ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] Applied Mathematical Modelling 0 (2016) 1–21 Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm Regularized solution of an inverse source problem for a time fractional diffusion equation Huy Tuan Nguyen a,b, Dinh Long Le b, Van Thinh Nguyen c,∗ a Department of Mathematics and Computer Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam Institute of Computational Science and Technology, Ho Chi Minh City, Vietnam c Department of Civil and Environmental Engineering, Seoul National University, Republic of Korea b a r t i c l e i n f o a b s t r a c t Article history: Received 17 November 2014 Revised 20 February 2016 Accepted 13 April 2016 Available online xxx In this paper, we study on an inverse problem to determine an unknown source term in a time fractional diffusion equation, whereby the data are obtained at the later time In general, this problem is illposed, therefore the Tikhonov regularization method is proposed to solve the problem In the theoretical results, a priori error estimate between the exact solution and its regularized solutions is obtained We also propose two methods, a priori and a posteriori parameter choice rules, to estimate the convergence rate of the regularized methods In addition, the proposed regularized methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions Eventually, from the numerical results it shows that the posteriori parameter choice rule method converges to the exact solution faster than the priori parameter choice rule method Keywords: Cauchy problem Ill-posed problem Convergence estimates © 2016 Elsevier Inc All rights reserved Introduction Diffusion equations with fractional order derivatives play an important role in modeling of contaminant diffusion processes One of such problems was raising by Adam and Gelhar [1]; during analyzing the field data of dispersion in a heterogeneous aquifer, they could not explain a long-tailed profile of spatial density distribution by a classical diffusion–advection equation with integer order derivatives The long-tailed profile is of asymptotic behavior of fundamental solution near t = 0, which can be described by a time-fractional diffusion equation as follows: Dtα u(x, t ) − ∂ ∂u D (x ) = F (x, t ), (x, t ) ∈ (0, π ) × (0, T ), ∂x ∂x (∗) where u = u(x, t ) denotes a concentration of contaminant at a position x and time t, α ∈ (0, 1) is a fractional order and an important parameter for anomaly of diffusion, F(x, t) is a source/sink function, D(x) is the diffusivity, and Dtα is the Caputo fractional derivative of order α defined by: Dtα u(t ) = ∗ t (1 − α ) u (s ) ds, (t − s )α Corresponding author Tel.: +82 28807355 E-mail address: vnguyen@snu.ac.kr (V.T Nguyen) http://dx.doi.org/10.1016/j.apm.2016.04.009 S0307-904X(16)30210-4/© 2016 Elsevier Inc All rights reserved Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 and (.) denotes the standard Gamma function Note that if the fractional order α tends to unity, the fractional derivative Dtα u converges to the canonical first-order derivative du [2], and thus the problem (∗) reproduces the canonical diffusion dt model See, e.g., [2,3] for the definition and properties of Caputos derivative The fractional initial boundary value problem was firstly considered by Nigmatullin [4] After that, several applications of the fractional calculus and derivatives in applied sciences were developed; and the research on theoretical analysis and numerical methods for solving direct problems, such as initial value and/or boundary value problems for the time fractional diffusion equation were growing For a well-posedness analysis, we refer to [5–9]; for numerical methods and simulations, see [10–16] and references therein Unfortunately, some input data and parameters of the diffusion equations in practical problems may be unknown, such as initial and boundary data, diffusion coefficients, and source terms; therefore we have to determine them by additional measurement data which can be yielded from a fractional diffusion inverse problem The fractional inverse problem provides an efficient tool for the modeling of the anomalous diffusion processes observed in various fields of science and engineering, such as in biology [17,18], physics [19,20], chemistry [21], and hydrology [22] Murio [23] considered an inverse problem of recovering boundary functions from transient data at an interior point for a 1D semi-infinite half-order time-fractional diffusion equation Liu and Yamamoto [24] applied a quasi-reversibility regularization method to solve a backward problem for the time-fractional diffusion equation Wei et al [25–27] studied an inverse source problem for a spatial fractional diffusion equation using quasi-boundary value and truncation methods Recently, Kirane et al [2,28] studied conditional well-posedness to determine a space dependent source for one-dimensional and two-dimensional time-fractional diffusion equations Rundell et al [29–31] considered an inverse problem for a onedimensional time-fractional diffusion problem However, there are only a few studies on determination of a sources term depending on both time and space for a time fractional diffusion equation In this work, we focus on an inverse problem for the following time-fractional diffusion equation: Dtα u(x, t ) + Au(x, t ) = F (x, t ), (x, t ) ∈ u(x, t ) = 0, x ∈ ∂ , u(x, ) = 0, x ∈ , where T > and < α < 1; given by: d Au(x, t ) = i=1 ∂ ∂ xi × ( 0, T ), is a bounded domain in Rd with the sufficient smooth boundary ∂ , and the operator A is d j (x ) j=1 ∂ u(x, t ) + B(x )u(x, t ), x ∈ · ∂xj where j = a ji , ≤ i, j ≤ d Moreover, we assume that the operator A is uniformly elliptic on smooth: there exists a constant μ > such that: μ d ξi2 ≤ i=1 d (1.1) a i j ( x ) ξi ξ j , x ∈ (1.2) and that its coefficients are , ξ ∈ Rd · (1.3) i, j=1 and the coefficients satisfy: j ∈ C ( ), B ∈ C ( ), B ( x ) ≤ 0, x ∈ · (1.4) Problem (1.1) is the forward problem when the source function F = F (x, t ) is given appropriately Whereas, an inverse source problem based on Problem (1.1) is to determine the source term F at a previous time from its value at the final time T as follows: u(x, T ) = h(x ), x ∈ (1.5) where h ∈ H ( ) ∩ H01 ( ) Assuming that the source term F = F (x, t ) can be split into a product of R(t)f(x), where R(t) is known in advance We assume the time-dependent source term R(t) is obtained from observation data R (t) in such a way that R (t ) − R(t ) L1 (0,T ) ≤ and is a noise level from a measurement The space-dependent source term f(x) is also determined from the observation of h(x) at the final data t = T by h ∈ L2 ( ) with the noise level of h −h L2 ( ) ≤ and satisfied: (1.6) It is known that the inverse source problem mentioned above is ill-posed in general, i.e., a solution does not always exist, and in the case of existence of a solution, which does not depend continuously on the given data In fact, from a small noise of a physical measurement, the corresponding solutions may have a large error This makes a troublesomeness for the numerical computation, hence a regularization is required If α = 1, the inverse source Problems (1.1) and (1.2) is a classical ill-posed problem and has been studied in [20,32] However, for the fractional inverse source problem, up-to-date, there are only very few works; for example, Sakamoto and Yamamoto [9] used the data u(x0 , t)(x0 ∈ ) to determine R(t) once f(x) was given, where the authors obtained a Lipschitz stability for R(t) Zhang and Wei [27] used the Fourier truncation method to solve an inverse source problem with R(t ) = in Problem (1.1) for one-dimensional problem with special coefficients The inverse source problem for the time-fractional diffusion equation with R(t) depended on time still has a limited achievement Actually, this problem recently has been Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 introduced by Jin and Rundell on Page 18 of [33], however a further regularized solution for this problem has not been focused in their paper Motivated by above reasons, in this study, we apply the Tikhonov regularization method to solve the fractional inverse source problem with variable coefficients in a general bounded domain We estimate a convergence rate under a priori bound assumption of the exact solution and a priori parameter choice rule Because the priori bound is difficult to obtain in practical application, so we also estimate a convergence rate under the posteriori parameter choice rule which is independent on the priori bound The paper is organized as follows In Section 2, we introduce some preliminary results The ill-posedness of the fractional inverse source Problem (1.1) and a conditional stability are provided in Section In Section 4, we propose a Tikhonov regularization method and give two convergence estimates under a priori assumption for the exact solution and two regularization parameter choice rules Finally two numerical examples to verify our proposed regularized methods are shown in Section Eventually, a conclusion is given in Section Preliminary results Throughout this paper, we use the following definition and lemmas Definition 2.1 (see [3]) The Mittag–Leffler function is: Eα ,β (z ) = ∞ k=0 zk , z∈C (α k + β ) where α > and β ∈ R are arbitrary constants Lemma 2.1 (see [3]) Let λ > 0, then we have: d Eα ,1 (−λt α ) = −λt α −1 Eα ,α (−λt α ), t > 0, < α < dt Lemma 2.2 (see [34]) For < α < 1, ρ > 0, we have < Eα ,1 (−ρ ) < Moreover, Eα ,1 (−ρ ) is completely monotonic, that is (−1 )m dm Eα ,1 (−ρ ) ≥ 0, dρ m ∀m ∈ N ∩ {0} Lemma 2.3 For < α < 1, ρ > 0, we have ≤ Eα ,α (−ρ ) ≤ ρ > (α ) Moreover, Eα ,α (−ρ ) is a monotonic decreasing function with Lemma 2.4 (see [3]) For α > and β ∈ R, we have: Eα ,β (z ) = zEα ,α +β (z ) + (β ) , z∈C Lemma 2.5 (see [35]) Assume R(t) ∈ L1 (0, T), for < α < and λ > 0, we have: Dtα t (t − s )α−1 Eα,α (−λ(t − s )α )R(s )ds = R(t ) − λ t (t − s )α−1 Eα,α (−λ(t − s )α )R(s )ds, ≤ t ≤ T Lemma 2.6 Let R : [0, T ] → R be a positive continuous function such that ∈ ft ∈ [0, T] |R(t)| > Set Then we have: inft∈[0,T ] |R(t )|[1 − Eα ,1 (−λ1 T α )] λp T ≤ Gα (s, λ p )R(s )ds ≤ R C[0,T ] λp R C[0,T ] = supt∈[0,T ] |R(t )| (2.7) where, λp is defined in Theorem 2.1 and Gα (s, λ p ) = (T − s )α −1 Eα ,α (−λ p (T − s )α )ds Proof Using Lemmas 2.1 and 2.3, we have: T Gα (s, λ p )ds = = = T T (T − s )α−1 Eα,α (−λ p (T − s )α )ds |sα−1 Eα,α (−λ p sα )|ds = − − Eα ,1 (−λ p T α ) λp ≤ λp T λp d Eα ,1 (−λ p sα )ds ds This implies that: T Gα (s, λ p )R(s )ds ≤ sup |R(t )| t∈[0,T ] T Gα (s, λ p )ds ≤ R C[0,T ] λp Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 and T Gα (s, λ p )R(s )ds ≥ inf t∈[0,T ] |R(t )| T Gα (s, λ p )ds ≥ inft∈[0,T ] |R(t )|[1 − Eα ,1 (−λ1 T α )] λp The proof is completed The inverse source problem First, we introduce a few properties of the eigenvalues of the operator A on an open, connected and bounded domain with Dirichlet boundary conditions (see also in Chapter of [36]) Theorem 2.1 (Eigenvalues of the Laplace operator) Each eigenvalues of A is real The family of eigenvalues {λ p }∞ p=1 satisfy ≤ λ1 ≤ λ2 ≤ λ3 ≤ , and λp → ∞ as p → ∞ There exists an orthonormal basis {φ p }∞ of L2 ( ), where φ p ∈ H01 ( ) is an eigenfunction corresponding to λ: p=1 Aφ p (x ) = −λ p φ p (x ), x ∈ φ p ( x ) = 0, x ∈ ∂ , (3.8) for p = 1, 2, Let = γ < ∞ By Hγ ( ) we denote the space of all functions g ∈ L2 ( ) with the property: ∞ ( + λ ) 2γ |g p |2 < ∞, (3.9) p=1 ∞ γ 2γ where g p = g(x )φ p (x )dx Then we also define g H γ ( ) = p=1 (1 + λ p ) |g p | If γ = then H ( ) is L ( ) This space is introduced by Brezis [37] (see Chapter V) and Feng et al [38] (see page 179) 3.1 The formula and uniqueness of the source term f Now we use the separation of variables to yield the solution of (1.1) Suppose that the exact u is defined by Fourier series: ∞ u(x, t ) = u p (t )φ p (x ), with u p (t ) = u(., t ), φ p (x ) (3.10) p=1 Then the eigenfunction expansions can be defined by the Fourier method That is, we multiply both sides of (1.1) by φ p (x) and integrate the equation with respect to x Using the Green formula and φ p |∂ = 0, we obtain an uncouple system of the initial value problem for the fractional differential equations for the unknown Fourier coefficient u p (t ) : Dtα u p (t ) = λ p u p (t ) + Fp (t ), < t < T u p (0 ) = u(x, ), ϕ p (x ) (3.11) where Fp (t ) = F (x, t ), φ p (x ) As Sakamoto and Yamamoto [9], the formula of the solution corresponding to the initial value problem for (3.11) is obtained as follows: u p (t ) = Eα ,1 (−λ pt α )u p (0 ) + t (t − s )α−1 Eα,α (−λ p (t − s )α )Fp (s )ds It follows from u(x, ) = and Fp (s ) = R(s ) f (x ), φ p (x ) that, h ( x ), φ p ( x ) = u p ( T ) = T (T − s )α−1 Eα,α (−λ p (T − s )α )R(s )ds f (x ), φ p (x ) A simple transformation gives: f (x ) = ∞ p=1 T h ( x ), φ p ( x ) φ p ( x ) (T − s )α−1 Eα,α (−λ p (T − s )α )R(s )ds (3.12) and by denote: Gα (s, λ p ) = (T − s )α −1 Eα ,α (−λ p (T − s )α ), (3.13) then we obtain the formula of the source function f : f (x ) = ∞ p=1 h ( x ), φ p ( x ) T φ p (x ) Gα (s, λ p )R(s )ds (3.14) In the following theorem, we provide the uniqueness property of the inverse source problem Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 Theorem 3.1 Let R : [0, T ] → R be as in Lemma 2.6, then the solution u(x, t), f(x) of Problems (1.1) and (1.2) is unique Proof Let f1 and f2 be the source functions corresponding to the final values h1 and h2 respectively Suppose that h1 = h2 then we prove that f1 = f2 In fact, it is well-known that Eα ,α (−λ p (t − s )α ) ≥ for s ≤ t Since R(t) ≥ R0 > for t ∈ [0, T], we have: T T Gα (s, λ p )R(s )ds ≥ R0 (T − s )α−1 Eα,α (−λ p (T − s )α )ds = R0 T α Eα ,α +1 (−λ p T α ) > (3.15) From (3.14) and (3.15), we get: ∞ f (x ) − f (x ) = h1 ( x ) − h2 ( x ), φ p ( x ) T p=1 φ p ( x ) = Gα (s, λ p )R(s )ds (3.16) The proof is completed 3.2 The ill-posedness of the inverse source problem Theorem 3.2 The inverse source problem is ill-posed Proof Defining a linear operator K: L2 ( ) → L2 ( ) as follows: ∞ K f (x ) = T Gα (s, λ p )R(s )ds p=1 f ( x ), φ p ( x ) φ p (x ) = k(x, ξ ) f (ξ )dξ , (3.17) where, ∞ k(x, ξ ) = T Gα (s, λ p )R(s )ds p=1 φ p ( x )φ p ( ξ ) Due to k(x, ξ ) = k(ξ , x ) we know K is self-adjoint operator Next, we are going to prove its compactness Defining the finite rank operators KN as follows: N T KN f (x ) = p=1 Gα (s, λ p )R(s )ds f ( x ), φ p ( x ) φ p ( x ) (3.18) Then, from (3.17) and (3.18), we have: ∞ KN f − K f T = p=N+1 ∞ R ≤ p=N+1 ≤ R Gα (s, λ p )R(s )ds C ([0,T ] ) p λ | f ( x ), φ p ( x ) |2 | f ( x ), φ p ( x ) |2 ∞ C ([0,T ] ) N p=N+1 λ | f ( x ), φ p ( x ) |2 (3.19) This implies that: KN f − K f ≤ R C[0,T ] N λ f L2 ( ) = R C[0,T ] λN f L2 ( ) Therefore, KN − K → in the sense of operator norm in L(L2 ( ); L2 ( )) as N → ∞ Also, K is a compact operator Next, the singular values for the linear self-adjoint compact operator K are: ψp = T Gα (s, λ p )R(s )ds, (3.20) and corresponding eigenvectors is φ p which is known as an orthonormal basis in L2 ( ) From (3.17), the inverse source problem introduced above can be formulated as an operator equation K f ( x ) = h ( x ), (3.21) Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 and by Kirsch [39], we conclude that it is ill-posed To illustrate an ill-posed problem, we introduce the following example Let us choose the input final data hm (x ) = φ√m (x ) By (3.14), the source term corresponding to hm is: λm ∞ f (x ) = hm ( x ), φ p ( x ) m T p=1 Gα (s, λ p )R(s )ds φ p (x ) = φ√ m (x ) , φ p (x ) λm ∞ T p=1 Gα (s, λ p )R(s )ds φ p (x ) = φm ( x ) λm T Gα (s, λm )R(s )ds (3.22) Let us choose other input final data g = By (3.14), the source term corresponding to g is f = An error in L2 norm between two input final data is: hm − g = L2 ( ) φm ( x ) λm L2 ( ) = λm Therefore, lim m→+∞ hm − g = lim L2 ( ) λm m→+∞ = (3.23) And an error in L2 norm between two corresponding source term is: fm − f = L2 ( ) φm ( x ) Gα (s, λm )R(s )ds λ T m T From (3.24) and using the inequality fm − f ≥ L2 ( ) λm R C[0,T ] = L2 ( ) Gα (s, λm )R(s )ds ≤ λ T m R C[0,T ] λm Gα (s, λm )R(s )ds (3.24) as in Lemma 2.6, we obtain: , This leads to: lim m→+∞ fm − f L2 ( ) λm > lim m→+∞ R C[0,T ] = +∞ (3.25) Combining (3.23) and (3.25), we conclude that the inverse source problem is ill-posed 3.3 Conditional stability of the source term f In this section, we introduce a conditional stability by the following theorem Theorem 3.3 If f L2 ( ) f Hγ ( ) ≤ M for M > then, ≤ C (γ , M ) h γ γ +1 L2 ( ) , where, C (γ , M ) = M γ +1 γ γ inft∈[0,T ] |R(t )| γ +1 [1 − Eα ,1 (−λ1 T α )] γ +1 Proof From (3.14) and Hölder inequality, we have: ∞ L2 ( ) f h ( x ), φ p ( x ) = p=1 T Gα (s, λ p )R(s )ds ∞ = ∞ ≤ | h ( x ), φ p ( x ) |2 T 2γ +2 p=1 | Gα (s, λ p )R (s )ds| ∞ ≤ | f ( x ), φ p ( x ) |2 T 2γ p=1 | Gα (s, λ p )R (s )ds| γ | h(x ), φ p (x ) | γ +1 | h(x ), φ p (x ) | γ +1 | 0T Gα (s, λ p )R(s )ds|2 p=1 γ +1 ∞ | h ( x ), φ p ( x ) | γ γ +1 p=1 γ +1 h 2γ γ +1 L2 ( ) (3.26) Using Lemma 2.6, we have: T Gα (s, λ p )R(s )ds 2γ ≥ inft∈[0,T ] |R(t )|2γ [1 − Eα ,1 (−λ1 T α )]2γ λ2pγ , Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 and this inequality leads to: λ2pγ | f (x ), φ p (x ) |2 inft∈[0,T ] |R(t )|2γ [1 − Eα ,1 (−λ1 T α )]2γ p=1 ∞ ∞ | f ( x ), φ p ( x ) |2 ≤ T 2γ p=1 | Gα (s, λ p )R (s )ds| f 2H γ inft∈[0,T ] |R(t )|2γ [1 − Eα ,1 (−λ1 T α )]2γ = (3.27) Combining (3.26) and (3.27), we obtain: f L2 ( ) f ≤ inft∈[0,T ] |R(t )| 2γ γ +1 γ +1 Hγ − Eα ,1 (−λ1 Tα) 2γ γ +1 h 2γ γ +1 L2 ( ) ≤ C ( γ , M )2 h 2γ γ +1 L2 ( ) Regularization of the inverse source problem using the Tikhonov method As mentioned above, applying the Tikhonov regularization method we solve the inverse source problem, which minimizes the function f in the following quantity in L2 ( ) Kf −h + β2 f , and its minimized value fβ satisfies: K ∗ K f β ( x ) + β f β ( x ) = K ∗ h ( x ) (4.28) Due to the singular value decomposition for the compact self-adjoint operator K as in (3.20), we have: f β (x ) = ∞ T Gα (s, λ p )R(s )ds p=1 β + | Gα (s, λ p )R(s )ds|2 T < h ( x ), φ p ( x ) > φ p ( x ) (4.29) If the measured data (R (t), h (x)) of (R(t), h(x)) with a noise level of h−h L2 ( ) ≤ , R−R C[0,T ] and satisfied: ≤ , (4.30) then we can present the regularized solution as follows: ∞ f β (x ) = p=1 T Gα (s, λ p )R (s )ds β +| T Gα (s, λ p )R (s )ds|2 < h ( x ), φ p ( x ) > φ p ( x ) (4.31) 4.1 A priori parameter choice Afterwards, we will give an error estimation for for the regularization parameter f (x ) − f β (x ) L2 ( ) and show convergence rate under a suitable choice Theorem 4.1 Let f be as Theorem and the noise assumption (4.30) hold Then, a If < γ ≤ 1, and choose β = ( M ) γ +1 , we have a convergence estimate: f (x ) − f β (x ) L2 ( ) ≤ 3M γ +1 + + inft∈[0,T ] |R (t )|λ1 M1 (R )2 + M γ +1 γ γ +1 b If γ > 1, choose β = ( M ) , we have a convergence estimate: f (x ) − f β (x ) L2 ( ) ≤ 3M 1 + + M2 ( R ) M 2 inft∈[0,T ] |R (t )|λ1 2 , where, M1 ( R ) = Tα inft∈[0,T ] |R(t )| − Eα ,1 (−λ1 T α ) , M2 ( R ) = −γ T α λ1 inft∈[0,T ] |R(t )|[1 − Eα ,1 (−T α )] Proof We first give two lemmas as follows: Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 Lemma 4.1 Assume that (4.30) holds Then we have the following estimate: f β (x ) − f β (x ) f L2 ( ) + inft∈[0,T ] |R (t )| 2β ≤ L2 ( ) (4.32) Proof From (4.29) and (4.31), we have: ∞ f β (x ) − f β (x ) = T Gα (s, λ p )R(s )ds β2 + | p=1 ∞ + T Gα (s, λ p )R(s )ds|2 T β2 β +| T p=1 ∞ T + β2 + | × h ( x ), φ p ( x ) ∞ T β2 + | p=1 Gα (s, λ p )R (s )ds|2 β2 + | T T Gα (s, λ p )R(s )ds|| β2 + | Gα (s, λ p )R(s )ds|2 φ p (x ) < h ( x ), φ p ( x ) > φ p ( x ) Gα (s, λ p )R (s )ds|2 T h ( x ), φ p ( x ) φ p (x ) Gα (s, λ p )(R(s ) − R (s ))ds Gα (s, λ p )R(s )ds|2 T T h ( x ) − h ( x ), φ p ( x ) Gα (s, λ p )(R(s ) − R (s ))ds| p=1 + T Gα (s, λ p )R (s )ds β2 + | Gα (s, λ p )R (s )ds|2 ∞ T Gα (s, λ p )R (s )ds T p=1 β + | = − T Gα (s, λ p )R (s )ds| Gα (s, λ p )R (s )ds|2 φ p (x ) Gα (s, λ p )R (s )ds T Gα (s, λ p )R (s )ds|2 h ( x ) − h ( x ), φ p ( x ) φ p ( x ) = A1 + A2 + A3 , (4.33) where, ∞ β2 A1 = β2 + | p=1 T Gα (s, λ p )(R(s ) − R (s ))ds Gα (s, λ p )R(s )ds|2 T β2 + | T Gα (s, λ p )R (s )ds|2 h ( x ), φ p ( x ) φ p ( x ), (4.34) Gα (s, λ p )(R(s ) − R (s ))ds T ∞ A2 = β2 + | p=1 ∞ T A3 = p=1 β +| T Gα (s, λ p )R(s )ds β2 + | Gα (s, λ p )R(s )ds|2 Gα (s, λ p )R (s )ds T T Gα (s, λ p )R (s )ds|2 T T Gα (s, λ p )R (s )ds h ( x ), φ p ( x ) Gα (s, λ p )R (s )ds|2 h ( x ) − h ( x ), φ p ( x ) φ p ( x ), φ p ( x ) We continue to estimate the error by diving into three steps Step Estimate A1 Using the a2 + b2 L2 ( ) ≥ 2ab, ∀a, b ≥ 0, ∞ A1 L2 ( ) β2 ≤ p=1 β +| ∞ |β ≤ p=1 ∞ p=1 Gα (s, λ p )R(s )ds| T | T inft∈[0,T ] |R (t )|2 | R−R ∞ C[0,T ] inft∈[0,T ] |R (t )| β +| T Gα (s, λ p )R (s )ds| Gα (s, λ p )(R(s ) − R (s ))ds|2 C[0,T ] p=1 Gα (s, λ p )(R(s ) − R (s ))ds Gα (s, λ p )R(s )ds|2 | R−R ≤ ≤ 4β | T T T T Gα (s, λ p )R (s )ds|2 | h ( x ), φ p ( x ) |2 | h ( x ), φ p ( x ) |2 Gα (s, λ p )ds|2 T | h ( x ), φ p ( x ) |2 Gα (s, λ p )ds|2 | Gα (s, λ p )R(s )ds|2 | f ( x ), φ p ( x ) |2 = T R−R C[0,T ] inft∈[0,T ] |R (t )|2 f L2 ( ) (4.35) Hence, A1 L2 ( ) ≤ Step Estimate A2 R − R C[0,T ] f L2 ( ) f L2 ( ) ≤ inft∈[0,T ] |R (t )| inft∈[0,T ] |R (t )| (4.36) L2 ( ) Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 We have: ∞ A2 L2 ( ) Gα (s, λ p )(R(s ) − R (s ))ds T ≤ β +| T p=1 ≤ Gα (s, λ p )R(s )ds| | p=1 T ∞ ≤ p=1 | ≤ | p=1 ∞ ≤ ≤ R−R | T Gα (s, λ p ) R(s ) − R (s ) ds T C[0,T ] T T T T Gα (s, λ p )R (s )ds Gα (s, λ p )R (s )ds| Gα (s, λ p )R(s )ds|2 | Gα (s, λ p )R(s )ds|4 | Gα (s, λ p )R(s )ds|2 | T R−R ∞ T β +| 2 Gα (s, λ p )(R(s ) − R (s ))ds T ∞ Gα (s, λ p )R(s )ds T T T 2 | h ( x ), φ p ( x ) |2 Gα (s, λ p )R (s )ds|2 Gα (s, λ p )R (s )ds|4 | h ( x ), φ p ( x ) |2 Gα (s, λ p )R (s )ds|2 | h ( x ), φ p ( x ) |2 Gα (s, λ p )ds | h ( x ), φ p ( x ) |2 | Gα (s, λ p )R(s )ds|2 Gα (s, λ p )R (s )ds|2 T C[0,T ] | h ( x ), φ p ( x ) |2 inft∈[0,T ] |R (t )| | Gα (s, λ p )R(s )ds|2 p=1 R−R ∞ C[0,T ] inft∈[0,T ] |R (t )| T | f ( x ), φ p ( x ) |2 = p=1 R−R C[0,T ] inft∈[0,T ] |R (t )|2 L2 ( ) f (4.37) Hence, A2 L2 ( ) ≤ R − R C[0,T ] f inft∈[0,T ] |R (t )| ∞ L2 ( ) T ≤ β2 + p=1 ≤ ≤ f L2 ( ) inft∈[0,T ] |R (t )| (4.38) L2 ( ) Step Estimate A3 We have: A3 L2 ( ) ∞ 4β 2 Gα (s, λ p )R (s )ds T Gα (s, λ p )R (s )ds | h ( x ) − h ( x ), φ p ( x ) |2 = p=1 h ( x ) − h ( x ), φ p ( x ) 4β h−h L2 ( ) ≤ 4β (4.39) Hence, A3 L2 ( ) ≤ 2β (4.40) Combining from (4.36) to (4.40), we obtain: f β (x ) − f β (x ) ≤ A1 L2 ( ) ≤ L2 ( ) + A2 L2 ( ) + A3 L2 ( ) f L2 ( ) + inft∈[0,T ] |R (t )| 2β (4.41) The proof of lemma is completed In order to obtain the boundedness of bias, we need the priori condition By Tikhonov’s theorem, the L−1 restricted to the continuous image of a compact set M Thus, we assume f is in a compact subset of L2 ( ) From now on, we assume that f H ( ) ≤ M for γ > Lemma 4.2 Suppose that f (x ) − f β (x ) L2 ( ) f ≤ Hγ ( ) ≤ M Then the following estimate holds: M M1 (R )2 + 1β γ , < γ ≤ M2 ( R )M β , γ > (4.42) Proof From (3.14) and (4.29) and using Parseval equality, we get: Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM 10 [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 +∞ f (x ) − f β (x ) L2 ( ) β | h ( x ), φ p ( x ) |2 = | p=1 T Gα (s, λ p )R(s )ds|2 β2 + | T Gα (s, λ p )R(s )ds|2 γ 2γ β λ−2 λ p | h ( x ), φ p ( x ) |2 p +∞ = | p=1 T Gα (s, λ p )R(s )ds|2 β2 + | T Gα (s, λ p )R(s )ds|2 λ2pγ | h(x ), φ p (x ) |2 = sup |D( p)|2 f T p∈N p=1 | Gα (s, λ p )R (s )ds| +∞ ≤ sup |D( p)|2 p∈N Hγ ( ) (4.43) Hence, D(p) is given by: D ( p) = β λ−p γ β + | 0T Gα (s, λ p )R(s )ds|2 (4.44) Next, we continue to estimate D(p) Infact, we have: D ( p) ≤ β λ−p γ β T α λ1p−γ ≤ T inft∈[0,T ] |R(t )| − Eα ,1 (−λ1 T α ) 2β Gα (s, λ p )R(s )ds (4.45) We divide into two following cases: Case 1: γ ≥ 1; in this case, we note that: λ1p−γ = λγp −1 ≤ λγ1 −1 −γ = λ1 (4.46) Combining (4.43), (4.45) and (4.46), we obtain: f (x ) − f β (x ) L2 ( ) ≤ β T α λ11−γ f inft∈[0,T ] |R(t )|[1 − Eα ,1 (−T α )] Hγ ( ) (4.47) Case 2: < γ < Choose any m such that m ∈ (0, 1) We rewrite N by N = P1 ∪ P2 where, −γ ≤ β −m }, −γ p > β −m }, P1 = { p ∈ N, λ p P2 = { p ∈ N, λ (4.48) From (4.43) and (4.47), we have: f (x ) − f β (x ) L2 ( ) β T α λ1p−γ inft∈[0,T ] |R(t )| − Eα ,1 (−λ1 T α ) = sup p∈P1 P2 + p=1 ≤ β λ−p γ β + | 0T Gα (s, λ p )R(s )ds|2 p∈P2 ≤ Choose m = − γ and from f (x ) − f β (x ) λ2pγ | f (x ), φ p (x ) |2 inft∈[0,T ] |R(t )| − Eα ,1 (−λ1 T α ) −2γ P2 β 2−2m f Hγ ( ) β 2−2m f Hγ ( ) λ2pγ | f (x ), φ p (x ) |2 p=1 Tα inft∈[0,T ] |R(t )| − Eα ,1 (−λ1 f (x ) λ2pγ | f (x ), φ p (x ) |2 p=1 Tα + sup λ p P1 Hγ ( ) Tα) mγ + β 1−γ f Hγ ( ) (4.49) ≤ M we have: L2 ( ) ≤ M1 ( R ) β 2γ M + β 2γ M = β 2γ M ( M1 ( R ) + ) L2 ( ) ≤ βγ M This implies that: f (x ) − f β (x ) M1 (R )2 + (4.50) Now, we continue to prove the theorem If ≤ γ ≤ then from Lemmas 4.1 and 4.2, we get: Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 f (x ) − f β (x ) L2 ( ) f (x ) − f β (x ) f L2 ( ) ≤ f β (x ) − f β (x ) L2 ( ) + 11 L2 ( ) + + M M1 (R )2 + 1β γ 2in f t∈[0,T ] |R (t )| 2β M ≤ + M M1 (R )2 + 1β γ γ + 2β 2in f t∈[0,T ] |R (t )|λ1 ≤ Since β = ( M ) γ +1 and f (x ) − f β (x ) f L2 ( ) ≤ L2 ( ) f λγ1 Hγ ( ) ≤ M λγ1 , we obtain: 3M + M γ +1 inft∈[0,T ] |R (t )|λ1 ≤ γ γ +1 M1 (R )2 + 1M γ +1 + 3M γ +1 + + inft∈[0,T ] |R (t )|λ1 ≤ M1 (R )2 + M γ +1 γ γ +1 γ γ +1 (4.51) If γ > then from Lemmas 4.1 and 4.2, we get: f (x ) − f β (x ) L2 ( ) f (x ) − f β (x ) ≤ L2 ( ) + f β (x ) − f β (x ) L2 ( ) f L2 ( ) + + M2 ( R )M β inft∈[0,T ] |R (t )| 2β ≤ (4.52) Since β = ( M ) and f (x ) − f β (x ) f L2 ( ) L2 ( ) ≤ f λγ1 Hγ ( ) ≤ M λγ1 , we obtain: 3M γ inft∈[0,T ] |R (t )|λ1 ≤ + 1 M2 2 + M2 ( R )M 2 3M 1 + M2 ( R ) M γ + 2 inft∈[0,T ] |R (t )|λ1 ≤ (4.53) 4.2 A posteriori parameter choice In this section, we consider a posteriori regularization parameter choice in Morozov’s discrepancy principle (see in [35]) As usual, it follows the lemma below Lemma 4.3 Set ρ (β ) = ∞ p=1 β2 β + T | h ( x ), φ p ( x ) |2 Gα (s, λ p )R (s )ds ) (4.54) If < < h L2 ( ) , then the following results hold: (a) ρ (β ) is a continuous function (b) ρ (β ) → as β → (c) ρ (β ) → h L2 ( ) as β → ∞ (d) ρ (β ) is a strictly increasing function Theorem 4.2 Assume the priori condition and the noise assumption hold, and there exists k > such that < k < h we choose a unique regularization parameter β > such that: Then, γ +1 If ≤ γ ≤ 1, since β = f (x ) − f β (x ) If γ > 1, since β = f (x ) − f β (x ) L2 ( ) M ≤ M L2 ( ) ≤ we have: γ γ +1 γ γ +1 √ M + α + T Eα ,α +1 (−λ1 T α )N λ1 N γ +1 γ γ γ +1 γ + M γ +1 K γ Nλ1 (4.55) we have: √ M + α + α λ1 N T Eα,α+1 (−λ1 T )N γ 1 + M2K γ , Nλ1 (4.56) where inft∈[0,T ] |R (t )| = N, supt∈[0,T ] |R(t )| = K Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM 12 [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 Firstly, we can receive the following estimation: ∞ k2 β2 β + | 0T Gα (s, λ p )R (s )ds|2 = p=1 ∞ β2 T β + | Gα (s, λ p )R (s )ds|2 ≤2 p=1 ∞ β2 +2 β2 + | p=1 T T | h ( x ), φ p ( x ) |2 | h ( x ) − h ( x ), φ p ( x ) |2 Gα (s, λ p )R(s )ds Gα (s, λ p )R (s )ds|2 λγp λ2pγ | h(x ), φ p (x ) |2 | 0T Gα (s, λ p )R(s )ds|2 (4.57) From (4.57), we get: ∞ k2 | h ( x ) − h ( x ), φ p ( x ) |2 ≤2 p=1 ∞ +2 p=1 β 0T Gα (s, λ p )R(s )ds T γ [β + | Gα (s, λ p )R (s )ds|2 ]λ p ∞ ≤2 ∞ +2 λ2pγ | h(x ), φ p (x ) |2 | 0T Gα (s, λ p )R(s )ds|2 β 0T Gα (s, λ p )R(s )ds T γ [β + | Gα (s, λ p )R (s )ds|2 ]λ p +2 p=1 ≤2 E ( p )2 p=1 λ2pγ | h(x ), φ p (x ) |2 | 0T Gα (s, λ p )R(s )ds|2 λ2pγ | h(x ), φ p (x ) |2 , | 0T Gα (s, λ p )R(s )ds|2 (4.58) where, β 0T Gα (s, λ p )R(s )ds β + | 0T Gα (s, λ p )R (s )ds|2 λγp E ( p) = (4.59) We estimate E(p) as follows: E ( p) = ≤ β 0T Gα (s, λ p )R(s )ds β + | 0T Gα (s, λ p )R (s )ds|2 λγp β 0T Gα (s, λ p )R(s )ds T γ 2λ p β Gα (s, λ p )R (s )ds β supt∈[0,T ] |R(t )| 0T Gα (s, λ p )ds T γ 2λ1 inft∈[0,T ] |R (t )| Gα (s, λ p )ds β supt∈[0,T ] |R(t )| ≤ γ 2λ1 inft∈[0,T ] |R (t )| ≤ (4.60) Therefore, combining (4.58) and (4.60), we conclude that: k2 ( k2 − ) 2 β supt∈[0,T ] |R(t )|2 f 2H γ ( ) 2γ 2λ1 inft∈[0,T ] |R (t )| β supt∈[0,T ] |R(t )|2 ≤ f 2H γ ( ) 2γ 2λ1 inft∈[0,T ] |R (t )| β supt∈[0,T ] |R(t )|2 ≤ f 2H γ ( ) , 2γ 2λ1 inft∈[0,T ] |R (t )|(k2 − )2 ≤2 + (4.61) which gives the required results Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 13 Secondly, we have estimate: ∞ f (x ) − f β (x ) h ( x ), φ p ( x ) ≤ L2 ( ) T p=1 ∞ h ( x ), φ p ( x ) + − φ p (x ) Gα (s, λ p )R (s )ds T p=1 ∞ φ p (x ) Gα (s, λ p )R(s )ds h ( x ), φ p ( x ) T − φ p (x ) Gα (s, λ p )R (s )ds T β + Gα (s, λ p )R (s )ds T L2 ( ) Gα (s, λ p )R (s )ds h ( x ), φ p ( x ) h ( x ), φ p ( x ) φ p (x ) L2 ( ) h ( x ), φ p ( x ) ≤ T p=1 ∞ φ p (x ) h ( x ), φ p ( x ) φ p ( x ) − T Gα (s, λ p )R(s )ds Gα (s, λ p )R (s )ds L2 ( ) h ( x ), φ p ( x ) + T p=1 ∞ φ p (x ) h ( x ), φ p ( x ) φ p ( x ) − T Gα (s, λ p )R (s )ds Gα (s, λ p )R (s )ds h ( x ), φ p ( x ) + T p=1 φ p (x ) − Gα (s, λ p )R (s )ds β2 + T L2 ( ) Gα (s, λ p )R (s )ds T Gα (s, λ p )R (s )ds φ p (x ) L2 ( ) = B1 + B2 + B3 , (4.62) where, ∞ B1 = p=1 ∞ B2 = p=1 ∞ B3 = p=1 h ( x ), φ p ( x ) T φ p (x ) h ( x ), φ p ( x ) T − Gα (s, λ p )R(s )ds φ p (x ) Gα (s, λ p )R (s )ds h ( x ), φ p ( x ) T φ p (x ) Gα (s, λ p )R (s )ds h ( x ), φ p ( x ) − T φ p (x ) Gα (s, λ p )R (s )ds h ( x ), φ p ( x ) φ p (x ) − T G ( s, λ ) R (s )ds β + α p T , L2 ( ) , L2 ( ) Gα (s, λ p )R (s )ds Gα (s, λ p )R (s )ds T h ( x ), φ p ( x ) φ p (x ) L2 ( ) (4.63) We continue to estimate the error by diving into three following steps Step Estimate B21 , we have: ∞ B21 ≤ | h ( x ), φ p ( x ) |2 | T p=1 | Gα (s, λ p )R (s )ds| T ∞ ≤ T p=1 ≤ Gα (s, λ p )ds R−R Gα (s, λ p )ds C[0,T ] inft∈[0,T ] |R (t )|2 f T Gα (s, λ p )(R(s ) − R (s ))ds|2 R−R | T Gα (s, λ p )R (s )ds|2 C[0,T ] | h ( x ), φ p ( x ) |2 | Gα (s, λ p )R(s )ds|2 inft∈[0,T ] |R (t )| T L2 ( ) (4.64) Hence, B1 ≤ R − R C[0,T ] f inft∈[0,T ] |R (t )| (4.65) L2 ( ) Setp Estimate B22 , we have: ∞ B22 ≤ | h ( x ) − h ( x ), φ p ( x ) |2 T p=1 | Gα (s, λ p )R (s )ds| ∞ ≤ | h ( x ) − h ( x ), φ p ( x ) |2 T 2 p=1 | Gα (s, λ p )ds| inft∈[0,T ] |R (t )| ∞ ≤ p=1 ≤ T 2α [E | h ( x ) − h ( x ), φ p ( x ) |2 α 2 α ,α +1 (−λ1 T )] inft∈[0,T ] |R (t )| h−h T 2α [E α ,α +1 (−λ1 L2 ( ) T α )]2 inf t∈[0,T ] |R (t )|2 (4.66) Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM 14 [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 Hence, B2 ≤ h − h L2 ( ) T α [Eα ,α +1 (−λ1 T α )] inft∈[0,T ] |R (t )| (4.67) Step Estimate B23 , we have: B23 ≤ β2 | h ( x ), φ p ( x ) | T β + | Gα (s, λ p )R (s )ds|2 | T Gα (s, λ p )R (s )ds|2 β2 | h ( x ), φ p ( x ) | T β + | Gα (s, λ p )R (s )ds|2 ≤ ≤ k2 2 (4.68) From (4.58) and (4.68), we get: B23 ≤ 2 + β supt∈[0,T ] |R(t )|2 f 2γ inft∈[0,T ] |R (t )|2 λ1 √ + (4.69) a2 + b2 ≤ a + b ∀a, b ≥ Hence, From (4.69) and using the inequality: B3 ≤ Hγ ( ) β supt ∈[0,T ]|R(t )| γ f inft∈[0,T ] |R (t )|λ1 (4.70) Hγ ( ) From (4.62), we receive: f (x ) − f β (x ) ≤ L2 ( ) R − R C[0,T ] f inft∈[0,T ] |R (t )| √ + + From (4.71) and f (x ) − f β (x ) f ≤ L2 ( ) f + β supt ∈[0,T ]|R(t )| T α [E γ inft∈[0,T ] |R (t )|λ1 Hγ ( ) f h − h L2 ( ) α α ,α +1 (−λ1 T )] inft∈[0,T ] |R (t )| Hγ ( ) (4.71) putting inft∈[0,T ] |R (t )| = N, supt∈[0,T ] |R(t )| = K, we get: √ M M K + α + +β γ γ α λ1 N T [Eα,α+1 (−λ1 T )]N λ1 N ≤ L2 ( ) λγ1 L2 ( ) Now, we continue to prove the theorem γ +1 If ≤ γ ≤ 1, since β = f (x ) − f β (x ) L2 ( ) If γ > 1, since β = f (x ) − f β (x ) M ≤ √ M + α + + γ α T [ E ( − λ T ) ] N λ1 N α ,α +1 ≤ γ +1 M γ γ +1 γ +1 γ γ +1 √ M + α + γ α T Eα ,α +1 (−λ1 T )N λ1 N γ M γ +1 K γ Nλ1 γ γ γ +1 + M γ +1 K γ Nλ1 (4.72) we have: ≤ L2 ( ) we have: ≤ √ M + α + + γ α λ1 N T [Eα,α+1 (−λ1 T )]N √ M + α + γ α λ1 N T Eα,α+1 (−λ1 T )N 2 1 M2 K γ Nλ1 1 + M2K γ Nλ1 (4.73) Numerical experiments To verify our proposed methods, we carry out numerically above regularization methods Two different numerical examples corresponding to T = 1, and α = 0.2 are shown in this section The couple of (h , R ) which are determined below, play as measured data with a random noise as follows: h ( ) = h ( ) + rand ( ) , h L2 ( ) R ( ) = R( ) + rand ( ), (5.74) Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 15 where rand() ∈ (−1, ) is a random number We can easily verify the validity of the inequality: h−h L2 ( ) ≤ , R−R C[0,T ] ≤ and In addition, we can take the regularization parameter for the priori parameter choice rule, β pri = ( M ) , where the value of M plays a role as the priori condition computed by f H (0,π ) Similarly, based on the choice of k we can take the regularization parameter for the posteriori parameter choice rule, β pos = satisfied (4.61) From (3.14), we can definite the exact solution, as follows: f (x ) = P π p=1 T Gα (s, λ p )R(s )ds M{k,R,R } , where M{k,R,R } is dependent on {k, h(x ), sin( px ) sin( px ) } (5.75) From (4.31), we can definite the regularized solution using Tikhonov method f β (x ) = π p=1 Gα (s, λ p )R (s )ds T P [ β ( )] + T Gα (s, λ p )R (s )ds h (x ), sin( px ) sin( px ) (5.76) where P is a large enough number, and plays as a truncation number 5.1 Example In this example, we consider the function f is an exact data function We particularly consider an one-dimensional case of the problem (1.1) as follows with λ p = p2 and T = ⎧ α ⎪ ⎨ut (x, t ) − uxx (x, t ) = R(t ) f (x ), (x, t ) ∈ (0, π ) × (0, ), u(0, t ) = u(π , t ) = 0, t ∈ [0, 1], ⎪ ⎩u(x, ) = 0, x ∈ (0, π ), u(x, ) = h(x ), x ∈ (0, π ) (5.77) whereby, t −α et − , (2 − α ) h(x ) = 10−1 e − sin(4x )(π − x ) (2 − α ) R(t ) = (5.78) From (3.12) and (5.78), we can deduce the exact solution, as follows: f (x ) = 10−1 P e−1 (2 − α ) p=1 [β ( ) ]2 Gα (s, p2 )R (s )ds + Gα (s, p2 )R (s )ds sin(4x )(π − x ) (5.79) and s −α es − + rand ( ), (2 − α ) R (s ) = Gα (s, p2 ) = (1 − s )α −1 Eα ,α (−p2 (1 − s )α ) (5.80) From (4.31) and (5.80), we can deduce the regularized solution, as follows: fβ (x ) = 10−1 e−1 (2 − α ) P × p=1 where [ β ( )] + Gα (s, p2 )R (s )ds = 1+ rand ( ) h L2 ( ) Gα (s, p2 )R (s )ds Gα (s, p2 )R (s )ds sin(4x )(π − x ) α −1 E α α ,α (−p (1 − s ) )R (1 − s ) ds = [1 p2 − Eα ,1 (−p2 )] (5.81) ( R ( si ) + |rand (si )| )ds Next, we establish the regularized solution according to composite Simpson’s rule In general, the whole numerical procedure is summarized in the following steps: Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM 16 [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 Table Error estimation between the exact solution and its regularized solution in Example β 1E−01 1E−02 1E−03 1E−04 1E−05 1E−06 1E−07 1E−08 1E−09 1E−10 β β β E1 pri E2 pri E1 pos E2 pos 8.15E−01 8.02E−02 7.64E−03 6.38E−04 6.15E−05 5.94E−06 5.75E−07 4.53E−08 3.54E−09 4.64E−10 9.04E−01 8.86E−02 8.50E−03 7.34E−04 8.20E−05 7.95E−06 6.84E−07 6.73E−08 5.25E−09 5.24E−10 6.97E−02 8.16E−03 7.24E−04 7.19E−05 5.16E−06 6.09E−07 4.09E−08 4.09E−09 3.85E−10 4.53E−11 7.97E−02 9.16E−03 8.58E−04 7.34E−05 6.84E−06 7.33E−07 5.43E−08 7.55E−09 4.45E−10 5.95E−11 Step Choose Q and L to generate the spatial and temporal discretization in such a manner as: xi = i x, x= π , i = 0, Q , Q t = , j = 0, L L t j = j t, (5.82) Of course, the higher value of Q and L will provide numerical results more accurate and stable, however in the following numerical examples Q = L = 100 are satisfied Step Setting fβ (xi ) = fβ ,i and f (xi ) = fi , constructing two vectors contained all discrete values of f β and f denoted by and , respectively β Q+1 , β = [ f β ,0 , f β ,1 , , f β ,Q ] ∈ R = [ f0 , f1 , , fQ ] ∈ RQ+1 (5.83) Step Error estimate between the exact and regularized solutions Relative error estimation: Q i=0 E1 = | fβ (xi ) − f (xi )|2L2 (0,π ) Q i=0 | f (xi )|2L2 (0,π ) (5.84) Absolute error estimation: E2 = Q +1 Q | fβ (xi ) − f (xi )|2L2 (0,π ) i=0 (5.85) In practice, it is very difficult to obtain the value of M for the priori parameter choice rule without having an exact solution We thus try to take rule, and β pos = supt∈[0,T ] |R (t )| M{k,R,R } 2γ 2λ1 (k2 −2 ) inft∈[0,T ] |R (t )| f f H2 ( ) ≤ M with M = 517 leading to β pri = ( M ) for the priori parameter choice with M{k,R,R } = 1858 for the posteriori parameter choice rule based on k = 1.5 and M{k,R,R } = Hγ ( ) Figs and below show a comparison between the exact solution and its regularized solutions for both parameter choice rule methods, the priori and posteriori, in Example 1, respectively Table shows the absolute and relative error estimates between the exact solution and its regularized solutions for both methods, the priori and posteriori parameter choice rules, in Example 5.2 Example In this example, we consider the function f is an exact data function, and a two-dimensional case of the problem (1.1) with λ p = p2 + q2 and T = as follows: ⎧ α ut (x, y, t ) − uxx (x, y, t ) − uyy (x, y, t ) = R(t ) f (x, y ), (x, y ) ∈ (0, π ) × (0, π ), t ∈ [0, 1], ⎪ ⎪ ⎨u(0, y, t ) = u(π , y, t ) = 0, (y, t ) ∈ (0, π ) × [0, 1], u(x, 0, t ) = u(x, π , t ) = 0, (x, t ) ∈ (0, π ) × [0, 1], ⎪ ⎪ ⎩u(x, y, ) = 0, (x, y ) ∈ (0, π ) × (0, π ), u(x, y, ) = h(x, y ), (x, y ) ∈ (0, π ) × (0, π ) (5.86) Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 17 Fig A comparison between the exact solution and its regularized solution for the priori parameter choice rule in Example whereby, t −α et − , (2 − α ) h(x, y ) = 2−1 e − sin(4x ) sin(π − y ) (2 − α ) R(t ) = (5.87) From (3.12) and (4.31), we can obtain the exact and regularized solutions, as follows: Q Q f (x, y ) = e−1 2−1 q=1 q=1 fβ (x, y ) = 2−1 Q e−1 p=1 q=1 where 1 Q Gα (s, p2 + q2 )R (s )ds h L2 ( )×L2 ( ) Gα (s, ( p2 + q2 ))R (s )ds [ β ( )] + sin(4x ) sin(π − y ), rand ( ) 1+ (2 − α ) × (2 − α ) Gα (s, p2 + q2 )R (s )ds = Gα (s, p2 + q2 )R (s )ds α −1 E α ,α ( − ( p (1 − s ) sin(4x ) sin(π − y ) + q2 )(1 − s )α )R ds = (5.88) [1 − Eα ,1 (−( p2 p2 +q2 + q2 ))] ( R ( si ) + |rand (si )| )ds Next, we establish the regularized solution according to composite Simpson’s rule In this example, we choose P and Q are large enough, unlike the first example, we can have f H (0,π ) < M with M = 112230 from the analytical solution, which implies that β pri = ( M1 ) for the priori parameter choice, and β pos = supt∈[0,T ] |R (t )| M{k,R,R } ori parameter choice rule based on k = 1.8 and M{k,R,R } = 2γ f 2λ1 (k −2 ) inft∈[0,T ] |R (t )| procedure is summarized in the following steps: with M{k,R,R } = 239587 for the posteriHγ ( ) In general, the whole numerical Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM 18 [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 Fig A comparison between the exact solution and its regularized solution for the posterior parameter choice rule in Example Step Choose Q and K (in our computations, Q = K = 100 are chosen) to discretize spatial and temporal domain, as follows: xi = i x, x= y j = j y, y= tk = k t, π Q , i = 0, Q , π , j = 0, K , K t = , k = 0, L L (5.89) Step Setting fβ )(i, j )= f (., , )β xi , y j and f(i, j)=f(., , 0)(xi , yj ), constructing two following vectors contained all discrete values of f(., , 0)β , and f(., , 0) denoted by β and , respectively ⎡ f β ( 0, ) ⎢ f β ( 1, ) ⎢ ⎢ f ( 2, ) β = ⎢ β ⎢ ⎣ ⎡ fβ (Q, ) f ( 0, ) ⎢ f ( 1, ) ⎢ f ( 2, ) =⎢ ⎢ ⎣ f (Q, ) f β ( 0, ) f β ( 1, ) f β ( 2, ) fβ (Q, ) f ( 0, ) f ( 1, ) f ( 2, ) f (Q, ) f β ( 0, K − ) f β ( 1, K − ) f β ( 2, K − ) ⎤ f β ( 0, K ) f β ( 1, K ) ⎥ ⎥ fβ (2, K ) ⎥ ∈ RQ+1 × RK+1 ⎥ ⎥ ⎦ fβ (Q, K − ) fβ (Q, K ) ⎤ f ( 0, K − ) f ( 0, K ) f ( 1, K − ) f ( 1, K ) ⎥ ⎥ f ( 2, K − ) f (2, K ) ⎥ ∈ RQ+1 × RK+1 ⎥ ⎦ f (Q, K − ) f (Q, K ) Step Error estimate between the exact and regularized solutions; Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ARTICLE IN PRESS JID: APM [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 19 Fig A comparison between exact solution and its regularized solution for the priori parameter choice rule method in Example Relative error estimation: E1 = Q i=0 K j=0 | f β ( xi , y j ) − f ( xi , y j )|2 Q i=0 K j=0 | f ( xi , y j )|2 (5.90) Absolute error estimation: E2 = 1 Q +1K+1 Q K i=0 j=0 | f β ( xi , y j ) − f ( xi , y j )|2 (5.91) Figs and below show a comparison between the exact solution and its regularized solutions for both parameter choice rule methods, the priori and posteriori, in Example 2, respectively Table shows the absolute and relative error estimates between the exact solution and its regularized solutions for both parameter choice rule methods in Example From Figs and combined with Table of the first example, it shows that the posterior parameter choice rule method converges to the exact solution faster than the prior parameter choice rule method This property has been confirmed again by Figs and combined with Table in the second example Particularly, the error estimate shown in Tables and2 show that the posterior parameter choice rule method converges to the exact solution with one order faster than the prior parameter choice rule method Nevertheless, from two examples, it also shows our proposed regularized methods have a very good convergence rate to the exact solution once tends to Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 JID: APM 20 ARTICLE IN PRESS [m3Gsc;May 10, 2016;11:58] H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 Fig A comparison between exact solution and its regularized solution for the posteriori parameter choice rule method in Example Table Error estimation between the exact solution and its regularized solution in Example β 1E−01 1E−02 1E−03 1E−04 1E−05 1E−06 1E−07 1E−08 1E−09 1E−10 β β β E1 pri E2 pri E1 pos E2 pos 7.12E−02 6.88E−03 6.58E−04 6.33E−05 7.10E−06 5.89E−07 5.07E−08 4.35E−09 4.21E−10 3.23E−11 9.70E−02 9.42E−03 8.66E−04 7.35E−05 8.81E−06 7.58E−07 6.36E−08 5.55E−09 3.12E−10 5.43E−11 8.14E−03 7.09E−04 7.92E−05 5.81E−06 6.16E−07 5.54E−08 4.90E−09 3.88E−10 3.36E−11 4.23E−12 7.64E−03 8.99E−04 8.39E−05 6.74E−06 5.97E−07 5.09E−08 4.41E−09 3.35E−10 2.16E−11 4.51E−12 Conclusion In this study, we solved the inverse problem to recover the source term for the time fractional diffusion equation with the time dependent coefficient by applying Tikhonov method In the theoretical results, we obtained the error estimates of two priori and posterior parameter choice rule methods based on the priori condition In the numerical results, it shows that the proposed regularized solutions are converged to the exact solutions Furthermore, it also shows that the posteriori parameter choice rule method is better than the priori parameter choice rule method in term of the convergence rate In the future work, we will continue to study some source terms for multiple diffusion equations Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 JID: APM ARTICLE IN PRESS H.T Nguyen et al / Applied Mathematical Modelling 000 (2016) 1–21 [m3Gsc;May 10, 2016;11:58] 21 Acknowledgments This work was supported by the Institute for Computational Science and Technology at Ho Chi Minh City, Vietnam under the project named Fractional Diffusion–Wave Equations and Application to Soil Contaminant, and the NRF Research Grant of Korea The authors also would like to thank the editors and anonymous reviewers for their very valuable constructive comments to improve this manuscript References [1] E.E Adams, W.L Gelhar, Field study of dispersion in a heterogeneous aquifer 2: Spatial moments analysis, Water Resour Res 28 (1992) 3293307 [2] M Kirane, A.S Malik, M.A Al-Gwaiz, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math Methods Appl Sci 36 (9) (2013) 1056-1069 [3] I Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering,, 198, Academic Press Inc., San Diego, CA, 1990 [4] R Nigmatulin, The realization of the generalized transfer equation in a medium with fractal geometry., Phys Stat Sol B 133 (1986) 425–430 [5] D.S Eidelman, N.A Kochubei, Cauchy problem for fractional diffusion equations, J Differ Equ 199 (2) (2004) 211–255 [6] A Hanyga, Multidimensional solutions of time-fractional diffusion–wave equations., R Soc Lond Proc Ser A Math Phys Eng Sci 458 (2020) (2002) 933–957 [7] Y Luchko, Maximum principle for the generalized time-fractional diffusion equation, J Math Anal Appl 351 (1) (2009) 218–223 [8] Y Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract Calc Appl Anal 15 (1) (2012) 141160 [9] K Sakamoto, M Yamamoto, Initial value/boundary value problems for fractional diffusion–wave equations and applications to some inverse problems, J Math Anal Appl 382 (1) (2011) 426–447 [10] F Liu, P Zhuang, I Turner, K Burrage, V Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl Math Model 38 (15-16) (2014) 3871–3878 [11] S Shen, F Liu, J Chen, I Turner, V Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl Math Comput 218 (22) (2012) 10861–10870 [12] W McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J 52 (2) (2010) 123–138 [13] K Mustapha, W McLean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J Numer Anal 51 (1) (2013) 491–515 [14] Y Zhang, Z.Z Sun, H.L Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J Comput Phys 265 (2014) 195210 [15] X Zhao, Z.Z Sun, Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium, J Sci Comput 62 (2015) 747–771 [16] X Zhao, Q Xu, Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient, J Appl Math Model 38 (2014) 3848–3859 [17] F Liu, K Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput Math Appl 62 (2011) 822–833 [18] B Yu, X Jiang, C Wang, Numerical algorithms to estimate relaxation parameters and caputo fractional derivative for a fractional thermal wave model in spherical composite medium, J Appl Math Comput 274 (2016) 106–118 [19] R Metzler, J Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys Rep 339 (20 0) 1–77 [20] W Fan, X Jiang, H Qi, Parameter estimation for the generalized fractional element network zener model based on the Bayesian method, Physica A 427 (2015) 40–49 [21] S.B Yuste, L Acedo, K Lindenberg, Reaction front in an a+b → c reaction–subdiffusion process, Phys Rev E 69 (2004) 036126 [22] F Liu, V Anh, I Turner, Numerical solution of the space fractional Fokker–Planck equation, J Comput Appl Math 166 (2004) 209–219 [23] D.A Murio, Stable numerical solution of fractional-diffusion inverse heat conduction problem, Comput Math Appl 53 (2007) 1492–1501 [24] J.J Liu, M Yamamoto, A backward problem for the time-fractional diffusion equation, Appl Anal 89 (2010) 1769–1788 [25] T Wei, J Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl Numer Math 78 (2014) 95–111 [26] J.G Wang, Y.B Zhou, T Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl Numer Math 68 (2013) 3957 [27] Z.Q Zhang, T Wei, Identifying an unknown source in time-fractional diffusion equation by a truncation method, Appl Math Comput 219 (11) (2013) 5972–5983 [28] M Kirane, A.S Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl Math Comput 218 (1) (2011) 163–170 [29] B Jin, W Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Probl 28 (7) (2012) 075010.19 pp [30] W Rundell, X Xu, L Zuo, The determination of an unknown boundary condition in a fractional diffusion equation, Appl Anal 92 (7) (2013) 1511–1526 [31] Y Luchko, W Rundell, M Yamamoto, L Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction–diffusion equation, Inverse Probl 29 (2013) 065019.16pp [32] H.E Roman, P.A Alemany, Continuous-time random walks and the fractional diffusion equation, J Phys A 27 (1994) 3407–3410 [33] B Jin, W Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Probl 31 (2015) 035003.40pp [34] R Metzler, J Klafter, Boundary value problems for fractional diffusion equations, Phys A 278 (20 0) 107–125 [35] A.A Kilbas, H.M Srivastava, J J.Trujillo, Theory and Application of Fractional differnetial equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V, Amsterdam, 2006 [36] L.C Evans, Partial Differential Equation, 19, American Mathematical Society, Providence, Rhode Island, 1997 [37] H Brezis, Analyse Fonctionelle, Masson, Paris, 1983 [38] X.L Feng, L Elden, C.L Fu, Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region, J Math Comp Simul 79 (2) (2008) 177–188 [39] A Kirsch, An introduction to the mathematical theory of inverse problem, second ed., 2011, Springer Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.04.009 ... property of the inverse source problem Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical... Lemmas 4.1 and 4.2, we get: Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling... estimate between the exact and regularized solutions; Please cite this article as: H.T Nguyen et al., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied