Comb model for the anomalous diffusion with dual

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Commun Nonlinear Sci Numer Simulat 63 (2018) 135–144

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Research paper

Comb model for the anomalous diffusion with dual-phase-lag constitutive relation Lin Liu a,∗, Liancun Zheng a, Yu Fan a, Yanping Chen a, Fawang Liu b a b

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia

a r t i c l e

i n f o

Article history: Received 31 January 2018 Revised 16 March 2018 Accepted 16 March 2018 Available online 17 March 2018 Keywords: Anomalous diffusion Constitutive equation Comb model Relaxation parameter

a b s t r a c t As a development of the Fick’s model, the dual-phase-lag constitutive relationship with macroscopic and microscopic relaxation characteristics is introduced to describe the anomalous diffusion in comb model. The Dirac delta function in the formulated governing equation represents the special spatial structure of comb model that the horizontal current only exists on the x axis. Solutions are obtained by analytical method with Laplace transform and Fourier transform. The dependence of concentration field and mean square displacement on different parameters are presented and discussed. Results show that the macroscopic and microscopic relaxation parameters have opposite effects on the particle distribution and mean square displacement. Furthermore, four significant results with constant 1/2 are concluded, namely the product of the particle number and the mean square displacement on the x axis equals to 1/2, the exponent of mean square displacement is 1/2 at the special case τ q = τ P , an asymptotic form of mean square displacement (MSD∼t1/2 as t→0, ∞) is obtained as well at the short time behavior and the long time behavior. © 2018 Published by Elsevier B.V.

1. Introduction Anomalous diffusion is a non-Markov and non-local movement with a fractional relationship x2 (t) ∼ tγ [1], where γ > 1 refers to the anomalous superdiffusion, γ < 1 denotes the anomalous subdiffusion while γ = 1 corresponds to the normal diffusion. The diffusion in comb model is a special case of anomalous diffusion. As Fig. 1 shows, the comb model consists of a one-dimensional backbone with fingers of infinite length. The special behavior [2–3] is that the displacement in the x direction is possible only along the x axis with diffusion coefficient D1 δ (y) while the diffusion along the y direction plays a role of traps with traditional diffusion coefficient D0 , here δ (y) refers to the Dirac delta function, D1 and D0 are nonnegative constant. It has attracted considerable attention due to its simplicity and ability to reproduce subdiffusive behaviors of disordered systems [4–5], such as the fractional transport of cancer cells due to self-entrapping [6], the reaction-subdiffusion front propagation in spiny dendrites [7], and the reaction front propagation of actin polymerization [8]. As the basis to study the diffusion process, the constitutive model refers to the relationship between the diffusion flux and concentration gradient. The classical constitutive relation [9–10] to describe the anomalous diffusion in comb model is



Corresponding author. E-mail address: [email protected] (L. Liu).

https://doi.org/10.1016/j.cnsns.2018.03.014 1007-5704/© 2018 Published by Elsevier B.V.

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L. Liu et al. / Commun Nonlinear Sci Numer Simulat 63 (2018) 135–144

Fig. 1. Schematic drawing of the comb model.

based on the Fick’s law with a linear form, given as:

− → J =



 ∂P ∂P −D1 δ (y ) , −D0 , ∂x ∂y

(1)

− → where the symbol J is the diffusion flux vector, P = P (x, y, t ) refers to the distribution function at the special positions (x, y) and time t. Here, x denotes the direction along the backbone of the comb structure while y measures the distance along the fingers away from the backbone. As is well known, the classical Fick’s model contains paradox with infinite propagation velocity [11–12]. In order to overcome this paradox, the macroscopic relaxation parameter is introduced by Cattaneo [13] to modify the classical model. As a development of the Fick’s model, the Cattaneo model has widespread applications in practical problem, such as the reaction-diffusion problem in a disordered porous medium [14], the non-isothermal incompressible fluid flow involving heat transfer and friction [15], the laser short-pulse heating of a solid surface [16]. However, the Cattaneo model only takes account the fast-transient effects. For a complete diffusion process, it should consider not only the macroscopic description but also the interactions between the microscopic structures [17]. As a generalization of the Cattaneo model, the dual-phase-lag model is proposed by Tzou [18–19] which contains both the macroscopic relaxation parameter and the microscopic one. It describes the actual problem more accurately and attracts many scholars to study. Among the researchers, Liu [20] analyzed the dual-phase-lag thermal behavior in two-layered thin films with an interface thermal resistance, concluding that the lagging thermal behavior depends on the magnitude of the macroscopic and microscopic relaxation parameters. Using the dual-phase-lag model, Ghazanfarian and Abbassi [21] simulated microand nano-scale heat conduction within a thin slab for Knudsen numbers more than 0.1 including phonon scattering boundary condition by the numerical and analytical methods. Abouelregal [22] investigated rayleigh waves in a thermoelastic solid half space using dual-phase-lag model by numerical method and the effect of the coupling parameter and phase-lags was showed graphically. More references related to the studies of the dual-phase-lag model can be seen in Refs. [23–25]. In actual, the study on the anomalous diffusion in comb model with modified constitutive equations is a promising direction and a lot of modified constitution relationships have been analyzed and discussed. Iomin [26] proposed the time fractional diffusion flux along the fingers to describe the anomalous diffusion in comb model. Tateishi [27–28] analyzed the anomalous diffusion in comb model by changing the integer derivative operator in Fick’s model as the fractional one. Liu et al. [29–31] analyzed the anomalous diffusion in comb model by adopting the fractional Cattaneo model and the fractional Cattaneo–Christov model. Méndez et al. [32–33] displayed a variety of macroscopic transport regimes and formulated the mesoscopic description of the random walk on the comb. Inspired by the studies mentioned above, in this paper, the dual-phase-lag model containing the macroscopic and microscopic relaxation parameters is introduced to study the anomalous diffusion in comb model. The governing equation with the highest order of two is solved by analytical method where the Laplace and Fourier transforms are applied. The product of the particle number and the mean square displacement on the x axis is computed. Besides, three special cases about the mean square displacement, such as the case of the equal macroscopic and microscopic relaxation parameters, the limit cases as time tends to zero and infinite are analyzed. In order to verify the correctness of analytical solution, the comparisons with the degenerated solutions are presented. The particle distribution and the mean square displacement with the influences of different parameters are discussed and the main results and conclusions are also analyzed.

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2. Mathematical formulation Based on Eq. (1), the dual-phase-lag model containing the macroscopic and microscopic relaxation parameters is given as:

  − →  − → ∂ J ∂ ∂P ∂P J + τq = 1 + τP −D1 δ (y ) , −D0 , ∂t ∂t ∂x ∂y

(2)

where τ q denotes the macroscopic relaxation parameter while τ P refers to the microscopic one. The energy conservation equation is:

− → ∂P +∇ · J =0 ∂t

(3)

Combining Eqs. (2) and (3), we obtain:

τq

∂ 2P ∂ P ∂ 2P ∂ 3P ∂ 2P ∂ 3P + = D 1 δ ( y ) 2 + D 1 τP δ ( y ) + D 0 2 + D 0 τP 2 2 ∂t ∂t ∂x ∂t∂ x ∂y ∂ t ∂ y2

(4)

The initial conditions and the boundary conditions are given as:

P (x, y, 0 ) = δ (x )δ (y ),

∂ P (x, y, 0) =0 ∂t

(5)

and

P (±∞, y, t ) = P (x, ±∞, t ) = 0,

(6)

respectively. The dimensionless quantities are introduced:

t→

D21 D30

t ∗, x →

D1 ∗ D1 ∗ x ,y→ y , D0 D0

τq →

D21 D30

τq∗ , τP →

D21 D30

τP∗ , P →

D20 D21

P∗

(7)

Then the governing equation with the initial and boundary conditions can be rewritten in the dimensionless forms (For simplicity, the superscript ∗ is omitted):

∂ 2P ∂ P ∂ 2P ∂ 3P ∂ 2P ∂ 3P + = δ ( y ) 2 + τP δ ( y ) + + τP , 2 2 2 ∂t ∂t ∂x ∂t∂ x ∂y ∂ t ∂ y2 ∂ P (x, y, 0) P (x, y, 0 ) = δ (x )δ (y ), = 0, ∂t τq

P (±∞, y, t ) = P (x, ±∞, t ) = 0.

(8) (9) (10)

The newly formulated governing Eq. (8) is a novel one with the highest order of two and Dirac delta function. It is a development and generalization of the classical diffusion equation. For τP = 0, the anomalous diffusion in the comb model based on the Cattaneo model [29] is obtained. For τq = 0 and τP = 0, it reduces to the transport in comb model based on the classical Fick’s model [6]. 3. Solutions of the governing equation by analytical method The solutions to Eq. (8) subject to the initial conditions (9) and boundary conditions (10) are obtained by analytical method with the Laplace transform and Fourier transform. The Laplace transform for the time derivative with respect to t [34] is given as:



 n−1  ∂ n P (x, y, t ) (i ) L ; s = sn P (x, y, s ) − sn−i−1 P (x, y, 0 ), n ∂t i=0

(11)

where n ∈ N, s is the Laplace variable and the one hat denotes the Laplace transform function. The Fourier transform for the space derivative versus x is given as [34]:



F

 ∂ 2 P (x, y, t ) ; k = −k2 P (k, y, t ) ∂ x2

where k is the Fourier variable and the two hats denote the Fourier transform function. By performing the Laplace transform for the time derivative in Eq. (8), yields:

(12)

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L. Liu et al. / Commun Nonlinear Sci Numer Simulat 63 (2018) 135–144

∂ 2 P¯ ∂ 2 P¯ − s τP δ ( y ) 2 ∂ x2 ∂x

s2 τq P¯ − sτq δ (x )δ (y ) + sP¯ − δ (x )δ (y ) − δ (y ) −

∂ 2 P¯ ∂ 2 P¯ − s τP 2 = 0 ∂ y2 ∂y

(13)

Performing the Fourier transform for the space derivative with respect to x in Eq. (13), we have:

s2 τq P¯¯ − sτq δ (y ) + sP¯¯ − δ (y ) + δ (y )k2 P¯¯ + k2 sτP δ (y )P¯¯ −

∂ 2 P¯¯ ∂ 2 P¯¯ − s τP 2 = 0 2 ∂y ∂y

(14)

The separate variable method is applied to get the exact solution in (x, y, t) space. Firstly, define P¯¯ (k, y, s ) = h(s, k )e−λ|y| [35], then the derivative of P¯¯ (k, y, s ) is given as:

  ∂ 2 P¯¯ (k, y, s) = h(s, k ) λ2 − 2λδ (y ) e−λ|y| 2 ∂y

(15)

By substituting the derivative item (15) into Eq. (14), the equation changes as:

s2 τq h(s, k )e−λ|y| − sτq δ (y ) + sh(s, k )e−λ|y| − δ (y ) + δ (y )k2 h(s, k )e−λ|y|



+k2 sτP δ (y )h(s, k )e−λ|y| − (1 + sτP )h(s, k )

 λ2 − 2λδ (y ) e−λ|y| = 0

(16)

By applying the characteristic of the Dirac delta function, the above equation can be separated into the following two equations:

k2 h(s, k )e−λ|y| δ (y ) + 2λ(1 + sτP )h(s, k )e−λ|y| δ (y ) + k2 sτP δ (y )h(s, k )e−λ|y| = δ (y ) + sτq δ (y )

(17)

s2 τq h(s, k )e−λ|y| + sh(s, k )e−λ|y| − λ2 (1 + sτP )h(s, k )e−λ|y| = 0

(18)

and

The solutions to λ and h(s, k) are obtained:



λ=

s 2 τq + s 1 + s τP

(19)

and

s τq + 1

h(s, k ) = k2

( 1 + s τP ) + 2



(20)



s 2 τq + s ( 1 + s τP )

respectively. Then the solution of P¯¯ (k, y, s ) is given as:

 2  s τq + s  − |y|

s τq + 1

P¯¯ (k, y, s ) = k2

( s τP + 1 ) + 2



s2



τq + s ( 1 + s τP )

e

1 + s τP

 2  s τq + s  − |y |

=

1 + s τP

(sτq + 1 )e ( s τP + 1 )



−2

+∞

e 0

 2  s τq + s  τ −k2 τ

1 + s τP



(21) Performing the inverse Fourier transform [36] with respect to k in Eq. (21), we have:

 2  s τq + s  − |y |

(sτq + 1 )e 1 + sτP P¯ (x, y, s ) = ( s τP + 1 )

 2  s τq + s  − |y |

=

(sτq + 1 )e 1 + sτP √ 2 π ( s τP + 1 )

 2  s τq + s  +∞ −2 τ

1 + s τP

e 0



+∞ 0

1 √ e

τ





2 F −1 e−k τ dτ

 2  s τq + s x2  −2 τ−

1 + s τP



where the symbol F −1 refers to the inverse Fourier transform operator.



(22)

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By performing the inverse Laplace transform [36] with respect to s in Eq. (22) and expanding the exponential function in Taylor series, the equation changes as:



P (x, y, t ) =

 +∞

1 √

2 π

0



 s 2 τq + s  x2 − ( 2 τ + |y | ) ⎢ − 1 + s τP s τ + 1 e e 4τ −1 ⎢ ( ) q √ L ⎢ ⎢ ( s τP + 1 ) τ





⎥ ⎥ ⎥dτ ⎥ ⎦ (23)



x n ⎤ − n  ∞ sτq + 1 2 +1 e 4τ  1 ⎦d τ √ (−1 )n (2τ + |y|)n L−1 ⎣s 2 s τP + 1 τ n=0 n! 2

=

2

1 √

π

 +∞ 0

where the symbol L−1 refers to the inverse Laplace transform operator. The particle distribution on the x axis is given as:

P (x, 0, t ) =

2



1 √

π

+∞ 0





x  sτ + 1  2 +1 ∞ e− 4τ  1 n q dτ √ (−1 )n (2τ )n L−1 s 2 n! s τP + 1 τ 2

n

(24)

n=0

By integrating with respect to x of the distribution function P(x, 0, t) on the interval[−∞, +∞], the total number of particles on the x axis is given as:



P =



+∞

−∞

P (x, 0, t )dx =



+∞

2

−∞



1 √

π



 s 2 τq + s  x2 −2τ ⎢ − s τP + 1 ⎢ 4 τ e (sτq + 1)e √ L−1 ⎢ ⎢ ( s τP + 1 ) τ

+∞



0



  ⎥ −1 ⎥ τ + s 1 q −1 ⎥dτ dx = L ⎥ 2 s τP + 1 ⎦

(25)

Prior to giving the expression of mean square displacement, we first calculate the following equation:





x2 P =

=

+∞

−∞

x2 P (t, x, 0 )dx =



+∞

2

−∞

1 √





π

+∞

2

− 4x τ

e x2 √

τ

0

L−1 ⎣

−2τ



(sτq + 1)e ( s τP + 1 )

s 2 τq + s 1 1+sτP



  ⎦dτ dx = 1 L−1 (sτq + 1) 2 s 2 τq + s

! "

1 −1 1 1 L = 2 s 2

(26)

Since the total number of particles on the x axis is variable, the expression for the mean square displacement [37–38] is given as:





x (t ) = 2

x2 P

P

1



= L−1

τq +s−1 sτP +1



(27)

For the expression of mean square displacement mentioned above, we can obtain four same constants 1/2 through the following cases. 3.1. Case 1 Through Eqs. (25)–(27), the first novel result for the anomalous diffusion in comb model is found that the product of the total number of particles and the mean square displacement on the x axis is equal to 1/2, namely,x2 (t ) × P  = 1/2. 3.2. Case 2 For a special case τq = τP , Eq. (27) reduces as:



x2 (t ) = L−1



1 sτ +1 s−1 sτPq +1

=

L−1



1 s−1/2



= π t 1/2

(28)

Form the above expression, the second novel result with exponent 1/2 of mean square displacement is found that the magnitude of the mean square displacement has nothing to do with the values of τ q and τ P when τq = τP . Besides, at the condition τq = τP , the expression of the mean square displacement for the anomalous diffusion in comb model with dualphase-lag model is same with the one for the classical Fick’s model [6] and all the processes correspond to a subdiffusion.

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3.3. Case 3 as t → 0 Defining f (t ) = L−1

f¯(s ) =



τq +s−1 sτP +1

 , we have:

τq + s−1 s τq + 1 −1/2 =s , s τP + 1 s τP + 1

(29)

where f¯(s ) refers to the Laplace transform of the function f(t). Prior to obtaining the limit case as t → 0, we first give three important equations [39]:

L[t f (t )] = −

d ¯ f (s ) ds

(30)

lim [sg¯ (s )] = lim g(t ) = g(0 )

(31)

lim [sg¯ (s )] = lim g(t ) = g(0 )

(32)

s→∞

t→0

s→0

t→∞

Introduce a new defined function g(t ) = t f (t ) and define L[g(t )] = g¯ (s ), then we obtain:

lim g(t ) = lim sg¯ (s ) = g(0 ) = 0

(33)

s→∞

t→0

and

g¯ (s ) = L[g(t )] = L[t f (t )] = −

d ¯ f (s ) ds

(34)

For the derivative of the function f¯ (s ), we have:





d f¯(s ) −3/2 lim /s =A ds

(35)

#

where A = −1/2 as s tends to zero while A = − τq /τP /2 as s tends to infinite. Then we can obtain g¯ (s ) = − d fds(s ) ∼ s−3/2 . According to the property (31) and adopting the definition for the Laplace transform of the Riemann–Liouville fractional integral [34], the limit of Dt−1/2 g(t ) as t → 0 is given as:



¯



lim Dt−1/2 g(t ) = lim sL Dt−1/2 g(t ) = lim ss−1/2 g¯ (s ) = lim s−1 = 0 s→∞

t→0

s→∞

(36)

s→∞

where Dt−1/2 refers to the Riemann–Liouville fractional integral operator. Exploiting the definition of Riemann–Liouville fractional derivative, property (31) and property (36), for the limit case t → 0, we have:







$

lim Dt1/2 g(t ) = lim sL Dt1/2 g(t ) = lim s s1/2 g¯ (s ) − Dt−1/2 g(t )$

#

t→0

=

τq / τP / 2

s→∞

s→∞

t=0



= lim ss1/2 g¯ (s ) s→∞

(37)

where Dt1/2 refers to the Riemann–Liouville fractional derivative operator. Using the definition of limitation, for any ɛ > 0, there exists a constant δ > 0, when 0 < t < δ , we have:

#

τq /τP /2 − ε < Dt1/2 g(t )
0, there exists M > 0, when t > M, we have:

1/2 − ε < Dt1/2 g(t ) < 1/2 + ε

(42)

By multiplying every items of the above inequality by Dt−1/2 , the above equation changes as:

( 1/2 − ε )

t −1/2 g(t ) t −1/2 < = f (t ) < (1/2 + ε ) (3/2 ) t (3/2)

(43)

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L. Liu et al. / Commun Nonlinear Sci Numer Simulat 63 (2018) 135–144

Fig. 4. The temporal evolution of mean square displacement on the x axis with the effects of different τ q when τP = 1.

Fig. 5. The temporal evolution of mean square displacement on the x axis with the effects of different τ P when τq = 1.

Then we have f (t ) ∼



x2 (t ) =

√1

π

t −1/2 when t > M. The expression for the mean square displacement is given as:

√ 1 ∼ π t 1/2 , t > M. f (t )

(44)

For the anomalous diffusion with Cattaneo model [11], the superdiffusion happens at the short time behavior while the subdiffusion happens at the long time behavior. Through the above analyses of case 3 and case 4 to study the limit cases of the anomalous diffusion in comb model with dual-phase-lag model, a novel and different result is found that it corresponds to the subdiffusion process with exponent 1/2 at both the short time behavior (0 < t < δ ) and the long time behavior (t > M). On the other hand, the exponent 1/2 with dual-phase-lag model is in agreement with the anomalous diffusion in comb model with classical Fick’s model [6]. However, more remarkable, the difference is that the coefficient A is variable.

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4. Comparison between the analytical solution and the previous results In order to verify the correctness of the analytical solution, the comparisons between the analytical solutions and the degenerate solutions are presented. For τP = 0, the expression for the distribution of mean square displacement on the x axis reduces as:





x2 (t ) =

x2 P

P

=

L−1

#

1

τq + s−1



(45)

which is in accordance with the result in Ref. [6]. For τq = τP = 0, the expression for the mean square displacement on the x axis reduces as:





x (t ) = 2

x2 P

P

=

1

L−1



s−1



= π t 1/2

(46)

which is similar with the result in Ref. [40]. Through the above discussions, we can verify the accuracy of computational results. 5. Results and discussion In this section, we mainly discuss and analyze the dynamic characteristics of the particle distribution and the mean square displacement with the effects of different involved parameters by graphically illustrations. The influences of macroscopic relaxation parameter τ q on the particle distribution are shown in Fig. 2. For τ q ≤ τ P , the distribution presents as a diffusion form and the position of the peak declines gradually with the increase of τ q . For τ q > τ P , the distribution presents as a wave form and the larger the τ q is, the stronger the wave characteristic will be. The reason is that the governing Eq. (8) possesses the wave characteristic since the highest order of governing equation is two. As Eq. (8) shows, the parameter τ q is added in the time item with the highest order of two which plays a role of wave characteristic while the parameter τ P is added in the diffusion item which plays a role of diffusion characteristic. For a fixed τ P , the particles transport faster and the wave characteristic becomes stronger with the increase of τ q . Besides, it should be noted that the distribution has a sharp peak for the condition τ q = τ P which is consistent with the reality that the bigger concentration difference makes the particles transport faster. The spatial evolution of the particle distribution with the influences of microscopic relaxation parameter τ P is depicted in Fig. 3. On the contrary, the peak of the distribution becomes higher and the distribution changes from wave form to diffusion one gradually with the increase of parameter τ P . The larger the τ P is, the slower the particles will transport and the weaker the wave characteristic will be. The graphical representations for the effects of macroscopic relaxation parameter on the mean square displacement versus t are shown in Fig. 4. For a larger τ q , it can be seen from the figure that the magnitude of the distribution is smaller at the short time behavior but the distribution increases rapidly. At the long time behavior, the magnitude of the distribution is larger for a larger τ q . As Fig. 5 shows, the larger the τ P is, the larger the magnitude of the distribution at the short time behavior while the smaller the magnitude at the long time behavior will be. Similar with the effects of macroscopic and microscopic relaxation parameters on the particle distributions as shown in Figs. 3 and 4, the effects of the macroscopic relaxation parameter and the microscopic one on the mean square displacement are also opposite. 6. Conclusions The dual-phase-lag model is introduced to describe the anomalous diffusion in comb model. The effects of the involved parameters on the particle distribution and the mean square displacement are shown and discussed graphically which are obtained by the analytical method with the Laplace and the Fourier transforms. It can be deduced that the influences of the macroscopic relaxation parameter and the microscopic one on the particle distribution and the mean square displacement are opposite. Results show that the particles transport faster and the wave characteristic becomes stronger for a larger macroscopic relaxation parameter or a smaller microscopic one. The magnitude of mean square displacement is larger at the short time behavior while smaller at the long time behavior for a smaller macroscopic relaxation parameter or a larger microscopic relaxation parameter. Most important of all, four same constants are obtained. Namely, the product of the total number of particles on the x axis and the mean square displacement equals to 1/2, the power exponent of mean square displacement equals to 1/2 when τq = τP . Besides, for the limit cases as time tends to zero and infinite, the power exponent of mean square displacement is constant 1/2 indicating a subdiffusion process which is similar with the classical diffusion in comb model while different with the diffusion process with Cattaneo model. Further researches with the anomalous diffusion in comb model with fractional constitution relationship need to be presented and analyzed. Acknowledgments The work is supported by the Project funded by China Postdoctoral Science Foundation (No. 2017M620602), the Fundamental Research Funds for the Central Universities (FRF-TP-17-067A1) and the National Natural Science Foundation s of China (Nos. 51406008, 11772046).

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