1 is given by the rule where The Formulas (14), ( 16) are valid for any t 1 ∈ (t 0 , T).

Solution Formulas in Case of Tempered and Atangana-Baleanu Derivatives
In this subsection, we derive the solution formulas for particular subcases of the generalized fractional derivative of the order n = 1.They are based on solution formulas derived for the usual fractional derivative and involve the operators F β,1 . Again, we assume that t 1 is an arbitrary number in the interval (t 0 , T).
Firstly, let us consider the tempered fractional derivatives of type I.
Then, the following assertions hold.
Finally, we consider the case of Atangana-Baleanu fractional derivative.
Similarly to Corollary 2, formulas of solutions of IP2 can be derived in cases of tempered and Atangana-Baleanu derivatives.

Conclusions
In this paper, two inverse problems were considered .The goal of IP1 was to reconstruct the history of a function based on its value and the value of its generalized fractional derivative on a final

Proof.
The function k can be extended as a complex analytic function k C in an open domain D ⊂ C containing the positive part of the real axis.Let us define w C