New Derivatives on Fractal Subset of Real-line

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on fractals subset of real-line lies in the fact that they are used for better modelling of processes with memory effect.


Introduction
The calculus involving arbitrary orders of derivatives and integrals is called the fractional calculus. Recently, the fractional calculus has found many applications in several areas of science and engineering [1][2][3][4]6]. The nonlocal property of the fractional derivatives and integrals is used to model the processes with memory effect [1,2]. For example, the fractional derivatives are used to model more appropriate the dynamics of the non-conservative systems in the Hamilton, Lagrange and Nambu mechanics [5,[7][8][9][10]. The continuous but non-differentiable functions admit the local fractional derivatives [11]. The local fractional derivative give the measure on fractal sets. Consequently, recently the F α -calculus on the fractal subset of real line and fractal curves is built as a framework [12,13]. Fractal analysis is established by many researchers by using different methods [14][15][16][17]. Using F α -calculus the Newton, Lagrange and Hamilton mechanics were built on fractal sets [18,19]. Also, the Schrödinger's equation on fractal curve was derived in [20][21][22]. Motivated by the above mentioned interesting results, in this work, we define the non-local derivative on fractal sets. These new derivatives can be successfully used to derive new mathematical models on fractal sets involving processes with memory.
We organize our manuscript as follows: In Section 2, we gave a brief exposition of F α -calculus and de-fined fractal Gamma and Beta functions. In Section 3 we defined the non-local derivative on fractals as a generalized Riemann-Liouville and Caputo fractional derivatives. In Section 4, Mittag-Leffler function and non-local Laplace fractional on fractal sets are introduced. We solved the non-local differential equations on fractal using the suggested methods. Section 5 was devoted to our conclusion.

A review of fractional local derivatives
In this section, we review the F α -calculus [12,13].

Calculus on fractal subset of real-line
The fractal geometry is the geometry of the real world [1]. Fractal shape is the object with the fractional dimension and the self similarity property [9,10]. In the seminal paper, Parvate and Gangal have established a calculus on fractals which similar to Riemann integration. The suggested framework become a mathematical model for many phenomena in fractal media [12,13]. We recall that the triadic Cantor set is a fractal that can be obtained by an iterative process. In figure 1 we show the Triadic Cantor set [14]. The integral staircase function for the triadic Cantor set is defined as [12,13].
where α is the γ-dimension of triadic Cantor set. In Figure 2 we plot the integral staircase function for triadic Cantor set. The definitions of F α -limit, F α -continuity and F α -integration are given in the ref. [12,13]. The F α -differentiation is denoted by D α F and it is defined as if the limit exists [12,13]. Definition 1. The Gamma function with the fractal support is defined as where Definition 2. The fractal Beta function on the fractal set is defined as follows which is called two-parameter (r, w) fractal integral, where (r) > 0 and (w) > 0. In the following we present some properties of fractal Beta function.
1) The fractal Beta function has a symmetry B α F (w, r) = B α F (r, w). Since, we have using the transformation S α F (x) = 1 − S α F (y), we conclude that 2) Using the transformation S α F (x) = sin 2 (S α F (θ)), we get following form for the fractal the Beta function 3) The Beta fractal function is related to the fractal Gamma function as Transforming to polar coordinates S α Thus, the proof is completed.

Non-local fractal derivative and integral
In this section, we define the non-local derivative for the functions with fractal support.
where if β = α then we have fractal integral whose order is equal the dimension of the fractal, and are called the analogous left sided and the right sided Riemann-Liouville fractal integral of order β. Definition 4. Let n − α ≤ β < n, then analogous left and right Riemann-Liouville fractal derivative are defined as follows: Also, the analogous right sided Caputo fractal derivative has the form Now, we give some important relations, namely . As a result we obtain Substituting Eqs. ( 21) and ( 22) in Eq. (20) we conclude that Then, we have In view of Eq. (5) we derive Applying Eq. (10) we get Now, we consider following formula Proof: By rewriting the Eq. (27) we get Now, we write some important composition relations, namely (31) Proof: Using the definitions we get Applying, n-times integration by part it leads to The similar proof works for the following formulas In figures 4 and 5 we compared the non-local standard derivative versus non-local fractal derivative and the generalized fractal integral.

Generalized functions in the non-local calculus on the fractal subset of real-line
In this section, we suggest the mathematical tools for solving the non-local fractal differential equations.

Gamma function on fractal subset of real line
Now, we define the Gamma function for the fractal calculus that will be used in non-local calculus on fractals.

Mittag-Leffler function on fractal subset of realline
It is well known that the exponential function has important role in the theory of standard differential equation. The generalized exponential function is called the Mittag-Leffer function and plays an important role for fractional differential equations [1]. Definition 6. The generalized two parameter η, ν Mittag-Liffler function on fractal F with α-dimension is defined as In the special case we have the following results, namely . (42)

Non-local Laplace transformation on fractal subset of real-line
The Laplace transformation is a very useful tool for solving standard linear differential equation with constant coefficients. The generalized Laplace transformation is applied to solve the fractional differential equations. Thus, in this section, we generalized the Laplace transformation for the function with fractal support which is utilized to solve the non-local differential equation on the fractal set [1]. Definition 7. Laplace transformation for the function f (x) is denoted by F (s) and it is defined as Now, we give the fractal Laplace transformation of some functions. If we define the fractal convolution of two function f (x) and g(x) as follows: the fractal Laplace transformation of power function of S α F (x) is Lemma: The Laplace transformation of the non-local fractal Riemann-Liouville integral is given by Proof: The Laplace transform of the fractal Riemann-Liouville integral is (47) Using the Eqs.(44) and (45) we arrive at The fractal Laplace transform of the non-local fractal Riemann-Liouville derivative of order β ∈ [0, 1) is given by where n = [β] + 1. The fractal Laplace transform of the nonlocal fractal Caputo derivative of order β ∈ [0, 1) is given by

Non-local fractal differential equations
In this section, we solve some illustrative examples.

Example 1. Consider the following linear fractal equation
with the initial condition where α = 0.6309 is Cantor set dimension. By applying 0 I 1 2 x on the both sides of the Eq. (52) we obtain Example 2. Consider a linear fractal differential equation with initial condition as By applying 0 I 1 2 x on the both sides of the Eq. (55) we arrive at In Figures, 7 and 6 we plot the solutions of Eqs.(54) and (51), respectively. Example 3. Consider a linear differential equation with the following initial condition, namely By inspection, the solution for the Eq. (57) becomes In Figure 8 we sketched the solution of Eq. (57) on the Cantor set and real-line.
After some calculations we obtain

Conclusion
In this work, we defined new non-local derivatives on fractal sets. These new type of non-local derivatives can describe better the dynamics of complex systems which possess memory effect on a fractal set. Four illustrative examples were solved in detail. Finally, one can recover the standard non-local fractional cases when put α = 1. All authors common finished the manuscript. All authors have read and approved the final manuscript.
The authors declare no conflict of interest.