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Inverse Problems Inverse Problems 32 (2016) 015011 (21pp) doi:10.1088/0266-5611/32/1/015011 An undetermined coefﬁcient problem for a fractional diffusion equation Zhidong Zhang Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA E-mail: [email protected] Received 24 July 2015, revised 9 November 2015 Accepted for publication 30 November 2015 Published 18 December 2015 Abstract We consider a fractional diffusion equation (FDE) CDta u = a (t ) uxx with an undetermined time-dependent diffusion coefﬁcient a(t). Firstly, for the direct problem part, we establish the existence, uniqueness and some regularity properties of the weak solution for this FDE with a ﬁxed a (t ). Secondly, for the inverse problem part, in order to recover a (t ), we introduce an operator and show its monotonicity. With this property, we establish the uniqueness of a(t) and create an efﬁcient reconstruction algorithm to recover this coefﬁcient. Keywords: fractional diffusion, fractional inverse problem, reconstruct diffusion coefﬁcient, uniqueness, monotonicity, iteration algorithm (Some ﬁgures may appear in colour only in the online journal) 1. Introduction In this paper, we consider the following fractional diffusion equation (FDE) model: ⎧ CD a u (x , t ) - a (t ) u xx (x , t ) = 0, 0 < t T , 0 < x < 1; t ⎪ ⎪ u (x , 0) = 0, 0 < x < 1; ⎨ 0 < t T; ⎪ u (1, t ) = 0, ⎪ 0 < t T , f (0) = 0; ⎩ u (0, t ) = f (t ) , (1.1) where 0 < a < 1 and T is ﬁxed. Model (1.1) is a diffusion model with time-dependent diffusion coefﬁcient a (t ). This model represents the direct problem—given f (t ), a (t ), solve for the ‘temperature’ in the rod. We are interested in the inverse problem of recovering a(t) from additional boundary data. Indeed, we impose the condition 0266-5611/16/015011+21$33.00 © 2016 IOP Publishing Ltd Printed in the UK 1 Inverse Problems 32 (2016) 015011 Z Zhang - a (t ) u x (0, t ) = g (t ) , 0 < t T , which corresponds to imposing the value of heat ﬂux at the end x = 0, and this inverse problem becomes given f (t ), g (t ), seek the coefﬁcient a (t ). It should be noted that our model is one of ‘anomalous’ diffusion, which is modelled by a time-fractional derivative CDta . Here, CDta (0 < a < 1) is the left-sided Djrbashian–Caputo αth order derivative with respect to time t. The deﬁnition of CDta is CD a u (x , t t) = 1 G (n - a ) t ò0 (t - t ) n - a - 1 dn u (x , t ) dt dt n with Gamma function G( · ) and the nearest integer n which is bigger than a. In our case, we are modeling subdiffusion process so that take n=1. The coefﬁcient function a(t) is positive and continuous. We denote the solution of model (1.1) by u (x, t; a ) to indicate its dependence on the coefﬁcient a(t). A considerable literature has been built over the last 20 years showing that under certain circumstance model (1.1) captures anomalous diffusion processes well, especially the subdiffusion process, in which the expected mean square path x 2 grows slower than that in the Gaussian process for large time. At the microscopic level, the subdiffusion process can be described by the continuous time random walk [23], of which the macroscopic counterpart is a time-fractional differential equation. There are some important applications of the anomalous diffusion processes and fractional derivative. For instance, the thermal diffusion in media with fractal geometry (see [24]), highly heterogeneous aquifer (see [1]), non-Fickian diffusion in geological formations (see [2]) and underground environmental problem (see [6]). Moreover, CDta can be applied to describe the viscoelasticity of some materials (see [21, 30, 31]). We note that in recent years, the direct problem of time-fractional diffusion has drawn much attention. Some classical papers by several authors have shown various aspects of this problem. For instance, Sakamoto and Yamamoto [27] studied some initial/boundary value problems for fractional diffusion-wave equations; meanwhile the same problems for multiterm time-fractional diffusion equations are discussed in [15]. Luchko established the maximum principle for time-fractional diffusion equation in case of 0 < a < 1 in [20]. In the numerical analysis aspect, Jin et al applied ﬁnite element method to analyze fractional subdiffusion equations and gave the corresponding error estimates in [7, 8]. Reference [18] offers some numerical estimates for the fractional advection–dispersion models which have fractional derivatives both on time and space. However, the research on inverse problem of time-fractional diffusion is still relatively scarce. Nonetheless, some interesting works have been exhibited. Cheng et al [4] established the uniqueness for an inverse problem of fractional diffusion equation, which was one of the ﬁrst mathematical works about fractional inverse problems. Li et al [14] discussed the reconstructions of the space-dependent coefﬁcient and the fractional order for a fractional diffusion equation. For the fractional diffusion equation with multi-term time-fractional orders has been proved in [16]. In [33], the authors used a Carleman estimate to deduce the stability in determining a zeroth-order coefﬁcient with one half order Caputo derivative. We also refer [3, 17, 32] for who are interested in Carleman estimate in time-fractional diffusion equations. Rundell et al recovered an unknown boundary condition for a time-fractional diffusion equation in [26]. Jin and Rundell [10] gave a tutorial on inverse problems for anomalous diffusion equations, summarizing some interesting results and pointing out several open problems in fractional inverse and direct problems. For model (1.1), we note that if α tends to 1, then it reproduces the classical diffusion equation with an undetermined coefﬁcient a (t ). The classical parabolic problem was 2 Inverse Problems 32 (2016) 015011 Z Zhang completed by Jones in [11, 12], which was one of the ﬁrst rigorous solutions of an undetermined coefﬁcient inverse problem. Jones used a variational substitution to create an integral equation, somewhat curiously, this was of fractional order. Also he showed the unique ﬁxed point of the integral equation is the unknown coefﬁcient a(t). Then he used monotonicity to prove existence and provided a complete stability and uniqueness analysis. Moreover, for the same model with a fractional a Î (0, 2), Lopushanskyi and Lopushanska [19] have used Green function to give the representation of solution u (x, t ) and an operator equation about a(t), which ensure the existence and uniqueness of (u (x, t ), a (t )). This method is really valuable since it can be extended to other time-fractional differential equations. This paper has two goals for the equation (1.1), which are analyzing the weak solution with a ﬁxed coefﬁcient a(t) and the unique recovery of a(t) from the data -a (t ) ux (0, t ) = g (t )(We prefer -a (t ) ux (0, t ) = g (t ) instead of the left end ﬂux data ux (0, t ) because physically, -a (t ) ux (0, t ) = g (t ) is usually measured rather than ux (0, t )). For the ﬁrst goal, which can be regarded as the direct problem, we establish the existence and uniqueness for the weak solution and deduce some regularity results. To the best of my knowledge, this is the ﬁrst rigorous theoretical work on the direct problem of model (1.1). For the second goal, i.e. the inverse problem, we create an operator whose ﬁxed point is a(t) and prove its monotonicity, uniqueness of its ﬁxed points and deduce a reconstruction algorithm. The rest of this paper has the following structure. In section 2, we deduce a modiﬁed model from model (1.1) by a simple variational transformation and collect some preliminary results of fractional calculus. In section 3, we consider the direct problem and give the spectral representation of the weak solution for the modiﬁed model, showing the existence, uniqueness and some regularity results of this weak solution. The inverse problem is discussed in section 4 by establishing an operator K of which the ﬁxed point is a(t). Then we prove some main properties of K, such as the monotonicity and the uniqueness of ﬁxed points which is followed from a reconstruction algorithm. In section 5, some numerical results are presented to illustrate the theories we build. 2. Preliminary 2.1. Modified model In order to simplify the boundary condition of model (1.1), we use the transformation v (x, t ) = u (x, t ) + (x - 1) f (t ) to deduce the following model: ⎧ CD a v (x , t ) - a (t ) vxx (x , t ) = (x - 1)CD a f (t ) , t ⎪ t ⎨ v (0, t ) = v (1, t ) = 0, 0 < t T ; ⎪ ( ⎩ v x , 0) = 0, 0 < x < 1, 0 < t T , 0 < x < 1; (2.1) with data - a (t ) ( vx (0, t ) - f (t ) ) = g (t ) , 0 < t T . It is easily shown that model (2.1) has Dirichlet boundary condition and the coefﬁcient a(t) in model (2.1) is equivalent to that in model (1.1). Following [27] we can analyze the direct problem of model (2.1) by spectral decomposition. In the next two subsections, we state some necessary results for the analysis. 3 Inverse Problems 32 (2016) 015011 Z Zhang 2.2. Mittag-Leffler function In this part, we describe the Mittag-Lefﬂer function which plays an important role in fractional diffusion equations. This is a two-parameter function deﬁned as Ea, b (z ) = ¥ å k=0 zk , z Î , G (ka + b ) which generalizes the natural exponential function in the sense that E1,1 (z ) = e z . We list some important properties of the Mittag-Lefﬂer function for future use. Lemma 2.1. For l > 0, a > 0, t > 0 and n Î N, we have dn Ea,1 ( - lt a ) = - lt a - nEa, a - n + 1 ( - lt a ). dt n In particular, if we set n = 1, then there holds d Ea,1 ( - lt a ) = - lt a - 1Ea, a ( - lt a ). dt Proof. See [27, lemma 3.2]. , Lemma 2.2. If 0 < a < 1 and z > 0, then Ea, a ( - z ) 0. Proof. This proof can be found in [22, 25, 29]. , Lemma 2.3. For 0 < a < 1, Ea,1 ( - t a ) is completely monotonic, that is, ( - 1)n Proof. dn Ea,1 ( - t a) 0, t > 0, n Î . dt n See [5]. , 2.3. Some results for CD αt The following results concern some properties of CDta , which are crucial for the future development. The proofs of the lemmas below can be found in [28] and [20, theorem 1], respectively. Lemma 2.4. Deﬁne Riemann–Liouville αth order integral Ita as Ita u (t ) = 1 G (a ) t ò0 (t - t )a- 1u (t ) dt . 4 Inverse Problems 32 (2016) 015011 Z Zhang For 0 < a < 1, u (t ), CDta u (t ) Î C [0, T ], we have CD a t ◦ Ita u (t ) = Ita ◦ CDta u (t ) = u (t ) - u (0) , t Î [0, T ] . Lemma 2.5. Fix 0 < a < 1 and given u (t ) Î C [0, T ] with CDta u Î C [0, T ] . If u(t) attains its maximum over the interval [0, T ] at the point t = t0 Î (0, T ], then CDta0 u 0. 3. Direct problem We call v (x, t; a ) is a weak solution to model (2.1) in L2 [0, 1] corresponding with coefﬁcient a(t) if v ( · , t; a ) Î H01 [0, 1] for t Î (0, T ] and for any y Î H 2 [0, 1] Ç H01 [0, 1], Deﬁnition 3.1. ⎧ CD a v (x , t ; a) , y (x ) - ( a (t ) v (x , t ; a) , y (x ) ) = ((x - 1) , y (x )) CD a f (t ) , t Î (0, T ] ; ) xx t ⎨( t ( ( ) y ( )) = v x , 0; a , x 0, ⎩ ⎪ ⎪ where ( · , · ) is the inner product in L2 (W). In this section, we consider the weak solution v (x, t; a ) of model (2.1) under deﬁnition 3.1. Speciﬁcally, we will establish the spectral representation, existence, uniqueness and some regularity results for v (x, t; a ). To this end, we state some assumptions on f(t) and the coefﬁcient a(t). Assumption 3.2. (Direct Problem). Suppose a(t) and f(t) satisfy the following assumptions: (a) f(t) and CDta f (t ) are continuous and non-negative on [0, T ] with f (0) = 0; (b) a (t ) Î C +[0, T ] ≔ {y Î C [0, T ] : y (t ) > 0, t Î [0, T ]}. We suppose assumption 3.2 be valid in this section. In order to illustrate assumption 3.2, the following remark is set up. Remark 3.1. Assumption 3.2(a ) and (b ) are crucial to deduce the regularity results of the weak solution v (x, t; a), which can be seen in the future work. Also, without assumption 3.2(a ), [13, theorem 3.25] can not be used, which leads to the fails of the existence and uniqueness of v (x, t; a ). We prefer the condition CDta f (t ) is continuous and non-negative on [0, T ] rather that f ¢ (t ) Î C +[0, T ] since the latter one is stronger that the former. It is not hard to show the former condition can be derived from f ¢ (t ) Î C +[0, T ]. 3.1. Spectral representation of the weak solution For the operator -Lv = -vxx with homogeneous Dirichlet boundary condition, {(n2p 2, sin npx ) : n Î +} is the eigensystem of -L and {sin npx : n Î +} forms an orthogonal basis of L2 [0, 1]. Multiplying sin npx on the both sides of model (2.1) and integrating it on x over [0, 1] give 5 Inverse Problems 32 (2016) 015011 CD a (v (x , t Z Zhang t ; a) , sin npx ) - a (t ) ( vxx (x , t ; a) , sin npx ) = ((x - 1) , sin npx ) CDta f (t ) . The zero boundary condition for v (x, t; a ) yields that - ( vxx (x , t ; a) , sin npx ) = n2p 2 (v (x , t ; a) , sin npx ) , which leads to CD a (v (x , t t ; a) , sin npx ) + a (t ) n2p 2 (v (x , t ; a) , sin npx ) = ((x - 1) , sin npx ) CDta f (t ) , n Î +. 1 Set vn (t; a ) = 2 (v (x, t; a ), sin npx ) because of (sin npx, sin npx ) = 2 . The completeness of {sin npx : n Î +} in L2 [0, 1] allows us to write v (x, t; a ) as v (x , t ; a ) = ¥ å vn (t; a) sin npx. n=1 1 Also, recall that (x - 1, sin npx ) = - np , then the following fractional ODE can be deduced 2 C a Dt f (t ) , n Î +. np The initial condition v (x, 0; a ) = 0 and the linear independence of {sin npx : n Î +} yield that vn (0; a ) = 0, n Î +. To sum up, the spectral representation of v (x, t; a ) is CD a v (t ; t n a) = - n2p 2a (t ) vn (t ; a) - v (x , t ; a ) = ¥ å vn (t; a) sin npx, (3.1) n=1 where vn (t; a ) satisﬁes CD a v (t ; t n a) + n2p 2a (t ) vn (t ; a) = - 2 C a Dt f (t ) , vn (0; a) = 0, n Î +. np (3.2) 3.2. Existence, uniqueness and some properties of the weak solution In this subsection, we show the existence and uniqueness of the weak solution (3.1). At ﬁrst, we state the following lemma, which is [13, theorem 3.25]. Lemma 3.1. For the Cauchy-type problem CD a y (t ) t = F (y , t ) , y (0) = c0 , if for any continuous y (t ), F (y, t ) Î C [0, T ], and $ A > 0 which is independent on y Î C [0, T ] and t Î [0, T ] s.t. F (t , y1 ) - F (t , y2 ) A y1 - y2 , then there exists a unique solution y (t ) Î C [0, T ] for the Cauchy-type problem which also satisﬁes CDta y (t ) Î C [0, T ]. The following lemma follows from lemma 3.1. Lemma 3.2. Suppose assumption 3.2 holds. For model (2.1), there exists a unique weak solution v (x, t; a ) each n Î +. represented by (3.1) 6 with vn (t; a ), CDta vn (t; a) Î C [0, T ] for Inverse Problems 32 (2016) 015011 Z Zhang Fix n Î +. Due to assumption 3.2, the right side of (3.2) satisﬁes the conditions of lemma 3.1. Hence, for any n Î +, there exists a unique continuous solution vn (t; a ) of (3.2) with CDta vn (t; a ) Î C [0, T ]. With the spectral representation (3.1), the existence and uniqueness of each vn (t; a ) lead to the existence and uniqueness of the weak solution v (x, t; a ). This completes the proof. , Proof. Furthermore, there are two lemmas about vn (t; a ) which play important roles for deriving the regularity of v (x, t; a ). Lemma 3.3. For any a (t ) Î C +[0, T ] , there holds vn (t ; a ) 0 on [0, T ] , n Î +. For each a (t ) Î C +[0, T ] and n Î +, lemma 3.2 yields that vn (t; a ) Î C [0, T ], which leads to vn (t; a ) attains its maximum over [0, T ] at t = t0 Î [0, T ]. If t0 = 0, with the fact that vn (0, a ) = 0 which comes from lemma 3.2, then we have 0 is the maximum value of vn (t; a ) on [0, T ], which is the desired result. If t0 Î (0, T ], lemma 3.2 gives that vn (t; a ), CDta vn (t; a) Î C [0, T ], which ensure we can apply lemma 2.5 to vn (t; a ). With lemma 2.5 and the deﬁnition of t0, we have CDta0 vn 0. Inserting CDta0 vn 0 into (3.2) 2 and recalling assumption 3.2 lead to n2p 2a (t0 ) vn (t0; a ) = - CDta0 vn - np CDta0 f 0, i.e. vn (t0; a ) 0 and complete the proof. , Proof. Lemma 3.4. Given a1, a 2 Î C +[0, T ] with a1 a 2 on [0, T ] , we have vn ( t ; a1 ) vn ( t ; a 2 ) 0, t Î [0, T ] , n Î +. Proof. Equation (3.2) yields for each n Î +, ⎧C a 2 C a 2 2 ⎪ ⎪ Dt vn ( t ; a1 ) = - n p a1 (t ) vn ( t ; a1 ) - np Dt f (t ) , ⎨ ⎪ CD a v ( t ; a ) = - n2p 2a (t ) v ( t ; a ) - 2 CD a f (t ) , n n 2 2 2 ⎪ t ⎩ t np i.e. CD a v ( t ; t n a1 ) + n2p 2a1 (t ) vn ( t ; a1 ) = CDta vn ( t ; a 2 ) + n2p 2a 2 (t ) vn ( t ; a 2 ) . Let w (t ) = vn (t; a1 ) - vn (t; a 2 ). Lemma 3.2 ensures w (t ), w (0) = 0. From (3.3), we deduce that CD a w (t ) t CD a w (t ) t (3.3) Î C [0, T ] and + n2p 2a1 (t ) w (t ) = n2p 2 ( a 2 (t ) - a1 (t ) ) vn ( t ; a 2 ) 0, t Î [0, T ] , (3.4) where the last equality follows from lemma 3.3 and the condition a1 a 2 . w (t ) Î C [0, T ] yields that w(t) attains its maximum over [0, T ] at some t = t0 Î [0, T ]. If t0 = 0, we have w (t ) w (0) = 0. For the case t0 Î (0, T ], applying lemma 2.5 to w(t) yields that CDta0 w 0, which together with (3.4) implies n2p 2a1 (t0 ) w (t0 ) - CDta0 w 0. With the fact a1 Î C +[0, T ], we deduce w (t0 ) 0 and w (t ) w (t0 ) 0. The desired result follows from w 0 and lemma 3.3. , 7 Inverse Problems 32 (2016) 015011 Z Zhang 3.3. Regularity results In this subsection, we deal with the regularity properties of v (x, t; a ). Through the next two lemmas, v (x, t; a )C ( [0, T ]; H 2 [0,1] ) is bounded by f C [0, T ] and CDta f C [0, T ]. Lemma 3.5. Suppose assumption 3.2 holds, then v (x, t ; a )C ( [0, T ]; L2 [0,1] ) C f C [0, T ] . Pick t Î [0, T ], take the fractional integral Ita on both sides of (3.2) and use lemma 2.4, then Proof. 2 f (t ) , t Î [0, T ] , n Î +. np Lemma 3.3 together with assumption 3.2 and the deﬁnition of Ita gives that vn (t; a ) and Ita [a (t ) vn (t; a )] are both nonpositive on [0, T ], which implies vn (t ; a) + n2p 2Ita [ a (t ) vn (t ; a) ] = - vn (t ; a) 2 f (t ) on [0, T ] , n Î +. np (3.5) Now, ﬁx t Î [0, T ], calculate the L2 norm of v (x, t; a ) with respect to x, then 2 = v (x , t ; a ) 2 ¥ å vn (t; a) sin npx n=1 L2 [0,1] = L2 [0,1] 1 2 ¥ å ( vn (t; a) )2 , (3.6) n=1 where the last equality is due to the orthogonality of {sin npx : n Î +}. Equation (3.5) and (3.6) yield 2 v (x , t ; a ) L2 [0,1] i.e. v (x, t; a ) result. L2 [0,1] ¥ 1 4 (f (t ))2 å 2 2 = C (f (t ))2 C f C2 [0, T ] , 2 n=1 n p C f C [0, T ] for each t Î [0, T ], which leads to the expected , Lemma 3.6. With assumption 3.2, we have v (x , t ; a)C ( [0, T ]; H 2 [0,1] ) Ca ( f C [0, T ] + CDta f C [0, T ] ), where Ca > 0 only depends on the coefﬁcient a (t ). Assumption 3.2 states a Î C +[0, T ], which implies there exists constants qa , Qa > 0 such that Proof. 0 < qa < a (t ) < Qa on [0, T ] . (3.7) We denote these constants by qa , Qa due to their dependence on a(t). Then by lemma 3.4, vn (t; qa ) vn (t; a) 0, which leads to vn ( t ; qa ) vn (t ; a) , t Î [0, T ] , n Î +, 8 (3.8) Inverse Problems 32 (2016) 015011 Z Zhang and vn (t; qa ) is the solution of CD a v ( t ; t n qa ) = - n2p 2qa vn ( t ; qa ) - 2 C a Dt f (t ) . np For the above fractional ODE, [27] gives t 2 a CD a f (t )(t - t )a - 1E 2 2 a, a ( - n p qa (t - t ) ) dt , t np 0 which allows us to deduce the regularity of vxx (x, t; a ) immediately. Fix t Î [0, T ]. The representation (3.1) yields that ò vn ( t ; qa ) = - ¥ ∣ v (x , t ; a)∣2H 2 [0,1] = å n2p 2vn (t ; a) sin npx2L2 [0,1] = n=1 1 2 (3.9) ¥ å n4p 4 ( vn (t; a) )2 , (3.10) n=1 which implies the following estimate together with (3.8) and (3.9) ∣ v (x , t ; a)∣2H 2 [0,1] 1 2 1 2 ¥ ⎡ t å ⎣⎢ ò0 n=1 ¥ ⎡ t å ⎢⎣ ò0 n=1 ⎤2 2 C a Dt f (t ) n2p 2 (t - t )a - 1Ea, a ( - n2p 2qa (t - t )a ) dt ⎥ ⎦ np 2 C a Dt f (t ) np C [0, T ] 2 ´ n2p 2 (t - t )a - 1Ea, a ( - n2p 2qa (t - t )a ) dt ⎤⎦ = t ò0 The term 0 t ò0 =- 1 2 ¥ ⎡ 2 n=1 t ò0 ⎤2 n2p 2t a - 1Ea, a ( - n2p 2qa t a ) dt ⎥ . ⎦ (3.11) n2p 2t a - 1Ea, a (-n2p 2qa t a ) dt is bounded by lemmas 2.1 and 2.2 n2p 2t a - 1Ea, a ( - n2p 2qa t a ) dt = 1 qa t å np CDta f (t )C2 [0, T ] ⎢⎣ ò0 t ò0 n2p 2t a - 1Ea, a ( - n2p 2qa t a ) dt d 1 Ea,1 ( - n2p 2qa t a ) dt = 1 - Ea,1 ( - n2p 2qa t a ) qa-1, dt qa ( ) (3.12) where the last inequality follows from lemma 2.3. Inserting (3.12) into (3.11) gives ∣ v (x , t ; a)∣2H 2 [0,1] 1 2 ¥ 4 å n 2p 2 CDta f (t )C2 [0, T ] qa-2 n=1 ⎛ ¥ 4 ⎞ 1 = qa-2 ⎜ å 2 2 ⎟ CDta f C2 [0, T ] , 2 ⎝ n=1 n p ⎠ i.e. ∣ v (x , t ; a)∣H 2 [0,1] Ca CDta f (t )C [0, T ] , (3.13) where Ca > 0 depends on a(t) only. For v (x, t; a ) H1 [0,1], the spectral representation (3.1) gives ¥ ∣ v (x , t ; a)∣2H1 [0,1] = å npvn (t ; a) cos npx2L2 [0,1] n=1 ¥ 1 = 2 å n2p 2 ( vn (t; a) )2 ∣v (x, t; a)∣2H n=1 9 2 [0,1] , Inverse Problems 32 (2016) 015011 Z Zhang where the last inequality is ensured by (3.10). Then (3.13) yields ∣ v (x , t ; a)∣H1 [0,1] Ca CDta f C [0, T ] . (3.14) With lemma 3.5, (3.13) and (3.14), we have v (x , t ; a)H 2 [0,1] Ca ( f C [0, T ] + CDta f C [0, T ] ), t Î [0, T ] , , which gives the expected conclusion. The regularity of CDta v (x, t; a ) is derived from lemma 3.6 directly. Corollary 3.1. Suppose assumption 3.2 be valid, then we have CDta v (x , t ; a)C ( [0, T ]; L2 [0,1] ) Ca CDta f C [0, T ] , where Ca is a positive constant which only depends on the coefﬁcient a(t). Proof. Calculate CDta v (x, t; a ) directly from (3.1) and (3.2) CD a v (x , t t ; a) = ¥ ⎡ å ⎣⎢ -n2p 2a (t ) vn (t; a) - n=1 ⎤ 2 C a Dt f (t ) ⎥ sin npx . ⎦ np Fix an arbitrary t Î [0, T ], the above equality yields that CDta v (x , t ; a)2L2 [0,1] = 1 2 ¥ ⎡ ⎤2 2 C a Dt f (t ) ⎥ ⎦ np ¥ 4 + å 2 2 ∣CDta f (t )∣2 n n=1 p å ⎢⎣ -n2p 2a (t ) vn (t; a) - n=1 ¥ n 4p 4 (a (t ))2 å n=1 ¥ ( vn (t; a) )2 å n4p 4 (a (t ))2 ( vn (t; a) )2 + C CDta f C2 [0, T ] . n=1 The bound of ¥ å n = 1 n4p 4 (a (t ))2 (vn (t; a))2 follows from (3.7), (3.10) and (3.13) ¥ ¥ n=1 n=1 å n4p 4 (a (t ))2 ( vn (t; a) )2 å n4p 4Qa2 ( vn (t; a) )2 = 2Qa2 ∣v (x, t; a)∣2H 2 [0,1] 2Qa2 Ca2 CDta f C2 [0, T ] . Consequently, we have CDta v (x, t; a)2L2 [0,1] (2Qa2 Ca2 + C ) CDta f C2 [0, T ] , i.e. CDta v (x , t ; a)L2 [0,1] Ca CDta f C [0, T ] , t Î [0, T ] , , which yields the desired result. Remark 3.2. Lemma 3.6 and corollary 3.1 guarantee that model (2.1) is well deﬁned in the space L2 [0, 1]. 3.4. Main theorem for the direct problem In this subsection, we state the main theorem for the direct problem of model (2.1), which follows from lemmas 3.2 and 3.6 and corollary 3.1. 10 Inverse Problems 32 (2016) 015011 Z Zhang Theorem 3.3 (Main theorem for the direct problem). Suppose assumption 3.2 holds, then there exists a unique weak solution v (x, t; a ) for model (2.1) with the spectral representation (3.1) and the following regularity estimates: ⎧ C a ⎪ v (x , t ; a )C ( [0, T ]; H 2 [0,1] ) Ca ( f C [0, T ] + Dt f C [0, T ] ) ; ⎨ C a ⎪ C a ⎩ Dt v (x , t ; a)C ( [0, T ]; L2 [0,1] ) Ca Dt f C [0, T ] , where Ca > 0 only depends on the coefﬁcient a(t). 4. Inverse problem In this section, we consider the inverse problem for model (2.1): using the data -a (t )(vx (0, t; a) - f (t )) = g (t ) to reconstruct the coefﬁcient a (t ). We only consider the case a (t ) Î C +(0, T ], which is regarded as the admissible set of a (t ). 4.1. Operator K and its well-definedness In order to reconstruct a (t ), we introduces an operator K : W C +(0, T ] as Ky (t ) ≔ g (t ) = f (t ) - vx (0, t ; y) f (t ) - g (t ) , ¥ å n= 1npvn (t; y) where the domain Ω is deﬁned as W ≔ { y Î C +(0, T ] : 0 < y (t ) g f , t Î (0, T ] }. By the boundary condition -a (t )(vx (0, t; a) - f (t )) = g (t ), we can see the coefﬁcient a(t) is one of the ﬁxed points of operator K. In order to analyze this K, we state some restrictions for the data f and g. Assumption 4.1. Suppose f and g satisfy the following restrictions: (a) f (t ) Î C [0, T ] Ç C +(0, T ] with f (0) = 0 and CDta f (t ) Î C [0, T ] is non-negative on [0, T ]; (b) g (t ) Î C [0, T ] Ç C +(0, T ] and g (0) = 0. The following two remarks explain the reasonableness and reason of assumption 4.1 respectively. Remark 4.1. The function f(t) is input data so that we can control it, which means assumption 4.1(a ) can be achieved. For assumption 4.1(b ), given a (t ) Î C +(0, T ], lemma 3.2 ensures the continuity of vn (t; a ) for each n Î +. This result and the continuity of f(t) and a(t) imply the continuity of ¥ g. Also, lemma 3.3 yields that -vx (0, t; a) = - å n = 1 npvn (t; a) 0 on (0, T ], which together with a > 0 and assumption 4.1(a ) leads to g > 0 on (0, T ]. At last, the zero initial condition and f (0) = 0 give vx (0, 0) - f (0) = 0, which implies g (0) = 0. The above arguments guarantees the reasonableness of assumption 4.1(b ). Assumption 4.1(a ) and (b ) guarantee the well-deﬁnedness of the space Ω in the sense that the upper bound g/f is strictly larger than the lower bound 0 on the interval (0, T ]. Remark 4.2. 11 Inverse Problems 32 (2016) 015011 Z Zhang Also, assumption 4.1(a ) will be used to show the monotonicity of K and the uniqueness of its ﬁxed points. Furthermore, it is clear that vx (0, 0) - f (0) = g (0) = 0 due to the zero initial condition and assumption 4.1. That means we can not reconstruct a(t) at t = 0, but only over the interval (0, T ], which can be seen from the deﬁnition of W. For operator K, theorem 3.3 indicates that given a (t ) Î W, there exists a unique Ka. ¥ Also, assumption 4.1 and the result -å n = 1 npvn (t; a ) 0 derived from lemma 3.3 yield the ¥ dominator f (t ) - å n = 1 npvn (t; a ) of Ka is strictly positive on (0, T ], which indicates there is no singularities for Ka on (0, T ]. In addition, the positivity and continuity of the dominator and numerator of Ka ensure Ka Î C +(0, T ]. Therefore, the operator K is well-deﬁned and maps Ω to C +(0, T ]. 4.2. Monotonicity We show the monotonicity of K in this part. Theorem 4.2 (Monotonicity). With assumption 4.1, the operator K is monotone, i.e. given a1, a 2 Î W with a1 a 2 on (0, T ], we have Ka1 (t ) Ka 2 (t ), t Î (0, T ]. Lemma 3.4 gives vn (t; a1 ) vn (t; a 2 ) 0 on (0, T ], n Î +, which together with ¥ ¥ assumption 4.1 yields f (t ) - å n = 1 npvn (t; a1 ) f (t ) - å n = 1 npvn (t; a 2 ) > 0. Consequently, with assumption 4.1(b ), Proof. g (t ) ¥ f (t ) - å n = 1npvn ( t ; a1 ) f (t ) - g (t ) å ¥ npvn ( t ; n=1 a2 ) on (0, T ] , , which implies the monotonicity. The following lemma concerns the upper bound g/f of Ω. Lemma 4.1. K (g f ) Î W. ¥ Lemmas 3.2 and 3.3 give that å n = 1 npvn (t; g f ) is nonpositive and continuous; while assumption 4.1 yields f , g Î C +(0, T ]. These results ensure the continuity of K (g f ) g (t ) and 0 < K (g f ) = f (t ) - ¥ npv (t; g f ) g f on (0, T ], which lead to the expected Proof. result. ån=1 n , 4.3. Uniqueness In this subsection, we deal with the uniqueness of ﬁxed points of K. To this end, we introduce a successive iteration as a 0 = g f , a n + 1 = Ka n , n Î . The reasonableness of the initial guess g/f is guaranteed by lemma 4.1 and this iteration generates a series {a n : n Î }. The next two lemmas state the above series will converge to a ﬁxed point of K if it exists. 12 Inverse Problems 32 (2016) 015011 Z Zhang Lemma 4.2. Suppose assumption 4.1 holds. If a1, a 2 Î W are two ﬁxed points of K with a1 a 2 , then a1 º a 2 on (0, T ]. Proof. If a Î W is a ﬁxed point of operator K, then ¥ å npa (t ) vn (t; a) = -g (t ) + a (t ) f (t ), t Î (0, T ]. n=1 Taking Ita on both sides of the above equality and recalling its linearity yield ¥ å npIta [ a (t ) vn (t; a) ] = -Ita g (t ) + Ita [a (t ) f (t )], t Î (0, T ]. (4.1) n=1 Furthermore, the following equality follows from (3.2) and lemma 2.4 Ita [ a (t ) vn (t ; a) ] = - 2 1 f (t ) - 2 2 vn (t ; a) , n3p 3 np which together with (4.1) leads to Ita [a (t ) f (t )] + ¥ ¥ 1 2 å np vn (t; a) = -f (t ) å n2p 2 n=1 + Ita g (t ) , t Î (0, T ] . (4.2) n=1 Now pick two ﬁxed points a1, a 2 of K with a1 a 2 . For ﬁxed points a1, a 2 , (4.2) holds respectively. Combining them together yields Ita [ a1 (t ) f (t ) ] + ¥ ¥ 1 1 å np vn ( t; a1) = Ita [ a 2 (t ) f (t ) ] + å np vn ( t; a 2 ). n=1 (4.3) n=1 Lemma 3.4 and a1 a 2 give vn (t; a1 ) vn (t; a 2 ), n Î +, i.e. ¥ 1 ¥ 1 å np vn ( t; a1) å np vn ( t; a 2 ), t Î (0, T ], n=1 n=1 to Ita [(a1 (t ) which together with (4.3) leads - a 2 (t )) f (t )] 0, t Î (0, T ]. Moreover, recall that and the deﬁnition of we have f > 0, a1 a 2 Ita , a Therefore, it holds It [(a1 (t ) - a 2 (t )) f (t )] 0, t Î (0, T ]. 0 Ita [(a1 (t ) - a 2 (t )) f (t )] 0, i.e. Ita [(a1 (t ) - a 2 (t )) f (t )] = 0 on (0, T ]. The deﬁnition of Ita and the non-positivity of (a1 - a 2 ) f imply (a1 (t ) - a 2 (t )) f (t ) = 0 on (0, t ] for each t Î (0, T ], which leads to a1 - a 2 º 0 on (0, T ] due to f > 0. , Suppose assumption 4.1 be valid. If there exists a ﬁxed point a (t ) Î W of operator K, then the series {a n : n Î } is contained by Ω and converges to a (t ). Lemma 4.3. Proof. a 0 is the upper bound of Ω gives a a 0 , which together with the monotonicity of K yields Ka Ka 0 , i. e. a a1. Applying the monotonicity again to this inequality, it holds Ka Ka1, i. e. a a 2 . Continuing this procedure, we deduce a a n , n Î , which implies a(t) is a lower bound of {a n : n Î }. On the other hand, lemma 4.1 gives a1 a 0 . Apply the monotonicity of K to it, there holds Ka1 Ka 0 , i.e. a 2 a1. Repeat this argument again and again, we obtain a n + 1 a n , n Î , which yields {a n : n Î } is decreasing. 13 Inverse Problems 32 (2016) 015011 Z Zhang Algorithm 1. Numerical algorithm. Successive iteration algorithm to recover the coefﬁcient a(t) 1: Set up f(t) and measure the data g (t ). f and g should satisfy assumption 4.1; 2: Set the initial guess a 0 = g f ; 3: for k=1, K, N do 4: Using the L1 time-stepping [8] to compute v (x, t; ak - 1), which is the solution of model (2.1) with coefﬁcient ak - 1; 5: Update the coefﬁcient ak - 1 by ak = Kak - 1; 6: Check stopping criterion ak - ak - 1L2 [0, T ] 0 for some 0 > 0; 7: end for 8: output the approximate coefﬁcient aN . With the facts that {a n : n Î } is decreasing, a 0 is the upper bound of Ω and a (t ) Î W is a lower bound of {a n : n Î }, the following results can be deduced: {a n : n Î } is included by Ω and convergent in it. Denote the limit of {a n : n Î } by a. Clearly a is one of the ﬁxed points of K and a (t ) a (t ) due to a(t) is a lower bound of {a n : n Î }. Then lemma 4.2 gives a (t ) = a (t ), which completes this proof. , The uniqueness of ﬁxed points of K follows from lemma 4.3 immediately. Theorem 4.3 (Uniqueness). With assumption 4.1, there is at most one ﬁxed point of K in W. Given two ﬁxed points a1, a 2 Î W, lemma 4.3 yields that a n a1 and a n a 2 , which leads to a1 = a 2 and completes this proof. , Proof. 4.4. Main theorem for the inverse problem and the reconstruction algorithm In this part, we state the main theorem for the inverse problem which follows from lemma 4.3 and theorem 4.3 immediately. (Main theorem for the inverse problem). Let assumption 4.1 be valid. If there exists a ﬁxed point of K in W, then it is unique and coincides with the limit of {a n : n Î }. Theorem 4.4 Based on theorem 4.4, the reconstruction algorithm 1 is deduced. This algorithm is wellposed, meaning that the solution (a (t ), v (x, t; a)) depends continuously on the measured data g, which can be seen from the deﬁnition of operator K. However, if we turn things around and assume that g is given as part of the model and the ‘temperature’ f = u (0, t ) is measured, then this problem is no longer such continuously dependence. Indeed, it can be easily shown that the C0-norm of a(t) depends on CDta f C [0, T ] and CDta contains the term f ¢ . This illposedness problem can be done by either Tikhonov regularization—adding a penalty term, or molliﬁcation—replacing the data f by a molliﬁer f . 14 Inverse Problems 32 (2016) 015011 Z Zhang Figure 1. a 0 (t ) , a1 (t ) , a2 (t ) and a3 (t ) with a = 0.9 and 0 = 10-6 . Figure 2. a (t ) and a(t) with a = 0.9 and 0 = 10-6 . Moreover, the above situation also occurs in this inverse problem for the parabolic case [11], i.e. a = 1, in the sense that the integral equation of a(t) created by the author contains the terms f ¢ and g. 5. Numerical results for the inverse problem In this section, we will present some numerical results of the reconstruction. 5.1. Numerical results for smooth coefficient In this case, we let T=1 and a = 0.9, then pick the smooth data f (t ) = t 2 + t and a (t ) = sin 4pt + 1.1. We regard this a(t) as the exact result. L1 time-stepping [8] yields the output data g (t ). As expected, the g(t) we get is continuous and positive on (0, T ], which satisﬁes assumption 4.1. Then we insert the data f(t) and g(t) into algorithm 1 and pick 0 = 10-6. Theorem 4.4 ensures that {a n : n Î } is decreasing and a (t ) = a (t ) = limn ¥ a n , which are presented by ﬁgures 1 and 2, respectively. From ﬁgure 2, we can see a (t ) is a good approximation for a(t). More precisely, we can provide the L2 error as 15 Inverse Problems 32 (2016) 015011 Z Zhang Table 2. a - a L2 [0, T ] for different 0 with a = 0.9. 0 a - a L2 [0, T ] » 10−3 10−4 10−5 10−6 10−7 8 ´ 10-4 9 ´ 10-5 5 ´ 10-6 5 ´ 10-7 6 ´ 10-8 Figure 3. a 0 (t ) , a1 (t ) , a2 (t ) and a3 (t ) for nonsmooth case with a = 0.9 and 0 = 10-6 . Table 3. The amounts of iterations for different α with 0 = 10-6 . α 0.1 0.3 0.5 0.7 0.9 Amount of iterations from a 0 (t ) 83 62 45 32 19 a - a L2 [0, T ] » 5 ´ 10-7 » 0, which implies the L2 error a - a L2 [0, T ] maybe bounded by 0. In order to verity this relation, we try some more 0 and present the L2 errors corresponding different 0 in table 2. This table gives a rough estimate as a - a L2 [0, T ] » 0. If we ﬁx 0 = 10-6 and choose a different a Î (0, 1) for model (2.1), the numerical results are still good, which means algorithm 1 does not strongly depend on α. However, the α can affect the amount of iterations with the relation that the less α is, the more iterations we need, which can be seen in table 3. For table 3, [9, ﬁgure 1] provides a reasonable interpretation. For time-fractional diffusion equation, the fundamental solution is in terms of Mittag-Lefﬂer function, which has the property that restricted α in (0, 1), the less α is, the slower the decay rate of Ea,1 (-z ) is as z +¥. This property explains the phenomenon in table 3. 5.2. Numerical results for nonsmooth coefficient Even though assumption 4.1 requires continuous data, algorithm 1 still works for the coefﬁcient in L¥ (0, T ]. This means it is likely to weaken the restrictions on data to a certain extent in numerical part. 16 Inverse Problems 32 (2016) 015011 Z Zhang Figure 4. a (t ) and a(t) for nonsmooth case with a = 0.9 and 0 = 10-6 . Figure 5. g(t) and gd (t ) with d = 5%. For this numerical experiment, we pick a = 0.9, T = 1, 0 = 10-6 and set 1 f (t ) = t 2 + t , a (t ) = c( 0, T ) + c⎡⎣ T , 2 T ) + c⎡⎣ 2 T , T ⎤⎦ , 3 3 2 3 3 where the exact coefﬁcient a(t) has two discontinuities. The ﬁgures 3 and 4 illustrate theorem 4.4 holds for this experiment in the sense that ﬁgure 3 demonstrates {a n : n Î } is decreasing, meanwhile ﬁgure 4 indicates the existence of a (t ) and a (t ) = a (t ). Similar to the smooth case, a (t ) is still a good approximation for nonsmooth a(t) with the L2 error as a - a L2 [0, T ] » 8 ´ 10-7 » 0. 5.3. Numerical results for noisy data In this subsection, we consider the experiment with noisy data since every measurement in practice should contain noise. Let g be the exact data, denote the noisy data by gδ with relatively noise level d, i.e. g - gd C [0,1] gC [0,1] d. Deﬁne the noisy operator Kδ as 17 Inverse Problems 32 (2016) 015011 Z Zhang Figure 6. a(t) and ad (t ) with d = 5% under the smooth case. Figure 7. a(t) and ad (t ) with d = 5% under the nonsmooth case. Table 4. a - ad L2 [0, T ] aL2 [0, T ] for different δ with 0 = 10-2 . δ 1% 2% 3% 4% 5% 0.012 0.014 0.027 0.030 0.038 gd (t ) = f (t ) - vx (0, t ; y) f (t ) - gd (t ) a - ad L2 [0, T ] aL2 [0, T ] » Kd y (t ) ≔ ¥ å n= 1npvn (t; y) , with domain Wd ≔ { y Î C +(0, T ] : 0 < y (t ) gd (t ) f (t ) , t Î (0, T ] }. Since δ is a slightly positive number, we can assume gδ satisfy assumption 4.1(b ), which means theorem 4.4 still holds for Kd , and the uniqueness of the ﬁxed point aδ of Kδ can be deduced. If aδ exists, deﬁne ad,0 = gd f and ad, n + 1 = Kad, n , n Î . The series {ad, n : n Î } converges to aδ decreasingly because of theorem 4.4 and we denote the limit by ad . Algorithm 1 can be used to recover ad after a slightly modiﬁcation—replacing g and K by gδ and Kd , respectively. Similar to the numerical experiment with smooth data, we ﬁx T = 1, a = 0.9, 0 = 10-2 and let f (t ) = t 2 + t, a (t ) = sin 4pt + 1.1. Then we add 5% 18 Inverse Problems 32 (2016) 015011 Z Zhang noise into the data g generated by L1 time-stepping and denote the data with noise by gd . By the way, the reason we do not need to pick small 0 is that with noise d = 5%, 0 = 10-2 and 0 = 10-6 will reproduce similar approximations and absolutely, the cost for 0 = 10-2 is much less. Figure 5 shows g and gδ with d = 5%. With modiﬁed algorithm 1, we can obtain the approximate coefﬁcient ad , which comes from one experiment and is presented by ﬁgure 6. In addition, the relative L2 error is a - ad L2 [0, T ] aL2 [0, T ] » 0.035 » d , which illustrates the well-possedness of algorithm 1, and the relative L2 errors corresponding different δ are presented in table 4, which yields the rough error estimate: a - adL2 [0, T ] aL2 [0, T ] » d . We also attempt a numerical experiment with noise for the nonsmooth data. Similar to the previous experiment, we pick a = 0.9, T = 1, 0 = 10-2 , d = 5% and 1 f (t ) = t 2 + t , a (t ) = c( 0, T ) + c⎡⎣ T , 2 T ) + c⎡⎣ 2 T , T ⎤⎦ . 3 3 2 3 3 Figure 7 shows ad (t ) and a(t) with the relative L2 error as a - adL2 [0, T ] aL2 [0, T ] » 0.038 » d . 6. Concluding remarks We have studied an undetermined coefﬁcient problem for a fractional diffusion equation. For the direct problem part, ﬁxing the coefﬁcient a(t), we provide the spectral representation of the weak solution and prove its existence, uniqueness and some regularity properties. For the inverse problem part, in order to recover the coefﬁcient a (t ), we deﬁne an operator K from the FDE model of which one of the ﬁxed points coincides with a(t). Then we establish the monotonicity of K, uniqueness of its ﬁxed points and a numerical algorithm to recover the coefﬁcient a(t). For this work, there are several questions deserving further investigation. For the direct problem part, we can extend this time-fractional diffusion equation to the one with inhomogeneous right-hand side and nontrivial initial condition. In inverse problem part, at ﬁrst, we do not show the existence of a(t) since the domain Ω is not a complete subspace of C +(0, T ]. However, it is possible to reﬁne the domain and use monotonicity to show existence as Jones did in [11]. 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