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New methodologies in fractional and fractal derivatives modeling
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
New methodologies in fractional and fractal derivatives modelingR Wen Chen∗, Yingjie Liang∗ State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, China
a r t i c l e
i n f o
Article history: Received 28 January 2017 Revised 22 March 2017 Accepted 27 March 2017 Available online xxx Keywords: Structural derivative Structural function Structural metric Structural fractal Implicit calculus equation modeling Ultraslow diffusion Inverse Mittag–Leffler function
a b s t r a c t This paper surveys the latest advances of the first author’s group on the three new methodologies of fractional and fractal derivatives modeling to meet the increasing and challenging demands in scientific and engineering communities. Firstly, the structural fractal was proposed as a generalization of the Euclidean distance. Using the structural metric, the structural derivative approach was derived as a significant extension of the global fractional calculus and the local fractional derivative approaches to tackle the perplexing modeling problems. The classical derivative describes the change rate of a certain physical variable with respect to time or space, which rarely takes into account the significant influence of mesoscopic time-space metric of a complex system on its physical behaviors. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space structural metric of physical systems. Secondly, we employed the fundamental solution or probability density function of statistical distribution which can describe the problem of interest to construct the implicit calculus governing equation. The ‘implicit’ suggests that the explicit calculus expression of this governing equation is difficult to derive and not required. The fundamental solution or potential function of calculus governing equation and corresponding boundary conditions are sufficient to do numerical simulation. We call this strategy the implicit calculus equation modeling. Thirdly, based on the implicit calculus equation modeling approach, we introduced the concept of fundamental solution on fractal and consequently defined the fractal differential operator to describe various mechanical behaviors of fractal materials. Fractal calculus operator significantly extends the application scope of the classical calculus modeling approach under the framework of continuum mechanics. This is also a step-forward advance of the fractal derivative proposed earlier by the first author. To demonstrate the structural derivative application, we applied the inverse Mittag-Leffler function as the structural function to model ultraslow diffusion of a random system of two interacting particles. On the other hand, this paper uses the fractional Riesz potential as the fundamental solution to establish the implicit calculus equation of fractional Laplacian modeling the power law behaviors of steady heat conduction in multiple phase material. Finally, by using the singular boundary method, we made numerical simulation of the fractal Laplacian equation for phenomenological modeling potential problems in fractal media. Numerical experiments show that all the three new methodologies are feasible mathematical tools to describe complex physical behaviors. © 2017 Published by Elsevier Ltd.
1. Introduction Recent decades have witnessed a growing number of complex scientific and engineering problems which are not easy to be described by the classical calculus modeling methodology [1–3]. To remedy this troublesome problem, a variety of phenomenological partial differential equation models including multiple empirical R This paper is a revised version of the report presented in the International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, July 18– 20, 2016. The physical mechanism of the structural derivative and more structural functions are added in this paper. ∗ Corresponding authors. E-mail addresses: [email protected] (W. Chen), [email protected] (Y. Liang).
parameters have been proposed in theoretical research and engineering practice [4,5]. In some cases, statistical models are used instead of calculus models [6,7]. These models are not clearly interpreted in physics and require more parameters in which the artificial parameters have no physical significance. The fractional calculus [8] and the fractal derivative [9] have been found in numerous studies as alternative competitive methodologies to tackle such modeling bottleneck. However, the fractional calculus and fractal derivative simply do not work for ultraslow diffusion [10]. To remedy this challenging problem, the structural derivative modeling approach [10,11] is derived based on the structural metric. The structural function, which is substantially a time-space transform, characterizes the
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Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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time-space structures [12] of the system of interest. Based on this, the causal relationship between mesoscopic time-space structural metric and certain physical behaviors can be described using the structural derivative with fewer parameters and lower computational costs. The structural function can be obtained using the fundamental solution or the probability density function. As an example, the inverse Mittag–Leffler (ML) function [13] was used as the structural function in the structural derivative to model the dynamics of two interacting particles in a disordered chain [14]. It is noted that the structural derivative can be either a global integraldifferential operator or a local fractal derivative fashion. To circumvent computationally expensive singularity in the classical definition of fractional derivative, Caputo and Fabrizio [15] recently proposed to replace the standard power function with an exponential kernel. This exponential function can be understood as a structural function from a pure mathematical point of view. But in their study, the exponential function has no physical significance and is aimed at simplifying numerical computation by remedying singularity of the kernel function in fractional derivative. This is different from the present structural derivative methodology, in which the structural function characterizes the influence of the time-space structure on physics behaviors. It is worthy of noting that in some cases the explicit calculus expression of governing equation to the physical problems is difficult to derive. Thus, the implicit calculus equation modeling is proposed to tackle this issue, in which the fundamental solution [16] or the probability density function [11] is used to construct the implicit calculus governing equation [17,18]. It is not required to know the explicit expression of the governing equation. Based on the fundamental solution and the corresponding boundary conditions, the numerical solution of the problems of interest can directly and easily be computed by a boundary-type numerical scheme [19]. The implicit calculus modeling can also be used to construct the structural derivative [11]. By using the fractional Riesz potential as the fundamental solution, the implicit calculus equation of fractional Laplacian was established to model the power law behaviors of steady heat conduction in multiple phase materials [18]. The first author of this paper introduced the fractal derivative [9] as an alternative approach to model complex problems. The fractal derivative in time has been successfully used in many applications [3,9,20], but the fractal derivative in space attracts much less attention. Based on the proposed implicit calculus modeling approach, the concept of fundamental solution on fractal and the fractal differential operator [21] are introduced to describe various mechanical behaviors of fractal materials. The fractal calculus operator characterizes the spatial geometric feature of the fractal materials, and significantly extends the application scope of the classical calculus model under the framework of continuum mechanics. By using the singular boundary method [19], the fractal Laplacian equation for phenomenological modeling the potential problems in fractal media was successfully simulated [21]. The rest of this paper is organized as follows. In Section 2, the structural derivative approach, the implicit calculus equation modeling, and the definition of the fractal differential operator are introduced. In Section 3, some applications of the three recent methodologies are tested to some problems, in particular the ultraslow diffusion. Finally, we draw some conclusions in Section 4. 2. Methodologies 2.1. Structural derivative The Hausdorff dimension in space is described by
M = |x − x |d ,
(1)
where r represents the Euclidean distance, d is the fractal dimension. The power law in Eq. (1) has since found many applications in science and engineering. However, it is observed that many real problems cannot well be characterized by the classical fractal concept [12]. Thus, this paper generalizes the Euclidean metric Eq. (1) to the structural metric as
M = |R ( x ) − R ( x )|d ,
(2)
where R is the structural function and can be an arbitrary function. The formula (2) characterizes the structural fractal, which includes the Euclidean and the non-Euclidean metrics. To illustrate the above derivative on structural fractal without a loss of generality, we consider the displacement x under power function time metric
x = vt α ,
(3)
where α is the index of time fractal. The corresponding derivative of the displacement in Eq. (3) is obtained as,
dx = vd (t α ),
(4)
then Eq. (4) can be rewritten as,
v=
dx . d (t α )
(5)
Consequently, the Hausdorff derivative on time fractal α is given by
dp(x, t ) p(x, t1 ) − p(x, t ) = lim . t1 →t dt t1α − t α
(6)
The fractal derivatives on space fractal metric are also proposed in Refs. [9,22–25]. By using the similar strategy, the displacement x under general structural metric can be derived by
x = vk(t ),
(7)
where k(t) is the structural function. The derivative of the displacement in Eq. (7) is written as
dx = vdk(t ).
(8)
Then Eq. (8) can be rewritten as
v=
dx . dk(t )
(9)
The definition of structural derivative by using the structural metric further extends the fractal derivative concept. On time structural metric, the local structural derivative is defined as [7]
dp(x, t ) p(x, t1 ) − p(x, t ) = lim , t1 →t ds t k(t1 ) − k(t )
(10)
where S denotes the structural derivative, k(t) is the structural function, ds means the structural derivative. When k(t ) = t α , it is the fractal derivative. In contrast, the global structural derivative in time is defined as [26]
δ p(x, t ) ∂ = δst ∂t
t
t1
k(t − τ ) p(x, τ )dτ .
(11)
−α
When k(t ) = (t1−α ) , Eq. (11) degenerates into the classical Riemann-Liouville fractional derivative. Fig. 1 shows the decay of four different structural functions including the exponential, power law, logarithmic and inverse ML functions. It is stressed that the structural function is not necessary a power function. Instead it can be any form of a function, such as the inverse Mittag–Leffler function, the probability density function, and the stretched exponential function [9]. Compared with the classical nonlinear calculus models, the structural derivative requires fewer parameters and lower computational costs in detecting the causal relationship between mesoscopic time-space structure and certain physical behaviors [11].
Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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3
1 0.9
Exponential function Power law function Logarithmic fucntion Inverse Mittag-Leffler function
0.8
Structural function
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10 t
12
14
16
18
20
Fig. 1. Decay of four different structural functions including the exponential, power law, logarithmic and inverse Mittag–Leffler functions from bottom to top.
2.2. Implicit calculus equation modeling
The time-dependent diffusion equation on fractal has the fundamental solution:
Chen and Pang [17,18] proposed the implicit calculus equation modeling approach, which does not require the explicit expression of the governing equation. To describe the physical problems, the fundamental solution, potential function, or the probability density function is used to construct the implicit calculus governing equation. By using the fundamental solution or the probability density function with the corresponding boundary conditions, the numerical solution of the problems of interest can directly and easily be computed by a boundary-type numerical scheme [19].
u∗d (r ) =
H (t )
(4π Dt )d/2
e−r
2
/4Dt
,
(15)
where D is the diffusion coefficient, t the time interval, H(t) the Heaviside function. Then the implicit calculus equation modeling approach can be used to investigate the mechanical behaviors of fractal materials in terms of the above fundamental solutions. 3. Applications 3.1. Structural derivative modeling ultraslow diffusion
2.3. Differential operator on fractal Based on the implicit calculus equation modeling approach, Chen, Wang and Yang [21] defined the differential operator on fractal and consequently gave the fundamental solutions of four classical calculus equations on fractal, which are listed as below. The fundamental solution of the Laplacian operator on fractal is
u∗d (r ) =
1
( d − 2 )Sd ( 1 )
r 2−d ,
(12)
where Sd (1) =2π d /2 / (d/2), r represents the Euclidean distance, d is the fractal dimension. The fundamental solutions of the Helmholtz and the modified Helmholtz equations on fractal are constructed as
u∗d
1 (r ) = 2π
−ik 2π r
(d/2)−1 K(d/2)−1 (−ikr ),
(13)
and
u∗d (r ) =
1 2π
k 2π r
(d/2)−1 K(d/2)−1 (kr ),
(14)
where K( d /2)−1 is the modified Bessel function of the second kind, k the constant, i2 = −1.
In this section, the inverse Mittag–Leffler (ML) function was used as the structural function in the structural derivative to model ultraslow diffusion of two interacting particles in a disordered chain [14]. The case is at long time t2 ≤ 105 with the localization length L1 = 36, and the length of chain L = 1024. The experimental data on the center of mass R(t) are used, which are assumed as the square root of the mean squared displacement (MSD) from the diffusion point of view, i.e., R(t ) = (< x2 (t ) > )1/2 [14]. In the structural derivative modeling ultraslow diffusion, the propagator p(x,t) satisfies the following diffusion equation.
dp(x, t ) ∂ 2 p(x, t ) = Dα , ds t ∂ x2
(16)
where Da is the generalized diffusion coefficient (m2 /sa ), and the structural derivative is defined by
dp(x, t ) p(x, t1 ) − p(x, t ) = lim −1 , −1 t1 →t E ds t α (t1 ) − Eα (t )
(17)
where Eα−1 (t ) is the inverse function of the single-parameter ML function with 0 < a ≤ 1. Eq. (16) can be restated as a normal diffusion equation, and yields a Gaussian distribution.
p(x, tˆ) =
1 4π Dα tˆ
exp −
x2 4Dα tˆ
,
(18)
Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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120
100
R(t)
80
60
Experimental data Inverse ML diffusion law Logarithmic diffusion law
40
D = 0.5, b/2 = 2.0 20
0
0
0.5
1
1.5
2
2.5 t2
3
3.5
4
4.5
5 x 10
4
Fig. 2. Plots of centers of mass R(t) with fits using the root square of the inverse Mittag–Leffler and logarithmic diffusion laws (This figure is from [10]).
where tˆ = Eα−1 (t ). The MSD of the ultraslow diffusion particle x(t) can be derived from Eq. (16) as
x2 (t ) = 2Dα Eα−1 (t ).
(19)
It is noted that in Eq. (19), the MSD is infinite when t = 0. To contain the classical Sinai diffusion law and its generalization [27], we use k(t ) = (Eα−1 (1 + t ))b as the structural function in Eq. (16), then Eq. (19) can be generalized into
x2 (t ) = 2Dα (Eα−1 (1 + t ))b,
(20)
where b > 0. Eq. (20) degenerates into a logarithmic diffusive law when a = 1,
x (t ) = 2D1 (ln(1 + t )) . 2
b
(21)
When b = 4, Eq. (21) becomes the Sinai diffusion law. By using the inverse ML and logarithmic diffusion laws in Eqs. (20) and (21), the experimental data with fits are compared as shown in Fig. 2. It is easily seen from Fig. 2 that both the diffusion laws are correct when the time range is very small. With the increasing time, the curve estimated by the logarithmic diffusion law, however, grows much faster than the data points, and encounters a lower accuracy compared with the results estimated via the inverse ML diffusion law. In addition, the structural derivative equation reflects the propagation of particles in regions with greater local interactions in the ultraslow diffusion than in the Brownian motion, which gives physics interpretation of the inverse ML diffusion. Thus, the structural derivative appears a feasible mathematical tool to describe such ultraslow diffusion. 3.2. Implicit calculus equation approach modeling non-local heat conduction Ref. [18] uses the fractional Riesz potential as the fundamental solution to establish the implicit calculus equation of fractional Laplacian modeling the non-local heat conduction within a cylinder, in which the radius and the height are 1 and 6, respectively. In this case, the temperature on the cylinder surface is given by
√ uˆ (x, y, z ) = e cos x
2 y cos 2
√ 2 z 2
+ 2.
(22)
The implicit fractional Laplacian (-)s /2 (s∈(1,2]) was defined by the integral-differential operator, and it is not needed to know the explicit expression of the fractional Laplacian. The underlying fundamental solution has the following form
u ∗ ( x ) = c1 ( s )
1
x 3−s
, x ∈ R3 ,
(23)
where c1 (s) is the normalizing constant,s ∈ (1, 2] the order, and || || the Euclidean norm. Eq. (23) satisfies the fractional Laplacian equation
(−)s/2 u(x ) = δ (x ), x ∈ R3 ,
(24)
where δ (x) denotes the Dirac function. By using the singular boundary method (SBM), the solutions on the characteristic line {(x,y,z)|x = y = 0, z∈[−3,3]} are shown in Fig. 3, in which the ordinates of the blue circles are the exact solutions at the boudnary points (0,0,±3). We can observe from Fig. 3 that the curves show the continuous variation of the solutions against s, and the temperature decreases as the order s decreases. Moreover, the accuracy of the numerical simulations at the boundary points becomes less accurate with the increasing order s. Thus, it is sufficient to do numerical simulation using the fundamental solution of the implicit calculus governing equation with the corresponding boundary conditions. 3.3. Fractal differential operator modeling fractal materials This section considers the heat conduction within four types of cubic fractal materials, which include thermal insulation, the granular, energetic materials and the bituminous pavement. The dimensionless side length of the cubic fractal material is 2. In this case, Eq. (23) is used as the fundamental solution of the Laplacian operator on fractal. The temperatures of the bottom surface and the top surface are 10 °C and 0 °C respectively, and the other surfaces are insulated. By using the singular boundary method, the numerical simulation of Eq. (24) for the characteristic line {(x,y,z)|x = y = 1, z∈[0,2]} are shown in Fig. 4. It can be observed from Fig. 4 that when the fractal dimension d approaches to 3, the solutions of the fractal dimension Laplacian equation converge to those of the classical Laplacian equation on 3D domain.
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5
3 s=2.0
2.8
s=1.9 s=1.8
Approximate solutions
2.6
s=1.7
2.4
s=1.6
2.2 2 1.8 Exact solution at the boundary point (0,0,3) 1.6 Exact solution at the boundary point (0,0,-3) 1.4 -3
-2
-1
0
1
2
3
z Fig. 3. √The temperature variations on the z-axis line {(x,y,z)|x = y = 0, z∈[−3,3]} given by the SBM solutions. The boundary conditions satisfy Eq. (22) uˆ (x = 0, y = 0, z = −3 ) = √ cos(− 3 2 2 ) + 2 and uˆ (x = 0, y = 0, z = 3 ) = cos( 3 2 2 ) + 2. The ordinates of the two blue circles are the exact solutions at the boundary points (0,0,±3) (This figure is from [18]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
10 9 8
Temperature u
7 6 5 4 3
d = 3.0 Traditional material d = 2.7 Thermal insulation material d = 2.5 Energetic material d = 2.43 Granular material d = 2.35 Bituminous pavement
2 1 0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Z Fig. 4. Variation of temperature u on the line {(x, y, z) | x = y = 1, z∈[0,2]} against the fractal dimension d (This figure is from [21]).
It is also noted that the temperature deceases linearly in the traditional 3D material, while in the fractal materials, the temperature decays more slowly at the bottom and top surfaces, and varies more dramatically at the central part. From the above application, it is clear that the structural function plays a central role in the structural derivative modeling real mathematical physical problems. The structural function can be any form of a function, such as the kernel functions in integral equations [28,29], the potential functions in molecular dynamics [30,31] or the probability density functions [32,33] in statistical modelings used to construct the structural derivative, as shown in Table 1. Table 1 gives the examples of three different kinds of functions used as potential structural functions in the structural deriva-
tive modeling strategy. It is noted that each function has clear physical meaning. For example, the Lévy stable and Mittag-Leffler distributions are frequently employed in characterizing the statistical behaviors of anomalous diffusion It should also be pointed out that some other types of functions can be chosen as the structural function according to the physical behaviors of the systems, e.g., the fundamental solution function [16]. 4. Conclusions In this paper, we summarize the structural derivative approach, the implicit calculus equation modeling, and the fractal calculus operator. The structural derivative is derived using the structural metric, which generalizes the Euclidean metric. The structural
Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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W. Chen, Y. Liang / Chaos, Solitons and Fractals 000 (2017) 1–6 Table 1 Structural functions selected from the kernel, potential or probability density functions. Structural functions Kernel Fredholm integral equation
Potential
|ln | x t
Volterra integral equation eλ(x−t )
Probability density
Lennard–Jones potential
A ri j 12
−
B ri j 6
Morse potential Ae−2B(ri j −r0 ) − 2Ae−B(ri j −r0 )
function can be any form of a function to construct the structural derivative, which characterizes the time-space structures of the system of interest. In the implicit calculus equation modeling, it is not necessary to know the explicit expression of the governing equation. The fundamental solution, potential, or the probability density function is used to construct the implicit calculus governing equation. By using the fundamental solution and the boundary conditions, the numerical solution of the problems of interest can directly and easily be computed by a boundary-type numerical scheme. The implicit calculus modeling can also be used to construct the structural derivative. The differential operator on fractal characterizes the spatial geometric feature of the fractal materials, and significantly extends the application scope of the classical calculus model under the framework of continuum mechanics. From the above applications of the three recent methodologies, it is found that the inverse ML diffusion law has clear physical mechanism, and its corresponding structural derivative is a feasible mathematical tool modeling the ultraslow diffusion. Using the implicit calculus equation modeling, the fundamental solution of calculus governing equation and corresponding boundary conditions are sufficient to do numerical simulation. The fractal differential operator and its Laplacian equation can be successfully used for phenomenological modeling potential problems in fractal media. Acknowledgment The work described in this paper was supported by the National Science Funds for Distinguished Young Scholars of China (11125208). References [1] Mongiovi MS, Zingales M. A non-local model of thermal energy transport: the fractional temperature equation. Int J Heat Mass Tran 2013;67:593–601. [2] Schumer R, Meerschaert MM, Baeumer B. Fractional advection dispersion equations for modeling transport at the Earth surface. J Geophys Res 20 09;114:F0 0A07. [3] Sun HG, Meerschaert MM, Zhang Y, Zhu J, Chen W. A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Adv Water Resour 2013;52:292–5. [4] Strunin DV, Suslov SA. Phenomenological approach to 3D spinning combustion waves: numerical experiments with a rectangular rod. Int J Self Prop High Temp Synth 2015;14:33–9. [5] Wio HS, Escudero C, Revelli JA, Deza RR, de MS. Recent developments on the Kardar–Parisi–Zhang surface-growth equation. Philos T R Soc A 2011;369:396–411. [6] Brown KS, Sethna JP. Statistical mechanical approaches to models with many poorly known parameters. Phys Rev E 2003;68:125–49. [7] Yablonskiy DA, Bretthorst GL, Ackerman JJH. Statistical model for diffusion attenuated MR signal. Magn Reson Med 2003;50:664–9.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
New methodologies in fractional and fractal derivatives modelingR Wen Chen∗, Yingjie Liang∗ State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, China
a r t i c l e
i n f o
Article history: Received 28 January 2017 Revised 22 March 2017 Accepted 27 March 2017 Available online xxx Keywords: Structural derivative Structural function Structural metric Structural fractal Implicit calculus equation modeling Ultraslow diffusion Inverse Mittag–Leffler function
a b s t r a c t This paper surveys the latest advances of the first author’s group on the three new methodologies of fractional and fractal derivatives modeling to meet the increasing and challenging demands in scientific and engineering communities. Firstly, the structural fractal was proposed as a generalization of the Euclidean distance. Using the structural metric, the structural derivative approach was derived as a significant extension of the global fractional calculus and the local fractional derivative approaches to tackle the perplexing modeling problems. The classical derivative describes the change rate of a certain physical variable with respect to time or space, which rarely takes into account the significant influence of mesoscopic time-space metric of a complex system on its physical behaviors. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space structural metric of physical systems. Secondly, we employed the fundamental solution or probability density function of statistical distribution which can describe the problem of interest to construct the implicit calculus governing equation. The ‘implicit’ suggests that the explicit calculus expression of this governing equation is difficult to derive and not required. The fundamental solution or potential function of calculus governing equation and corresponding boundary conditions are sufficient to do numerical simulation. We call this strategy the implicit calculus equation modeling. Thirdly, based on the implicit calculus equation modeling approach, we introduced the concept of fundamental solution on fractal and consequently defined the fractal differential operator to describe various mechanical behaviors of fractal materials. Fractal calculus operator significantly extends the application scope of the classical calculus modeling approach under the framework of continuum mechanics. This is also a step-forward advance of the fractal derivative proposed earlier by the first author. To demonstrate the structural derivative application, we applied the inverse Mittag-Leffler function as the structural function to model ultraslow diffusion of a random system of two interacting particles. On the other hand, this paper uses the fractional Riesz potential as the fundamental solution to establish the implicit calculus equation of fractional Laplacian modeling the power law behaviors of steady heat conduction in multiple phase material. Finally, by using the singular boundary method, we made numerical simulation of the fractal Laplacian equation for phenomenological modeling potential problems in fractal media. Numerical experiments show that all the three new methodologies are feasible mathematical tools to describe complex physical behaviors. © 2017 Published by Elsevier Ltd.
1. Introduction Recent decades have witnessed a growing number of complex scientific and engineering problems which are not easy to be described by the classical calculus modeling methodology [1–3]. To remedy this troublesome problem, a variety of phenomenological partial differential equation models including multiple empirical R This paper is a revised version of the report presented in the International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, July 18– 20, 2016. The physical mechanism of the structural derivative and more structural functions are added in this paper. ∗ Corresponding authors. E-mail addresses: [email protected] (W. Chen), [email protected] (Y. Liang).
parameters have been proposed in theoretical research and engineering practice [4,5]. In some cases, statistical models are used instead of calculus models [6,7]. These models are not clearly interpreted in physics and require more parameters in which the artificial parameters have no physical significance. The fractional calculus [8] and the fractal derivative [9] have been found in numerous studies as alternative competitive methodologies to tackle such modeling bottleneck. However, the fractional calculus and fractal derivative simply do not work for ultraslow diffusion [10]. To remedy this challenging problem, the structural derivative modeling approach [10,11] is derived based on the structural metric. The structural function, which is substantially a time-space transform, characterizes the
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Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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time-space structures [12] of the system of interest. Based on this, the causal relationship between mesoscopic time-space structural metric and certain physical behaviors can be described using the structural derivative with fewer parameters and lower computational costs. The structural function can be obtained using the fundamental solution or the probability density function. As an example, the inverse Mittag–Leffler (ML) function [13] was used as the structural function in the structural derivative to model the dynamics of two interacting particles in a disordered chain [14]. It is noted that the structural derivative can be either a global integraldifferential operator or a local fractal derivative fashion. To circumvent computationally expensive singularity in the classical definition of fractional derivative, Caputo and Fabrizio [15] recently proposed to replace the standard power function with an exponential kernel. This exponential function can be understood as a structural function from a pure mathematical point of view. But in their study, the exponential function has no physical significance and is aimed at simplifying numerical computation by remedying singularity of the kernel function in fractional derivative. This is different from the present structural derivative methodology, in which the structural function characterizes the influence of the time-space structure on physics behaviors. It is worthy of noting that in some cases the explicit calculus expression of governing equation to the physical problems is difficult to derive. Thus, the implicit calculus equation modeling is proposed to tackle this issue, in which the fundamental solution [16] or the probability density function [11] is used to construct the implicit calculus governing equation [17,18]. It is not required to know the explicit expression of the governing equation. Based on the fundamental solution and the corresponding boundary conditions, the numerical solution of the problems of interest can directly and easily be computed by a boundary-type numerical scheme [19]. The implicit calculus modeling can also be used to construct the structural derivative [11]. By using the fractional Riesz potential as the fundamental solution, the implicit calculus equation of fractional Laplacian was established to model the power law behaviors of steady heat conduction in multiple phase materials [18]. The first author of this paper introduced the fractal derivative [9] as an alternative approach to model complex problems. The fractal derivative in time has been successfully used in many applications [3,9,20], but the fractal derivative in space attracts much less attention. Based on the proposed implicit calculus modeling approach, the concept of fundamental solution on fractal and the fractal differential operator [21] are introduced to describe various mechanical behaviors of fractal materials. The fractal calculus operator characterizes the spatial geometric feature of the fractal materials, and significantly extends the application scope of the classical calculus model under the framework of continuum mechanics. By using the singular boundary method [19], the fractal Laplacian equation for phenomenological modeling the potential problems in fractal media was successfully simulated [21]. The rest of this paper is organized as follows. In Section 2, the structural derivative approach, the implicit calculus equation modeling, and the definition of the fractal differential operator are introduced. In Section 3, some applications of the three recent methodologies are tested to some problems, in particular the ultraslow diffusion. Finally, we draw some conclusions in Section 4. 2. Methodologies 2.1. Structural derivative The Hausdorff dimension in space is described by
M = |x − x |d ,
(1)
where r represents the Euclidean distance, d is the fractal dimension. The power law in Eq. (1) has since found many applications in science and engineering. However, it is observed that many real problems cannot well be characterized by the classical fractal concept [12]. Thus, this paper generalizes the Euclidean metric Eq. (1) to the structural metric as
M = |R ( x ) − R ( x )|d ,
(2)
where R is the structural function and can be an arbitrary function. The formula (2) characterizes the structural fractal, which includes the Euclidean and the non-Euclidean metrics. To illustrate the above derivative on structural fractal without a loss of generality, we consider the displacement x under power function time metric
x = vt α ,
(3)
where α is the index of time fractal. The corresponding derivative of the displacement in Eq. (3) is obtained as,
dx = vd (t α ),
(4)
then Eq. (4) can be rewritten as,
v=
dx . d (t α )
(5)
Consequently, the Hausdorff derivative on time fractal α is given by
dp(x, t ) p(x, t1 ) − p(x, t ) = lim . t1 →t dt t1α − t α
(6)
The fractal derivatives on space fractal metric are also proposed in Refs. [9,22–25]. By using the similar strategy, the displacement x under general structural metric can be derived by
x = vk(t ),
(7)
where k(t) is the structural function. The derivative of the displacement in Eq. (7) is written as
dx = vdk(t ).
(8)
Then Eq. (8) can be rewritten as
v=
dx . dk(t )
(9)
The definition of structural derivative by using the structural metric further extends the fractal derivative concept. On time structural metric, the local structural derivative is defined as [7]
dp(x, t ) p(x, t1 ) − p(x, t ) = lim , t1 →t ds t k(t1 ) − k(t )
(10)
where S denotes the structural derivative, k(t) is the structural function, ds means the structural derivative. When k(t ) = t α , it is the fractal derivative. In contrast, the global structural derivative in time is defined as [26]
δ p(x, t ) ∂ = δst ∂t
t
t1
k(t − τ ) p(x, τ )dτ .
(11)
−α
When k(t ) = (t1−α ) , Eq. (11) degenerates into the classical Riemann-Liouville fractional derivative. Fig. 1 shows the decay of four different structural functions including the exponential, power law, logarithmic and inverse ML functions. It is stressed that the structural function is not necessary a power function. Instead it can be any form of a function, such as the inverse Mittag–Leffler function, the probability density function, and the stretched exponential function [9]. Compared with the classical nonlinear calculus models, the structural derivative requires fewer parameters and lower computational costs in detecting the causal relationship between mesoscopic time-space structure and certain physical behaviors [11].
Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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3
1 0.9
Exponential function Power law function Logarithmic fucntion Inverse Mittag-Leffler function
0.8
Structural function
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
2
4
6
8
10 t
12
14
16
18
20
Fig. 1. Decay of four different structural functions including the exponential, power law, logarithmic and inverse Mittag–Leffler functions from bottom to top.
2.2. Implicit calculus equation modeling
The time-dependent diffusion equation on fractal has the fundamental solution:
Chen and Pang [17,18] proposed the implicit calculus equation modeling approach, which does not require the explicit expression of the governing equation. To describe the physical problems, the fundamental solution, potential function, or the probability density function is used to construct the implicit calculus governing equation. By using the fundamental solution or the probability density function with the corresponding boundary conditions, the numerical solution of the problems of interest can directly and easily be computed by a boundary-type numerical scheme [19].
u∗d (r ) =
H (t )
(4π Dt )d/2
e−r
2
/4Dt
,
(15)
where D is the diffusion coefficient, t the time interval, H(t) the Heaviside function. Then the implicit calculus equation modeling approach can be used to investigate the mechanical behaviors of fractal materials in terms of the above fundamental solutions. 3. Applications 3.1. Structural derivative modeling ultraslow diffusion
2.3. Differential operator on fractal Based on the implicit calculus equation modeling approach, Chen, Wang and Yang [21] defined the differential operator on fractal and consequently gave the fundamental solutions of four classical calculus equations on fractal, which are listed as below. The fundamental solution of the Laplacian operator on fractal is
u∗d (r ) =
1
( d − 2 )Sd ( 1 )
r 2−d ,
(12)
where Sd (1) =2π d /2 / (d/2), r represents the Euclidean distance, d is the fractal dimension. The fundamental solutions of the Helmholtz and the modified Helmholtz equations on fractal are constructed as
u∗d
1 (r ) = 2π
−ik 2π r
(d/2)−1 K(d/2)−1 (−ikr ),
(13)
and
u∗d (r ) =
1 2π
k 2π r
(d/2)−1 K(d/2)−1 (kr ),
(14)
where K( d /2)−1 is the modified Bessel function of the second kind, k the constant, i2 = −1.
In this section, the inverse Mittag–Leffler (ML) function was used as the structural function in the structural derivative to model ultraslow diffusion of two interacting particles in a disordered chain [14]. The case is at long time t2 ≤ 105 with the localization length L1 = 36, and the length of chain L = 1024. The experimental data on the center of mass R(t) are used, which are assumed as the square root of the mean squared displacement (MSD) from the diffusion point of view, i.e., R(t ) = (< x2 (t ) > )1/2 [14]. In the structural derivative modeling ultraslow diffusion, the propagator p(x,t) satisfies the following diffusion equation.
dp(x, t ) ∂ 2 p(x, t ) = Dα , ds t ∂ x2
(16)
where Da is the generalized diffusion coefficient (m2 /sa ), and the structural derivative is defined by
dp(x, t ) p(x, t1 ) − p(x, t ) = lim −1 , −1 t1 →t E ds t α (t1 ) − Eα (t )
(17)
where Eα−1 (t ) is the inverse function of the single-parameter ML function with 0 < a ≤ 1. Eq. (16) can be restated as a normal diffusion equation, and yields a Gaussian distribution.
p(x, tˆ) =
1 4π Dα tˆ
exp −
x2 4Dα tˆ
,
(18)
Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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120
100
R(t)
80
60
Experimental data Inverse ML diffusion law Logarithmic diffusion law
40
D = 0.5, b/2 = 2.0 20
0
0
0.5
1
1.5
2
2.5 t2
3
3.5
4
4.5
5 x 10
4
Fig. 2. Plots of centers of mass R(t) with fits using the root square of the inverse Mittag–Leffler and logarithmic diffusion laws (This figure is from [10]).
where tˆ = Eα−1 (t ). The MSD of the ultraslow diffusion particle x(t) can be derived from Eq. (16) as
x2 (t ) = 2Dα Eα−1 (t ).
(19)
It is noted that in Eq. (19), the MSD is infinite when t = 0. To contain the classical Sinai diffusion law and its generalization [27], we use k(t ) = (Eα−1 (1 + t ))b as the structural function in Eq. (16), then Eq. (19) can be generalized into
x2 (t ) = 2Dα (Eα−1 (1 + t ))b,
(20)
where b > 0. Eq. (20) degenerates into a logarithmic diffusive law when a = 1,
x (t ) = 2D1 (ln(1 + t )) . 2
b
(21)
When b = 4, Eq. (21) becomes the Sinai diffusion law. By using the inverse ML and logarithmic diffusion laws in Eqs. (20) and (21), the experimental data with fits are compared as shown in Fig. 2. It is easily seen from Fig. 2 that both the diffusion laws are correct when the time range is very small. With the increasing time, the curve estimated by the logarithmic diffusion law, however, grows much faster than the data points, and encounters a lower accuracy compared with the results estimated via the inverse ML diffusion law. In addition, the structural derivative equation reflects the propagation of particles in regions with greater local interactions in the ultraslow diffusion than in the Brownian motion, which gives physics interpretation of the inverse ML diffusion. Thus, the structural derivative appears a feasible mathematical tool to describe such ultraslow diffusion. 3.2. Implicit calculus equation approach modeling non-local heat conduction Ref. [18] uses the fractional Riesz potential as the fundamental solution to establish the implicit calculus equation of fractional Laplacian modeling the non-local heat conduction within a cylinder, in which the radius and the height are 1 and 6, respectively. In this case, the temperature on the cylinder surface is given by
√ uˆ (x, y, z ) = e cos x
2 y cos 2
√ 2 z 2
+ 2.
(22)
The implicit fractional Laplacian (-)s /2 (s∈(1,2]) was defined by the integral-differential operator, and it is not needed to know the explicit expression of the fractional Laplacian. The underlying fundamental solution has the following form
u ∗ ( x ) = c1 ( s )
1
x 3−s
, x ∈ R3 ,
(23)
where c1 (s) is the normalizing constant,s ∈ (1, 2] the order, and || || the Euclidean norm. Eq. (23) satisfies the fractional Laplacian equation
(−)s/2 u(x ) = δ (x ), x ∈ R3 ,
(24)
where δ (x) denotes the Dirac function. By using the singular boundary method (SBM), the solutions on the characteristic line {(x,y,z)|x = y = 0, z∈[−3,3]} are shown in Fig. 3, in which the ordinates of the blue circles are the exact solutions at the boudnary points (0,0,±3). We can observe from Fig. 3 that the curves show the continuous variation of the solutions against s, and the temperature decreases as the order s decreases. Moreover, the accuracy of the numerical simulations at the boundary points becomes less accurate with the increasing order s. Thus, it is sufficient to do numerical simulation using the fundamental solution of the implicit calculus governing equation with the corresponding boundary conditions. 3.3. Fractal differential operator modeling fractal materials This section considers the heat conduction within four types of cubic fractal materials, which include thermal insulation, the granular, energetic materials and the bituminous pavement. The dimensionless side length of the cubic fractal material is 2. In this case, Eq. (23) is used as the fundamental solution of the Laplacian operator on fractal. The temperatures of the bottom surface and the top surface are 10 °C and 0 °C respectively, and the other surfaces are insulated. By using the singular boundary method, the numerical simulation of Eq. (24) for the characteristic line {(x,y,z)|x = y = 1, z∈[0,2]} are shown in Fig. 4. It can be observed from Fig. 4 that when the fractal dimension d approaches to 3, the solutions of the fractal dimension Laplacian equation converge to those of the classical Laplacian equation on 3D domain.
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5
3 s=2.0
2.8
s=1.9 s=1.8
Approximate solutions
2.6
s=1.7
2.4
s=1.6
2.2 2 1.8 Exact solution at the boundary point (0,0,3) 1.6 Exact solution at the boundary point (0,0,-3) 1.4 -3
-2
-1
0
1
2
3
z Fig. 3. √The temperature variations on the z-axis line {(x,y,z)|x = y = 0, z∈[−3,3]} given by the SBM solutions. The boundary conditions satisfy Eq. (22) uˆ (x = 0, y = 0, z = −3 ) = √ cos(− 3 2 2 ) + 2 and uˆ (x = 0, y = 0, z = 3 ) = cos( 3 2 2 ) + 2. The ordinates of the two blue circles are the exact solutions at the boundary points (0,0,±3) (This figure is from [18]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
10 9 8
Temperature u
7 6 5 4 3
d = 3.0 Traditional material d = 2.7 Thermal insulation material d = 2.5 Energetic material d = 2.43 Granular material d = 2.35 Bituminous pavement
2 1 0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Z Fig. 4. Variation of temperature u on the line {(x, y, z) | x = y = 1, z∈[0,2]} against the fractal dimension d (This figure is from [21]).
It is also noted that the temperature deceases linearly in the traditional 3D material, while in the fractal materials, the temperature decays more slowly at the bottom and top surfaces, and varies more dramatically at the central part. From the above application, it is clear that the structural function plays a central role in the structural derivative modeling real mathematical physical problems. The structural function can be any form of a function, such as the kernel functions in integral equations [28,29], the potential functions in molecular dynamics [30,31] or the probability density functions [32,33] in statistical modelings used to construct the structural derivative, as shown in Table 1. Table 1 gives the examples of three different kinds of functions used as potential structural functions in the structural deriva-
tive modeling strategy. It is noted that each function has clear physical meaning. For example, the Lévy stable and Mittag-Leffler distributions are frequently employed in characterizing the statistical behaviors of anomalous diffusion It should also be pointed out that some other types of functions can be chosen as the structural function according to the physical behaviors of the systems, e.g., the fundamental solution function [16]. 4. Conclusions In this paper, we summarize the structural derivative approach, the implicit calculus equation modeling, and the fractal calculus operator. The structural derivative is derived using the structural metric, which generalizes the Euclidean metric. The structural
Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066
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W. Chen, Y. Liang / Chaos, Solitons and Fractals 000 (2017) 1–6 Table 1 Structural functions selected from the kernel, potential or probability density functions. Structural functions Kernel Fredholm integral equation
Potential
|ln | x t
Volterra integral equation eλ(x−t )
Probability density
Lennard–Jones potential
A ri j 12
−
B ri j 6
Morse potential Ae−2B(ri j −r0 ) − 2Ae−B(ri j −r0 )
function can be any form of a function to construct the structural derivative, which characterizes the time-space structures of the system of interest. In the implicit calculus equation modeling, it is not necessary to know the explicit expression of the governing equation. The fundamental solution, potential, or the probability density function is used to construct the implicit calculus governing equation. By using the fundamental solution and the boundary conditions, the numerical solution of the problems of interest can directly and easily be computed by a boundary-type numerical scheme. The implicit calculus modeling can also be used to construct the structural derivative. The differential operator on fractal characterizes the spatial geometric feature of the fractal materials, and significantly extends the application scope of the classical calculus model under the framework of continuum mechanics. From the above applications of the three recent methodologies, it is found that the inverse ML diffusion law has clear physical mechanism, and its corresponding structural derivative is a feasible mathematical tool modeling the ultraslow diffusion. Using the implicit calculus equation modeling, the fundamental solution of calculus governing equation and corresponding boundary conditions are sufficient to do numerical simulation. The fractal differential operator and its Laplacian equation can be successfully used for phenomenological modeling potential problems in fractal media. Acknowledgment The work described in this paper was supported by the National Science Funds for Distinguished Young Scholars of China (11125208). References [1] Mongiovi MS, Zingales M. A non-local model of thermal energy transport: the fractional temperature equation. Int J Heat Mass Tran 2013;67:593–601. [2] Schumer R, Meerschaert MM, Baeumer B. Fractional advection dispersion equations for modeling transport at the Earth surface. J Geophys Res 20 09;114:F0 0A07. [3] Sun HG, Meerschaert MM, Zhang Y, Zhu J, Chen W. A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Adv Water Resour 2013;52:292–5. [4] Strunin DV, Suslov SA. Phenomenological approach to 3D spinning combustion waves: numerical experiments with a rectangular rod. Int J Self Prop High Temp Synth 2015;14:33–9. [5] Wio HS, Escudero C, Revelli JA, Deza RR, de MS. Recent developments on the Kardar–Parisi–Zhang surface-growth equation. Philos T R Soc A 2011;369:396–411. [6] Brown KS, Sethna JP. Statistical mechanical approaches to models with many poorly known parameters. Phys Rev E 2003;68:125–49. [7] Yablonskiy DA, Bretthorst GL, Ackerman JJH. Statistical model for diffusion attenuated MR signal. Magn Reson Med 2003;50:664–9.
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Please cite this article as: W. Chen, Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.03.066