Generalized Tikhonov methods for an inverse source problem of the time-fractional diffusion equation

Chaos, Solitons and Fractals 108 (2018) 39–48

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Generalized Tikhonov methods for an inverse source problem of the time-fractional diffusion equation Yong-Ki Ma a, P. Prakash b, A. Deiveegan b,∗ a b

Department of Applied Mathematics, Kongju National University, Chungcheongnam-do 32588, Republic of Korea Department of Mathematics, Periyar University, Salem 636 011, India

a r t i c l e

i n f o

Article history: Received 31 July 2017 Revised 13 December 2017 Accepted 3 January 2018

MSC: 35R30 35R11 65M32

a b s t r a c t In this paper, we identify the unknown space-dependent source term in a time-fractional diffusion equation with variable coefficients in a bounded domain where additional data are consider at a fixed time. Using the generalized and revised generalized Tikhonov regularization methods, we construct regularized solutions. Convergence estimates for both methods under an a-priori and a-posteriori regularization parameter choice rules are given, respectively. Numerical example shows that the proposed methods are effective and stable. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Inverse problem Fractional diffusion equation Tikhonov regularization Convergence analysis

1. Introduction Fractional diffusion equations have attracted great attention in last few decades. The fractional diffusion equation is a generalization of the classical diffusion equation which models anomalous diffusive phenomena. However, for a few practical issues, the initial information, or a part of boundary information, or diffusion coefficients, or source term might not be given and that we need to recover them by extra measuring information which is able to yield to some fractional diffusion inverse problems. In recent years, inverse problems for time-fractional diffusion equation have become very active, interdisciplinary research area and have wide application in science, engineering, industry, medicine, finance as well as in life and earth sciences. In this paper, we consider the inverse problem of determining the unknown source term f(x) in time fractional diffusion equation. Let  ⊂ Rd be a bounded domain with sufficient smooth boundary ∂ . We consider the fractional diffusion equation of the

form

⎫ ∂ α u(x, t ) − (Lu )(x, t ) = f (x )q(t ), x ∈ , t ∈ (0, T ), ⎪ ⎬ u(x, t ) = 0, x ∈ ∂ , t ∈ (0, T ), (1.1) ⎪ ⎭ u(x, 0 ) = 0, x ∈ , 0 t

where 0 ∂tα u is the left-sided Caputo fractional derivative of order α (0 < α < 1) defined by α 0 ∂t u =

1 (1 − α )



t 0

1

(t − s )α

∂ u(x, s ) ds, 0 < α < 1, ∂s

(1.2)

and −L is a symmetric uniformly elliptic operator defined on D(−L ) = H 2 () ∩ H01 () by

Lu(x ) =

d  i=1

the

 d ∂  ∂ a (x ) u(x ) + c (x )u(x ), x ∈ , ∂ xi j=1 ij ∂ x j

coefficients

satisfy

¯ , ai j = a ji ∈ C 1 ()

d 

(1.3)

a i j ξi ξ j ≥

i, j=1

θ

d  i=1

∗

Corresponding author. E-mail address: [email protected] (A. Deiveegan).

https://doi.org/10.1016/j.chaos.2018.01.003 0960-0779/© 2018 Elsevier Ltd. All rights reserved.

¯ . |ξi |2 (θ > 0 ), c(x ) ≤ 0, c(x ) ∈ C ()

The inverse problem is to find f(x) from a final data u(x, T ) = g(x ). Since the data g(x) is measured, there must be measurement errors and we assume the measured data function gδ (x) ∈ L2 ()

40

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

Lemma 2.3. [12, Lemma 2.1] Let λ > 0, then we have:

which satisfies



g − gδ ≤ δ

(1.4)

where  ·  denotes the norm and the constant δ > 0 represents noise level. According to the Hadamard requirements (existence, uniqueness and stability of the solution), the inverse problem is ill-posed mathematically. For stable reconstruction, we require some regularization techniques see [1]. If α = 1, the problem is inverse problem for standard diffusion equation and has been studied in [2–4]. However, for the fractional inverse source problem, there are only very few works; [5] obtained the uniqueness in determining diffusion coefficient on the basis of Gel’fand-Levitan theory. Sakamoto and Yamamoto [6] derived the regularity and qualitative properties of solution to fractional diffusion-wave equation. Jiang et al. [7] proved uniqueness and Fujishiro and Kian [8] studied stability of the inverse problem of determining a source term with interior measurements in fractional diffusion equation. When q(t ) = 1, a number of regularization techniques including Tikhonov regularization and a simplified Tikhonov regularization method [9], quasi-reversibility method [10], truncation method [11] have been applied for the inverse source problem for fractional diffusion equation. According to our knowledge, there are few articles dealing the source term with q(t) such as modified quasi-boundary value method [12], Tikhonov regularization method [13] and the Fourier transform method [14]. The generalized Tikhonov regularization [3] and the revised generalized Tikhonov regularization [4] have been proposed for solving the inverse problems for usual partial differential equations.Yang et al. [15] constructed generalized regularization method for inverse source problem for space-fractional diffusion equation. Zhang and Zhang [16] used generalized Tikhonov method to solve backward time-fractional diffusion problem. Motivated by above reasons, in this article, we propose generalized and revised generalized Tikhonov regularization methods for inverse source problem for time-fractional diffusion equation with variable coefficients in a general bounded domain. We establish a convergence estimates under an a-priori and a-posteriori regularization parameter choice rules. All the numerical results are based on the a posteriori parameter choice rule which is independent of the a priori bound of the exact solution. It is more useful in practical issues. The paper is organized as follows. In Section 2, we simply recall some preliminaries. In Section 3 we construct the regularized solutions by the generalized Tikhonov regularization method and give convergence estimates under an a-priori and a-posteriori regularization parameter choice rules. In Section 4 we establish the revised generalized Tikhonov regularization method and give convergence estimates under two regularization parameter choice rules. Finally numerical example and its simulation are exploited to demonstrate the usefulness and effectiveness of the methods. L2

2. Preliminaries In this section, we recall basic definitions and lemmas. Definition 2.1. [17,18] The Mittag–Leffler function is defined as

Eα ,β (z ) =

∞  k=0

zk , (α k + β )

z ∈ C,

where α > 0 and β ∈ R are arbitrary constants.

d Eα ,1 (−λt α ) = −λt α −1 Eα ,α (−λt α ), dt

1

(β )

, z ∈ C.

0 < α < 1.

Lemma 2.4. [12, Lemma 2.3] For 0 < α < 1, η > 0, we have 0 ≤ Eα ,α (−η ) ≤ (1α ) . Moreover, Eα ,α (−η ) is a monotonic decreasing function with η > 0. Lemma 2.5. [12, Lemma 2.6] For any λn satisfying λn ≥ λ1 > 0, there exist positive constants C, depending on α , T, λ1 such that,

C

λn T α

≤ Eα ,α +1 (−λn T α ) ≤

1

λn T α

Lemma 2.6. [12, Remark 3.1] If q(t) ∈ C[0, T] satisfying q(t) ≥ q0 > 0 for all t ∈ [0, T], set qC[0,T ] = sup |q(t )|. Then we have: t∈[0,T ]

q0 C

λn

 ≤

T

0

q(τ )(T − τ )α −1 Eα ,α (−λn (T − τ )α )dτ ≤

qC[0,T ] λn

(2.1)

Proof. By using previous lemmas it is easy to prove (2.1). We omit the details here.  Lemma 2.7. For constants p > 0, μ > 0, s ≥ λ1 > 0, we have

F (s ) =

1 s ≤ C1 (q0C , p)μ− p+1 , q0C (1 + μs p+1 )

(2.2)

G (s ) =

p μs 2 +1 ≤ C2 ( p)μ 2 p+2 , 1 + μs p+1

(2.3)

H (s ) =

p+2 qC[0,T ] μs 2 ≤ C3 (qC[0,T ] , p)μ 2 p+2 . p+1 1 + μs

(2.4)

p

p

Proof. We

know

that,

lim F (s ) = lim F (s ) = 0, s→∞

s→0

thus

F (s ) ≤

sup F (s ) ≤ F (s0 ), where s0 > 0 such that F  (s0 ) = 0. It is easy s>0

to prove that s0 =

F ( s ) ≤ F ( s0 ) =





1 1/p+1 pμ

> 0, then we have

1 1 1 p p− p+1 μ− p+1 = C1 (q0C , p)μ− p+1 . q0 C ( p + 1 )

Similarly, we can prove (2.3) and (2.4).



3. A generalized Tikhonov regularization method Denote the eigenvalues of the operator −L as λn which satisfy 0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λn ≤ · · · , limn→∞ λn = +∞, and the corresponding eigenfunctions as Xn (x ) ∈ H 2 () ∩ H01 () form an orthonormal basis in L2 (). As in [6], the fractional power (−L )γ is defined for γ ∈ 1

R and for example D((−L ) 2 ) = H01 (). We set  uD( (−L )γ ) =

(−L )γ uL2 () . We note that the norm uD((−L)γ ) is stronger than uL2 () for γ > 0. Define



γ

D (−L )



=

ψ ∈ L2 ();

∞ 

 2γ n

λ |(ψ , Xn )| < ∞ , 2

n=1

where ( · , · ) is the inner product in L2 (), then D((−L )γ ) is a Hilbert space with the norm



Lemma 2.2. [12, Lemma 2.4] For α > 0 and β ∈ R, we have:

Eα ,β (z ) = zEα ,α +β (z ) +

t > 0,

 ψ D((−L)γ ) =

∞  n=1

12 2γ n

λ |(ψ , Xn )|

2

.

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

From equation (3.5) of [12], we know that the formal solution for (1.1) can be written as

u(x, t ) =

∞ 



t

fn 0

n=1

Qn (t ) =

and



τ )α −1 Eα ,α (−λn (t − τ )α )dτ ; then letting t = T , we have u(x, T ) = g(x ) = and

∞ 

t 0

q(τ )(t −

fn Qn (T )Xn (x ),

n=1

(3.1)

= fn Qn (T ).

gn

f (x ) =

n=1



T

fn

n=1

 K fμδ ,GEN − gδ 2 +  fμδ ,GEN 2D −L 2p  ( ) 1  δ 1 ≤ J fμ,GEN ≤ J( f ) μ μ 1 ≤  Kf − gδ 2 +  f 2D −L 2p  ( ) μ μ

0

(3.2)

q(τ )(T − τ )α −1 Eα ,α



−λn (T − τ )α dτ Xn (x )

= g(x ), (3.3) we know that K is a linear self-adjoint compact operator and the problem (3.3) is ill-posed. From [16], we have a stable solution to problem (3.3) with noisy data gδ by generalized Tikhonov regularization method which minimizes the quantity

J ( f ) = K f − gδ 2 + μ f 2

p

D((−L ) 2 )

,



δ 2

fμ,GEN

p

D((−L ) 2 )

≤

E2

δ2

δ 2 + E 2 ≤ 2E 2 ,

K ∗ K fμδ ,GEN + μ(−L ) p fμδ ,GEN = K ∗ gδ .

(3.5)

By singular value decomposition for compact self-adjoint operator, we have ∞ 

Qn ( T ) gδ Xn (x ). 2 + μλ p n | Q ( T ) | n n n=1

(3.6)

To obtain the convergence estimates, we usually need some ap priori bounded condition for f ∈ D((−L ) 2 ) :

 f D((−L) 2p ) ≤ E, p > 0,

Furthermore, we get

 fμδ ,GEN − f D −L 

√

(3.7)

By (3.8), we have

 fμδ ,GEN − f ≤ (q0C )− p+2  K fμδ ,GEN −g p/( p+2)  fμδ ,GEN − f 2/( p+2)p  D (−L ) 2 √ p − p+2 p/( p+2 ) 2/( p+2 ) ≤ 2 + 1 ( q0 C ) δ E . p

The proof is completed.



3.2. Convergence estimates of generalized Tikhonov regularization method under a-posteriori parameter choice rules According to the Morozov’s discrepancy principle [1], we adopt the regularization parameter μ as the solution of the equation

δ

K fμ,GEN − gδ = τ δ,

(3.10)

where τ > 1 is a constant.





δ δ δ

Lemma 3.3. Let ρ (μ ) = K fμ ,GEN (x ) − g (x ) . If 0 < δ < g , then the following results hold:

(a) ρ (μ) is a continuous function; (b) lim ρ (μ ) = 0; μ→ 0

(c)



lim ρ (μ ) = gδ ;

μ→ + ∞

(d) For μ ∈ (0, +∞ ), ρ (μ ) is a strictly increasing function.

where E > 0 is a constant. From (3.2), we obtain the following conditional stability.

Proof. The above results are straightforward by setting

Theorem 3.1. [12, Theorem 3.2] If q(t) ∈ C[0, T] satisfying p q(t) ≥ q0 > 0, t ∈ [0, T] and f (x ) ∈ D((−L ) 2 ) satisfy an a-priori bounded condition (3.7), then

ρ (μ ) = ⎝

p

2

 f ≤ C4 E p+2  g p+2 , p > 0, where C4 = (q0C )

p − p+2

(3.8)

is a constant depending on α , T, p, λ1 .

3.1. Convergence estimates of generalized Tikhonov regularization method under a-priori parameter choice rules Theorem 3.2. Let q(t) ∈ C[0, T] satisfying q(t) ≥ q0 > 0, t ∈ [0, T]. Suppose the a-priori bounded condition (3.7) and the noise assumption  (1.4) hold. If we choose the regularization parameter μ = Eδ 2 , we have the convergence estimate

δ

p 2

fμ,GEN (x ) − f (x ) ≤ C5 E p+2 δ p+2 , where C5 =

√



2 + 1 ( q0 C )

−

p p+2

(3.9)

.

δ Proof. Since fμ is the minimizer of J(f) defined by (3.4), we can ,GEN

obtain



 ≤ fδ     2 +1 E, μ,GEN D (−L ) 2p +  f D (−L ) 2p ≤ ( ) √  K fμδ ,GEN − g  ≤  K fμδ ,GEN − gδ  +  gδ − g ≤ 2 + 1 δ. p 2

(3.4)

δ where p ∈ R+ . Let fμ ,GEN be a solution of the problem (3.4) which satisfies the following normal equation

fμδ ,GEN (x ) =



From K f = g, g − gδ ≤ δ and (3.7), we have

E

From (3.1), the problem of finding f(x) can be formulated as an operator equation:

(Kf )(x ) =

 ≤ 1

 δ2  K fμδ ,GEN − gδ 2 ≤ J fμδ ,GEN ≤ J ( f ) ≤ δ 2 + 2 E 2 ≤ 2δ 2 .

1 gn Xn (x ). Qn ( T )

∞ 

p

)2

Similarly, we get

consequently, we get, ∞ 

(

q(τ )(t − τ )α −1 Eα ,α (−λn (t − τ )α )dτ Xn (x ).

fn = ( f, Xn ), gn = (g, Xn )

Denote

 fμδ ,GEN 2D −L

41

⎛

∞  n=1



μλ  Qn (T )|2 + μλnp p n

2

 δ 2 gn

⎞1/2 ⎠

(3.11)

 Lemma 3.3 indicates that there exists a unique solution for (3.10) if 0 < τ δ < gδ . Theorem 3.4. Let q(t) ∈ C[0, T] satisfying q(t) ≥ q0 > 0, t ∈ [0, T]. Suppose the a-priori condition (3.7) and the noise assumption (1.4) hold. The regularization parameter μ > 0 is chosen by Morozov’s discrepancy principle (3.10). Then we have the following convergence estimates

δ

fμ,GEN − f ≤ C6 δ p/( p+2) E 2/( p+2) ,

where C6 = 22/( p+2 ) (τ + 1 ) p/( p+2 ) (q0C )−p/p+2 δ Proof. Since fμ be the minimizer of J(f) defined by (3.4), we ,GEN can obtain



δ



K fμ,GEN − gδ 2 + μ fμδ ,GEN 2 p = J ( fμδ ,GEN ) D((−L ) 2 )

2 2 p ≤ J ( f ) = g − gδ + μ f  D((−L ) 2 )

42

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

Consequently, it has

 fμδ ,GEN 2D −L (



 g − gδ 2 − τ 2 δ 2 μ 1 2 2   ≤ E2 p p ≤  f + 1 −τ 2 δ2 <  f   D (−L ) 2 D (−L ) 2 μ  f 2D −L

≤

p

)2

(

p

)2

+ 1

 fμ,REV (x ) − f (x )

2

n=1

(

 K fμδ ,GEN − g 

p

)2

 fμδ ,GEN D −L

 ≤

(

p

)2

+

 f D −L (

p

)2

 ≤ 2E,

 K fμδ ,GEN − gδ  +  gδ − g ≤ (τ + 1 )δ.

≤

It follows from (3.8) that the assertion of this theorem is true.



4. A revised generalized Tikhonov regularization method From (3.6), we have

fμδ ,GEN (x ) = =

Qn ( T ) gδ Xn (x ) 2 + μλ p n | Q ( T ) | n n n=1 1 Qn ( T ) n 1 + |Qμλ 2 n ( T )| p

n=1

gδn Xn (x ).

(4.1)

Comparing (3.2) and (4.1), we note that the generalized Tikhonov regularization procedure is to supply some appropriate filter fac1 tors. By this idea, we can use a much better filter p+1 to replace the filter

1 μλnp 1+ |Qn ( T )|2

1+μλn

and propose a revised generalized

δ Tikhonov regularized solution fμ (x ) for noisy data gδ as ,REV

fμδ ,REV (x ) =

∞ 

1

n=1

Qn (T )(1 + μλnp+1 )

gδ X

n n

( x ).

(4.2)

p+2

λnp fn2

From (4.4) – (4.6), with μ = obtained. 

p+2  δ 2p+2

E

(4.6)

, the estimate (4.3) can be

As per discrepancy principle we choose the regularization parameter μ as the solution of the equation

δ

K fμ,REV (x ) − gδ = τ δ,

(4.7)

where τ > 1 is a constant.





δ Lemma 4.2. Let ρ (μ ) = K fμ (x ) − gδ (x ) . If 0 < δ < gδ , then ,REV the following results hold:

(a) ρ (μ) is a continuous function; (b) lim ρ (μ ) = 0;



lim ρ (μ ) = gδ ;

μ→ + ∞

(d) For μ ∈ (0, +∞ ), ρ (μ ) is a strictly increasing function. Proof. The above results are straightforward by setting

∞ 

1

n=1

Qn (T )(1 + μλnp+1 )



gn Xn (x ).

ρ (μ ) =



μλnp+1 δ gn 1 + μλnp+1

 2

1 2

.



, we have the convergence estimate

δ

p 2

fμ,REV (x ) − f (x ) ≤ D¯ E p+2 δ p+2 ,

(4.3)

where D¯ = C1 (q0C , p) + C2 ( p).

Theorem 4.3. Suppose that the a-priori condition (3.7) and the noise assumption (1.4) hold, and there exists τ > 1 such that 0 < τ δ < gδ . The regularization parameter μ > 0 is chosen by Morozov’s discrepancy principle (4.7). Then we have the following convergence estimate

δ

p 2

fμ,REV (x ) − f (x ) ≤ D˜ E p+2 δ p+2 ,

where D˜ = D(q0C , qC[0,T ] , p, τ ) + C4 (τ + 1 )

Proof. We know

(4.4)

From (1.4), Lemmas 2.6 and 2.7 , we have

=

∞ 

 ≤

n=1

 ≤

2



q0 C

λn

gδn − gn



1 1 + μλnp+1



2

 2 1 δ 2 C1 (q0C , p)μ− p+1

1 then,  fμδ ,REV (x ) − fμ,REV (x )  ≤ δC1 (q0C , p)μ− p+1 .

On the other hand, from Lemma 2.7, we get

 fμ,REV (x ) − f (x )  .

From (3.7), (4.7) and Lemma 2.7, there holds

1

 Qn (T ) 1 + μλnp+1 2

gδn − gn

∞ 

.

 fμδ ,REV (x ) − f (x )  ≤  fμδ ,REV (x ) − fμ,REV (x )  +

2



n=1

(4.8) p p+2

Proof. Similar to (4.4), we have

 fμδ ,REV (x ) − f (x )  ≤  fμδ ,REV (x ) − fμ,REV (x )  +  fμ,REV (x ) − f (x )  .  fμδ ,REV (x ) − fμ,REV (x )2

∞  n=1

Theorem 4.1. Suppose that the a-priori condition (3.7) and the noise assumption (1.4) hold. If we choose the regularization parameter μ = E



p

μ→ 0

4.1. Convergence estimates of revised generalized Tikhonov regularization method under a-priori parameter choice rules

 δ 2 p+2

∞ 

and  fμ,REV (x ) − f (x )  ≤ C2 ( p)E μ 2 p+2 .

(c)

and for exact data g as

fμ,REV (x ) =

2  2 −μλnp+1 1 p λn p+1 λnp 1 + μλn 2 p

gn Qn ( T )

4.2. Convergence estimates of revised generalized Tikhonov regularization method under a-posteriori parameter choice rules

∞ 

∞ 



μλn2 +1 1 + μλnp+1 n=1  2 p ≤ C2 ( p)μ 2 p+2 E =

This leads to

 fμδ ,GEN − f D −L

=

∞ 

(4.5)





∞ μλnp+1 δ



τδ =

g X x ( )

n

n=1 1 + μλnp+1 n







∞ μλnp+1  δ



≤

g − g X x ( )

n n n

n=1 1 + μλnp+1







∞ μλnp+1



+

g X (x )

p+1 n n

n=1 1 + μλn



p



+1 ∞ p

μλn2 Qn (T ) gn

2 λ X x (τ − 1 )δ ≤

( )

n n

n=1 1 + μλnp+1 Qn (T )

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

43

Table 1 The numerical results for different α with ε = 0.01.

p=1 p=2

α

0.05

0.1

0.3

0.7

0.9

0.95

e(fGEN , ε ) er (fGEN , ε ) e(fGEN , ε ) er (fGEN , ε )

0.0173 0.0267 0.0182 0.0280

0.0164 0.0276 0.0173 0.0292

0.0121 0.0291 0.0139 0.0336

0.0069 0.0329 0.0075 0.0356

0.0051 0.0336 0.0057 0.0373

0.0048 0.0338 0.0054 0.0384

Table 2 The numerical results for different ε with α = 0.6.

p=1 p=2

ε

0.0 0 05

0.001

0.002

0.004

0.008

0.016

0.032

0.064

e(fGEN , ε ) er (fGEN , ε ) e(fGEN , ε ) er (fGEN , ε )

0.0031 0.0127 0.0036 0.0147

0.0040 0.0163 0.0048 0.0195

0.0052 0.0208 0.0065 0.0261

0.0066 0.0264 0.0088 0.0356

0.0085 0.0341 0.0122 0.0489

0.0110 0.0444 0.0169 0.0681

0.0144 0.0578 0.0234 0.0942

0.0187 0.0754 0.0321 0.1291

Table 3 The numerical results for different α with ε = 0.01.

p=1 p=2

α

0.05

0.1

0.3

0.7

0.9

0.95

e(fREV , ε ) er (fREV , ε ) e(fREV , ε ) er (fREV , ε )

0.0347 0.0535 0.0168 0.0258

0.0331 0.0559 0.0162 0.0274

0.0233 0.0563 0.0135 0.0327

0.0126 0.0600 0.0084 0.0398

0.0119 0.0776 0.0077 0.0506

0.0114 0.0810 0.0075 0.0532

Table 4 The numerical results for different ε with α = 0.6.

p=1 p=2

ε

0.0 0 05

0.001

0.002

0.004

0.008

0.016

0.032

0.064

e(fREV , ε ) er (fREV , ε ) e(fREV , ε ) er (fREV , ε )

0.0019 0.0076 0.0026 0.0103

0.0027 0.0109 0.0038 0.0153

0.0040 0.0163 0.0043 0.0174

0.0059 0.0240 0.0058 0.0235

0.0097 0.0390 0.0081 0.0325

0.0141 0.0566 0.0108 0.0437

0.0211 0.0849 0.0144 0.0578

0.0326 0.1311 0.0194 0.0782



 p q



∞ p μλn2 +1 λCn[0,T ] 



2 ≤ sup λ f X x ( )

n n n

n

1 + μλnp+1 n=1  p+2 ≤ C3  qC [0,T ] , p E μ 2 p+2 .

This yields,

1

μ

 ≤

C3 (qC[0,T ] , p) (τ − 1 )

 2 p+2 p+2

 E  2 p+2 δ

p+2

.

From (4.11), (4.12) and the conditional stability (3.8), we deduce that



p p 2

fμ,REV (x ) − f (x ) ≤ C4 (τ + 1 ) p+2 E p+2 δ p+2 .

Combining (4.10) with (4.13), the convergence estimate can be established. 

(4.9) 5. Numerical example

Using (4.5) and (4.9), we obtain

δ

p 2

fμ,REV (x ) − fμ,REV (x ) ≤ D(q0C , qC[0,T ] , p, τ )E p+2 δ p+2 . (4.10) On the other hand





K fμ,REV (x ) − f (x ) = =

∞  n=1 ∞ 

 

n=1 ∞ 

+

−μλnp+1 1 + μλnp+1 −μλnp+1

gn Xn (x )



n=1

1 + μλnp+1

0 t

gδn Xn (x ).

2 ⎞ 12 ⎠

with the given data q(t)f(x) by a finite difference scheme [12]. In our computations, we consider d = 1 case. We choose  = (0, 1 ) and the grid sizes for time and space variables are t = NT 1 and x = M respectively. The grid points in the time interval [0, T] are labeled tn = nt, n = 0, 1, . . . , N; the grid points in the space interval [0, 1] are xi = ix, i = 0, 1, 2, . . . , M, and set uni = u(xi , tn ). The time-fractional derivative and the value of Lu are approximated by

(4.12)

(5.2)

Using (4.7), we get,



K ( fμ,REV (x ) − f (x )) ≤ δ + τ δ.

(4.11)

Applying the a-priori bounded condition (3.7) for f(x), we obtain

⎛  fμ,REV (x ) − f (x )D −L (

p

)2

 =

⎝

 ≤

∞  n=1

∞  n=1



p gn −μλnp+1  λ2 Qn (T ) 1 + μλnp+1 n

λnp fn2

After obtaining the theoretical results, we propose the numerical schemes for the inverse problem. In [19,20], the authors designed numerical scheme for forward time-fractional diffusion equation. As the analytic solution of problem (1.1) is difficult to derive, we construct the final data g(x) by solving the following forward problem

⎫ ∂ α u(x, t ) − (Lu )(x, t ) = q(t ) f (x ), x ∈ , t ∈ (0, T ),⎬ u(x, t ) = 0, x ∈ ∂ , t ∈ (0, T ), (5.1) ⎭ ¯, u(x, 0 ) = 0, x∈

 gn − gδn Xn (x ) p+1

1 + μλn −μλnp+1

(4.13)

12

≤ E.

  n−1  −α   t ( ) α un − w − wn− j uij − wn−1 u0i , 0 ∂t u (xi , tn ) ≈ (2 − α ) i j=1 n− j−1

44

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

Fig. 1. Exact and regularized solutions for ε = 0.001.

Lu(xi , tn ) ≈

where

( j )1−α

1

 a

1

(x )2 i+ 2 + c (xi )uni ,

uni+1 − (ai+ 1 + ai− 1 )uni + ai− 1 uni−1

i = 1, 2, . . . , M − 1,

2

2

2

(5.3) n = 1, 2, . . . , N,

and ai+ 1 = a(xi+ 1 ) with xi+ 1 = 2

2



2

xi +xi+1 . 2

w j = ( j + 1 )1−α −

Applying (5.2) and (5.3) in (5.1), we can get numerical solution to the direct problem. From this, we take g = uN as the exact final i data. The noisy data is generated by adding a random perturbation, that is, gδ = g + ε g(2 rand(size(g)) − 1 ). The corresponding noise level is calculated by δ = ε g.

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

45

Fig. 2. Exact and regularized solutions for ε = 0.05.

To compute Qn (T), first we establish,

   1   Qn ( T ) = − Eα ,1 (−λn (τ )α )q(T − τ )τ =T − q(T − τ )τ =0 λn   T α − Eα ,1 (−λn (τ ) )q(T − τ )dτ ,

In our numerical experiments, we use the a-posteriori regularization parameter choice rules (3.10) and (4.7) to find the regularization parameters of the two methods with τ = 1.1, we always fix T = 1, M = 50, N = 100. To test the accuracy of our methods, we compute the L2 error denoted by

and apply composite Simpson’s rule.

e( f• , ε ) =  f (x ) − fμδ ,• (x ),

0

(5.4)

46

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

Fig. 3. Exact and regularized solutions for ε = 0.001.

and the relative L2 error denoted by

er ( f • , ε ) =

 f (x ) − fμδ ,• (x ) .  f ( x )

(5.5)

Example 5.1. Let a(x ) = x2 + 1, c (x ) = −(x + 1 ). Take a source function q(t ) = e−t and f (x ) = (x(1 − x ))α sin(5π x ).

In Tables 1–2, we show the numerical results for generalized Tikhonov regularization method. The numerical results for revised generalized Tikhonov regularization method are given in Tables 3–4. For the two methods, the numerical errors are given for different α with p = 1, 2 and ε = 0.01 in Tables 1, 3 and for various ε with p = 1, 2, and α = 0.6, in Tables 2, 4. It can be seen that the numerical results depend on α and p. From Tables 2 and 4, it can

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

47

Fig. 4. Exact and regularized solutions for ε = 0.05.

be noted that the computational effect is satisfying and the error is decreasing as ε becomes smaller. Figs. 1 and 2 illustrate the exact source term and the regularized approximations given by generalized Tikhonov regularization method with p = 1, 2, 3 and ε = 0.001, 0.05 in case of α = 0.2, 0.8. Fig. 2 indicates that when p is increasing we can not get better computational results especially for large value of ε . Figs. 3 and 4 show the exact source term and the regularized approximations given by revised generalized Tikhonov regu-

larization method with p = 1, 2, 3 and ε = 0.001, 0.05 in case of α = 0.2, 0.8. In particular, when p is increasing we get better computational results for large value of ε . 6. Conclusion In this paper, we have solved an inverse source problem for a time-fractional diffusion equation with variable coefficients by using the generalized and revised generalized Tikhonov regularization methods. Based on the properties of Mittag–Leffler function

48

Y.-K. Ma et al. / Chaos, Solitons and Fractals 108 (2018) 39–48

and conditional stability we have derived convergence estimates for the two methods under a-priori and a-posteriori regularization parameter choice rules. Numerical example shows that the proposed methods are effective and stable. Acknowledgments This work was supported by University Grants Commission, New Delhi, India, Major Research Project 41-798/2012 (SR). The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No.2015R1C1A1A01054663). The second author was supported by the University Grants Commission, New Delhi, India under Special Assistance Programme (F.510/7/DRS-1/2016(SAP-I)). The authors would like to thank the referees for their valuable suggestions to improve the paper. References [1] Kirsch A. An introduction to the mathematical theory of inverse problem. New York.: Springer; 1999. [2] Yang F, Fu CL. A simplified Tikhonov regularization method for determining the heat source. Appl Math Model 2010;34:3286–99. [3] Qian A, Li Y. Optimal error bound and generalized Tikhonov regularization for identifying an unknown source in the heat equation. J Math Chem 2011;49:765–75. [4] Yang F, Fu CL. The revised generalized Tikhonov regularization for the inverse time-dependent heat source problem. J Appl Math Comput 2013;41:81–98. [5] Cheng J, Nakagawa J, Yamamoto M, Yamazaki T. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl 20 09;25:115 0 02. [6] Sakamoto K, Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math Anal Appl 2011;382:426–47.

[7] Jiang D, Li Z, Liu Y, Yamamoto M. Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations. Inverse Probl 2017;33:055 013. [8] Fujishiro K, Kian Y. Determination of time dependent factors of coefficients in fractional diffusion equations. Math Control Relat Fields 2016;6:251–69. [9] Wang JG, Zhou YB, Wei T. Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation. Appl Numer Math 2013;68:39–57. [10] Wang JG, Wei T. Quasi-reversibility method to identify a space-dependent source for the time-fractional diffusion equation. Appl Math Model 2015;39:6139–49. [11] Zhang Z, Wei T. Identifying an unknown source in time-fractional diffusion equation by a truncation method. Appl Math Comput 2013;219:5972–83. [12] Wei T, Wang JG. A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion problem. Appl Numer Math 2014;78:95–111. [13] Nguyen HT, Le DL, Nguyen VT. Regularized solution of an inverse source problem for a time fractional diffusion equation. Appl Math Model 2016;40:8244–64. [14] Tuan NH, Kirane M, Hoan LVC, Long LD. Identification and regularization for unknown source for a time-fractional diffusion equation. Comput Math Appl 2017;73:931–50. [15] Yang F, Fu CL, Li XX. Identifying an unknown source in space-fractional diffusion equation. Acta Math Sci 2014;34B:1012–24. [16] Zhang H, Zhang X. Generalized Tikhonov method for the final value problem of time-fractional diffusion equation. Int J Comput Math 2017;94:66–78. [17] Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations. New York.: Elsevier; 2006. [18] Podlubny I. Fractional differential equations, mathematics in science and engineering. New York.: Academic Press; 1999. [19] Gao GH, Sun ZZ. A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys 2011;230:586–95. [20] Sudha Priya G, Prakash P, Nieto JJ, Kayar Z. Higher order numerical scheme for fractional heat equation with Dirichlet and Neumann boundary conditions. Numer Heat Transfer B-Fundam 2013;63:540–59.