Abstract
In this paper, we discuss non-local derivatives on fractal Cantor sets. The scaling properties are given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared. Related physical models are also suggested.
1 Introduction
Fractional calculus became an important tool which applied successfully in many branches of science, engineering etc [1–5]. The models based on fractional derivatives are crucial for describing processes with memory effects [6]. Local fractional has been defined on the real-line [7]. As it is well known the integer, fractional and complex order derivatives and integrals are defined on the real-line. Fractal analysis have been conducted by many researchers [8–10]. The fractal curves and the functions on fractal space are not differentiable in the sense of standard calculus. Using this as motivation recently a seminal paper has suggested Fα-calculus as a framework for the fractal sets and fractal curves [11–14]. Fα-calculus is generalized and applied in physics as a new and useful tool for modelling processes on fractals. Newtonian mechanics and Schrödinger equation on the fractal sets and curves are given [15–17]. The gauge integral is utilized to generalized Fα-calculus for unbound and singular functions [18]. The fractal grating is modeled by Fα-calculus and corresponding diffraction is presented [18]. One of the important aspects of fractional calculus was transferred recently to fractal derivatives. The concept of non-local fractal derivatives was introduced in [20].In this manuscript our main aim is to define the fractal non-local derivatives and study their properties.
The outline of this work is as follows:
In Section 2 we summarize the basic definitions and properties of the the local fractional derivatives. In Section 3 the scaling properties of local and non-local derivatives are derived. We develop the theory of fractal local and non-local Laplace transformations in Section 4. In Section 5 the comparison of local and non-local linear fractal differential equations are presented. In Section 6 we indicate some illustrative applications. Section 7 contains our conclusion.
2 Preliminaries
In this section we recall some basic definitions and properties of the local fractal calculus (LFC) and non-local fractal calculus (NLFC) [11, 20].
2.1 Local fractal calculus
In the seminal paper local Fα-calculus is built on fractal Cantor set which is shown in Figure [1][11].
The integral staircase function
where a0 is an arbitrary real number. The plot of the integral staircase function is depicted in Figure [2].
if the limit exists. For more details we refer the reader to [11].
2.2 Non-local fractal calculus
In this section, we review the non-local derivatives and basic definitions [20].
A function
Here and subsequently, we define the fractal left-sided Riemann-Liouville integral as follows
where
The fractal left-sided Riemann-Liouville derivative is defined as
For A
The fractal Grünwald and Marchaud derivative of a function f(x) with support of fractal sets is defined as
The generalized fractal standard Mittag-Leffler functions is defined as [20]
The fractal two parameter η, ν Mittag-Liffler function is defined as
For a given function
where
In view of the above conditions the fractal Laplace transform exists for all
We denote that if we choose β = α then we have
3 Scale properties of fractal local and non-local fractal calculus
In this section we study the scale properties of the LFC and NLFC.
3.1 Scale change on the local fractal derivatives
A function
where for some m and for all λ. The fractals have self-similar properties, namely for the case of function with the fractal Cantor set support we choose m=1 and λ=⅓n, n=1,2,... then
where α = 0.6 is the dimension of triadic Cantor set. The fractal derivative of the fractal homogenous function
3.2 Scale change on the non-local fractal derivatives
By a scale change of the fractal function
and using Eq. (5) and choosing a = 0 we derive
which is called scale change on the non-local fractal derivatives.
4 Laplace transformation on fractals
We provide some important lemmas that are useful for finding the fractal Laplace transforms of function
The fractal Laplace transform of the non-local fractal Caputo derivative of order m α − α < β ≤ mα,m ∈ N is
We first compute the Laplace fractal transform of the fractal Caputo fractional derivative of order β as follows
In view of Eq. (28) which completes the proof.
For a given
Using the series expansion we have
The inverse fractal Laplace transform of Eq. (20) leads to
Suppose
Let us use following expression
Therefore we can write
The proof is complete.
For
Since we can write
according to the Lemma 3. the proof is complete.
Some important formulas of the local fractal calculus are given below :[11, 20]
and
If we choose α = 1 we obtain the standard result.
The important formulas of non-local fractal calculus are as follows[20]:
where c is constant.
If we choose β = α then we arrive at to the local fractal derivative whose order is equal the dimension of the fractal.
5 Comparison between the local fractal differential and non-local fractal differential
In this section, we compare the local and non-local fractal differential equations.
Consider the linear local fractal differential equation
with initial-value
Hence the solution to Eq. (30) is
where α =0.6309 is the γ-dimension of the triadic Cantor set [11, 20].
In Figure 3 we give the graph of Eq. (32).
Consider linear non-local fractal differential equation as
with the initial condition
In view of Eq. (17) we have
Applying the fractal Laplace transformation on both sides of Eq. (33) and using Eq. (17) we obtain
It follows that
using fractal inverse Laplace transform Eq. (19) we arrive at the solution of Eq. (33) as follows
In Figure 4 we present the graph of Eq.(38).
6 Application of non-local fractal differential equations
In this section we provide the applications and new models to non-local fractal derivatives [20].
Fractal Abel’s tautochrone:
As a first example we generalized Abel’s problem which is the curve of quick descent on the fractal time-space. Using the conservation of energy in the fractal space the differential equation of the motion a particle is
where
Let us consider
so that we have
Utilizing
It follows
The solution of Eq.(44) is called the fractal cycloid.
Fractal models for the viscoelasticity:We generalize the viscoelasticity models to the fractal mediums in the case of ideal solids and ideal liquids. Namely, the fractal ideal solids described by
which is called Hooke’s Law of fractal elasticity. Where
The fractal ideal fluid can be modeled and described by Newton’s Law of fractal viscosity as follows
where
which is called fractal Blair’s model. Here, we suggest the fractional non-local order fractal derivative β as an index of memory. Namely, if we choose $β=0$ the process is equivalent to “nothing forgotten” and the case of β = α the process is memoryless. Hence if we choose 0 < β < α it shows the processes with memory on fractals.
If we choose
where
In Figure 6 we show the graph of
If we choose β = 0 and β = α in Eq. (47) we will have the fractal stress and the fractal strain relations for the cases of fractal ideal solids and the fractal ideal fluids, respectively.
7 Conclusion
In this paper we generalized fractal calculus involving the non-local derivatives. The scaling properties of local and non-local derivatives are studied because they are important in physical applications. Using an illustrative example we compared the local and non-local linear fractal differential equations. We also suggested some applications for the new non-local fractal differential equations.
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© 2016 Alireza K. Golmankhaneh and D. Baleanu
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