Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

doi:10.1016/j.chaos.2017.04.027

Chaos, Solitons & Fractals

Volume 102, September 2017, Pages 396-406

https://doi.org/10.1016/j.chaos.2017.04.027 Get rights and content

Highlights

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Fractal-fractional differentiation.
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Fractal-fractional integration.
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New numerical scheme for fractal-fractional operators.
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New model of Darcy scale describing flow in a dual medium.

Abstract

New operators of differentiation have been introduced in this paper as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative. The new operators will be referred as fractal-fractional differential and integral operators. The new operators aimed to attract more non-local natural problems that display at the same time fractal behaviors. Some new properties are presented, the numerical approximation of these new operators are also presented with some applications to real world problem.

Introduction

In the past years, the concept of non-local operators of differentiation has embarked many researchers from almost all branches of sciences, technology and engineering due to their capabilities of including more complex natural into mathematical equations. Within this field, three dominants were suggested including the power law, exponential decay law and the generalized Mittag-Leffler law [1], [2], [3], [4], [5], [6]. To distinguish them some names were associated to them including Riemann-Liouville and Caputo fractional operators for power law or non-local and singular kernel type; Caputo-Fabrizio for non-singular local type and Atangana-Baleanu for non-local and non-singular type. It was demonstrated that, the kernel Mittag-Leffler function is more general than power law and exponential decay function; therefore both Riemann-Liouville and Caputo-Fabrizio are special case of Atangana-Baleanu fractional operators [7], [8], [9], [10]. Some researchers suggested to have a general kernel, however the question one will ask is what that kernel is? What will be the properties of this kernel? So far there is no answer to that question, and we believe the idea of having a general kernel is not practical. One must realize that in case of Caputo type, we have a convolution of a local derivative either exponential function, power law or Mittag-Leffler function. The concept of local operator of differentiation are not very much suitable for modeling complex real world problems as was indicated in many published research papers in the past decades. For instance, those physical occurrences that display fractal behaviors. Within the framework of applied mathematics and mathematics, there exists a nonstandard kind of derivative known as fractal derivative in which the variable is a scaled according to t^a. This nonstandard derivative was introduced to possible model those physical problems for which classical physical law for instance the well-known Darcy's, Fourier's Law and Fick's law are no longer appropriate, such physical problem are believed to be based on the Euclidean geometry and cannot be applicable to the media of non-integral fractal dimension. Among other we quote real world problems such as Porous media, aquifer, turbulence and more other media commonly display fractal properties [11], [12], [13], [14], [15]. Numerous real-world phenomena display limited or statistical fractal properties. Although this field faces some challenges including measuring fractal dimension are effected by numerous mechanical concerns, also the searching to numerical and experimental noise and restrictions in aggregate of data. However, this field has been attracted several researchers, it is also rapidly growing as appraised fractal dimensions for statistically self-similar singularities may have practical applications in multi-fields for instance: Electrochemical processes, physics, diagnostic imagining, neuroscience, image analysis, acoustic, physiology and Riemann zeta zeros [11], [12], [13], [14], [15]. Apart of fractal dimensions for statistically self-similar, one can consider a direct measurement which is to consider mathematical models that could possibly resembles formation of a real-world fractal object. In this paper new concept of differentiation that takes into account the fractal effect and also the memory, non-locality and elasticity will be introduced. Another important natural problem that could be handled by these new operators is the super-diffusion and sub-diffusion problems that occur in complex media for instance heterogeneous self-similar aquifers. These operators will be constructed using the power law, the generalized Mittag-Leffler function and also the exponential decay law.

Section snippets

New concept of differentiation

More complex physical problems required more complex mathematical operators of differentiation. In this section we introduce a new concept of differentiation that combines the concept of fractional differentiation and the concept of fractal derivative.

Definition 1

Let f(t) be differentiable in opened interval (a, b), if f is fractal differentiable on (a, b)with order β then, the Fractal-Fractional derivative of fof order α in Caputo sense with power law is given as: $\begin{matrix} _{a}^{F F P} D_{t}^{α, β} f (t) & = & \frac{1}{Γ [n - α]} \int_{a}^{t} \frac{d f (y)}{d y^{β}} {(t - y)}^{n - α -} \end{matrix}$

New concept of integration

Although the concept of fractal differentiation has been used intensively in the literature with great success, however, we do not find the associate integral of this operator, therefore the fractal calculus is incomplete. In this section, we shall construct first a fractal integral then provide fractal-fractional integral associate to those differential operators that were introduced in Section 2 above.

We start with this section by solving the following fractal differential equation: $\frac{d f (t)}{d t^{α}} = u$

Numerical approximation for fractal-fractional operators

In this section, we present the numerical approximation of the new fractal-fractional differential operators. We shall start with that of Fractal-fractional derivative in Caputo sense. Without lost of generality, we consider the case with Mittag-Leffler function: Here we divide the interval as follows: $t_{0} = a < t_{1} < t_{2} < . . . . . . . < t_{n - 1} = t$ $\begin{matrix} _{a}^{F F M} D_{t}^{α, β} f (t) & = & \frac{A B (α)}{(1 - α)} \frac{d}{d t^{β}} \int_{a}^{t} f (y) E_{α} [- \frac{α}{1 - α} {(t - y)}^{α}] d y, \\ = & \frac{A B (α)}{(1 - α) β} t^{1 - β} \frac{d}{d t} \int_{a}^{t} f (y) E_{α} [- \frac{α}{1 - α} {(t - y)}^{α}] d y, \\ _{a}^{F F M} D_{t}^{α, β} f (t_{n}) & = & \frac{A B (α)}{Δ t (1 - α) β} {t_{n}}^{1 - β} {\int_{a}^{t_{n}} f (y) E_{α} [- \frac{α}{1 - α} {(t_{n} - y)}^{α}] d y \\ + \int_{a}^{t_{n - 1}} f (y) E_{α} [ \end{matrix}$

Numerical approximation for fractal-fractional integral

In this section, we present a numerical approximation of the fractal-fractional integral introduced in this paper. However without loss of generality, we present that of fractal-fractional with Atangana-Baleanu fractional integral. In this approximation, we shall use both 3/8 Simpson rule and Boole's rule quadratures. $\begin{matrix} _{a}^{F F M} I_{t}^{α, β} f (t) = \frac{β (1 - α) t^{β - 1} f (t)}{A B (α)} \\ + \frac{β α}{A B (α) Γ (α)} \int_{a}^{t} y^{β - 1} f (y) {(t - y)}^{α - 1} d y = F (t) + T (t) . \end{matrix}$

We present the approximation of with 3/8 Simpson rule T(t) [16,17]. $\begin{matrix} T (t) & = & \frac{β α}{A B (α) Γ (α)} \int_{a}^{t} y^{β - 1} f (y) {(t - y)}^{α - 1} \end{matrix}$

Applications

In this section, we present the solution of some simple fractal-fractional differential equation using analytical and numerical methods. We shall start with the following: $_{0}^{F F M R} D_{t}^{α, β} f (t) = a$ To solve Eq. (41), we apply the fractal-Laplace on both sides to obtain: $\begin{matrix} \frac{A B (α) p^{α} L (f (t))}{(1 - α) p^{α} + α} & = & L (β a t^{β - 1}), \\ L (f (t)) & = & \frac{α + (1 - α) p^{α}}{p^{α} A B (α)} L (β a t^{β - 1}), \\ f (t) & = & \frac{(1 - α) β a t^{β - 1}}{A B (α)} + \frac{α Γ (β + 1) a t^{α + β - 1}}{A B (α) Γ (α + β)} \end{matrix}$ We also consider the following equation: $_{0}^{F F M R} D_{t}^{α, β} f (t) = t^{a}, a > 0$ Again to solve Eq. (43), we apply the fractal-Laplace on both

Conclusion

A new concept of differentiation and integration are introduced in this paper. The new concept takes into account the memory effect, the heterogeneity, and elasco-viscosity of the medium and also the fractal geometry of the dynamic system. The new differentiation is a combining of fractal differentiation and fractional differentiation. New properties are presented and some new theorems established. Numerical approximations of the new fractal-fractional integral are presented using Boole's rule

References (18)

B.S.T. Alkahtani
Chua's circuit model with Atangana–Baleanu derivative with fractional order
Chaos Solitons Fract
(2016)
A. Atangana et al.
Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order
Chaos Solitons Fract
(2016)
W. Chen
Time–space fabric underlying anomalous diffusion
Chaos Solitons Fract
(2006)
R. Kanno
Representation of random walk in fractal space-time
Physica A
(1998)
W. Chen et al.
Anomalous diffusion modeling by fractal and fractional derivatives
Comput Math Appl
(2010)
J. Leveinen
Composite model with fractional flow dimensions for well test analysis in fractured rocks
J. Hydrol
(2000)
O. Jefain et al.
Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model
Chaos Solitons Fract
(2016)
M Caputo
Linear model of dissipation whose Q is almost frequency independent-II
Geophys J R Astron Soc
(1967)
J. Hadamard
Essai sur l'étude des fonctions données par leur développement de Taylor
J Pure Appl Math
(1892)

There are more references available in the full text version of this article.

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Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

Highlights

Abstract

Introduction

Section snippets

New concept of differentiation

New concept of integration

Numerical approximation for fractal-fractional operators

Numerical approximation for fractal-fractional integral

Applications

Conclusion

Chaos Solitons Fract

Chaos Solitons Fract

Chaos Solitons Fract

Physica A

Comput Math Appl

J. Hydrol

Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model

Chaos Solitons Fract

Linear model of dissipation whose Q is almost frequency independent-II

Geophys J R Astron Soc

Essai sur l'étude des fonctions données par leur développement de Taylor

J Pure Appl Math