Stability of solution of a backward problem of a time-fractional diffusion equation with perturbed order

The aim of this paper is of studying the stability of solution of a backward problem of a timefractional diffusion equation with perturbed order. We investigate the well-posedness of the backwardproblemwithperturbedorder for t>0. The results on theunique existence and continuitywith respect to the fractional order, the source term as well as the final value of the solution are given. At t=0 the backward problem is ill-posed and we introduce a truncated method to regularize the backward problem with respect to inexact fractional order. Some error estimates are provided in Holder type.


INTRODUCTION
Let T > 0, α ∈ (0, 1), Ω = (0; π) and be the standard Laplace operator, we consider the inhomogeneous time-fractional diffusion equation x ∈ Ω, where D α t (.) is the Caputo fractional derivative with respect to t of the order define as ∫ t 0 (t − τ) −α u τ (x, τ)dτ, 0 < α < 1 u t (x, t), α = 1 As is known, when α = 1 the problem (1.1) -(1.3) is ill-posed for any 0 ≤ t < Tand which was studied in many papers such as 1,2 . In the last decade, the fractional backward problem with 0 < α < 1 was investigated. In this case, the fractional linear backward problem is stable for 0 < t < T and instable at t = 0 which is differential from the case. Hence, regularization of solution at is in order. Ting Wei et al. 3 and Tuan et al. 4 used the Tikhonov method to regularizing the homogeneous and nonhomogeneous problem. Yang et al. 5 also regularize the nonhomogeneous problem by the quasi-reversibility method. These papers used spectral method to obtain an explicit formula for the solution and gave regularization directly on that formula. In the listed paper, the fractional order is assume to be known exactly. But in the real world problem, the parameter is defined by experiments. Hence, we only know its values inexactly. Even if the parameters are known exactly but are irrational, then we only have its approximate values to compute. Thus, a natural question that arises in numerical computing is whether the solution of a problem is stable with such approximate parameters. To the best of our knowledge, this question has still not been considered much. We can list here some papers. Li and Yamamoto 6 investigated the solution of a forward problem with Neumann condition. Trong et al. 2 studied the continuity of solutions of some linear fractional PDEs with perturbed orders. In our knowledge, until now, we do not find another paper which considers the backward problem with respect to the inexact order. Base on the discussion above, we will The remainder of the present paper is organized as follows. The second section provides mathematical preliminaries, notations and lemmas which are used throughout the rest of this paper. In the third section, we investigate for the well-posedness of the problem (1.1) -(1.3) when 0 < t < T. Lastly, we give a method to regularization the problem (1.1) -(1.3) at t = 0

MATHEMATICAL PRELIMINARIES
In this section we set up some notations and some Lemma which use to proof the main results of the paper. First, we list some properties of the Mittag-Leffler function , z ∈ C where α, β ∈ C and Re(α) > 0. For short, we also denote E α,1 (z) = E α (z) Lemma 2.1 7 Letting α, λ > 0 and k ∈ N, we have Lemma 2.2 ( 8 Let 0 < α * < α * < 1 and let α, α ′ ∈ [α * , α * ] then there exists a constant C > 0 which dependent only on α * , α * such that

THE WELL-POSEDNESS OF THE BACKWARD PROBLEM WITH t > 0
In this section, we give a condition to the backward problem have a unique solution and we also prove that the solution is dependent continuously on the fractional order and the final data. As is known, by Fourier series the problem (1.1)-(1.2) corresponding to the initial data u(x, 0) = ξ (x) can be transform to the integral equation as follows Letting t = T and then by direct computation, we obtain From now on, we denote the solution of the backward problem (1.1)-(1.3) which satisfy (3.1) by u α,g, f to emphasize the relationship of function u with the data α, g, f In the following lemma, we give some estimates for G f ,α , H f ,α (t). Lemma 3.1. Let α ∈ (0, 1) Let g be the final data such that g ∈ H r (Ω) and the source function f ∈ L ∞ (0, T; H r (Ω)) then we have due to the Lemma 1 we have

The latter inequality yields
This implies the inequality (3.2). To prove the inequality (3.3), we note that this follows This completed the proof of the Lemma. ) which is given by where G k,f,g,α , H k,f,α (t) are defined in ( ) (ii). If r > 0 then, for any t > 0 we have The proof of Part (i) can be found in 4 .
(ii). The proof is subdivided into two steps.
Using the triangle inequality and combining Step 1 with Step 2 we obtain the desired. Indeed, from Step 1 and Step 2, we choose p such that p = [ |α − α ′ | 1 8+2r ] + 1, then ) Therefore, we only prove Step 1 and Step 2 in detail. The proof of Step 1. Using the Cauchy-Schwarz inequality, we have and H k,f,α (t), G k,f,α are defined in (3.1). Estimating for I 1 . We can use Lemma 2.2 to obtain due to λ k ≥ λ 1 for any k ∈ N, which imply that .

(3.5)
Estimating for I 2 . From the Lemma 2.2, we have , therefore, from (3.5) and (3.6), we obtain Estimating for I 3 . From the Lemma 2.2, for any p > 1 we have where C 31 is independent of |α − α ′ | and p Thus we get Combining (3.6) with the latter inequalities, we deduce Using Lemma 3.1, we have due to ln λ p ≤ λ p . Since 1 ≤ p ≤ λ p , then from (3.5), (3.7) and (3.8), we obtain This completed the proof of Step 1. We now proof Step 2. The proof of Step 2.
We can use the Lemma 2.1 and (3.6) to obtain ) .
This completed the proof of Step 2 and the proof of the Theorem.

REGULARIZATION AND ERROR ESTIMATES FOR BACKWARD PROBLEM AT t = 0
In this section, we propose a regularization method to regularize solution of the backward problem at t=0 we will give some error estimates in the case of inexact order. Let ε ∈ (0, 1), and α ε ∈ (0, 1), g ε ∈ H r (Ω), f ε ∈ L ∞ (0, T; H r (Ω)) be measurement data such that the following condition We approximate the solution of the backward problem at t=0 by the problem where p is the regularization parameter and G k, f ,g,α is defined in (3.1). First, we prove that the problem (4.2) is well-posed with respect to the fractional order. Theorem 4.1 Let 0 < α * < α * < 1 and let α, α ε ∈ [α * , α * ]. Let g, g ε ∈ H r (Ω) and f, f ε ∈ L ∞ (0, T; H r (Ω)) .