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New Derivatives on Fractal Subset of Real-line
Alireza Khalili Golmankhaneh, Dumitru Baleanu
TL;DR
The paper addresses non-local fractional calculus for functions defined on fractal subsets of the real line. It defines fractal special functions and generalized Riemann-Liouville and Caputo derivatives, then applies a fractal Laplace transformation to differential equations. The resulting framework is presented as a way to model memory-effect dynamics on fractal sets, while recovering standard non-local fractional cases at α = 1.
Problem
Existing fractional calculus models memory effects, but this work focuses on extending non-local derivatives to functions defined on fractal sets.
Method
The paper defines fractal Gamma and Beta functions, generalized Riemann-Liouville and Caputo derivatives, and a non-local Laplace transformation for fractal-supported functions.
Results
The proposed tools are applied to solve non-local differential equations on fractals, with solutions illustrated on the Cantor set and real line.
Takeaways & Limitations
The new derivatives are presented as tools for modelling complex systems with memory effects on fractal sets, and α = 1 recovers the standard non-local fractional cases.
Takeaways & Limitations
The composition results assume an α-order differentiable function on [a, b] and β > 0.
Abstract
from arXiv · showhide
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.
1 Introduction
The manuscript motivates non-local fractional derivatives on fractal sets for modelling memory-dependent processes and outlines the mathematical tools developed to support them.
- Fractional calculus studies derivatives and integrals of arbitrary order and is used to model processes with memory effects.
- The paper extends F α-calculus by defining non-local derivatives for functions supported on fractal sets.
- The manuscript reviews fractal Gamma and Beta functions, defines generalized Riemann-Liouville and Caputo derivatives, and introduces fractal Mittag-Leffler and Laplace tools.
- These tools are applied to solve non-local differential equations on fractals, including linear and non-linear cases.
2 A review of fractional local derivatives
This section reviews calculus on fractal subsets of the real line, using the triadic Cantor set and introducing fractal integration, differentiation, Gamma, and Beta functions.
- The triadic Cantor set provides the fractal support used to illustrate calculus on a fractal subset of the real line.It is constructed through an iterative process.
- The integral staircase function represents integration on the triadic Cantor set.Its construction and plotted behavior are introduced for the Cantor fractal.
- F α-differentiation is denoted by Dα_F and is defined through the fractal calculus framework.
- The section defines Gamma and Beta functions with fractal support and presents properties including a symmetry relation for the Beta function.
3 Non-local fractal derivative and integral
The paper defines non-local Riemann-Liouville and Caputo-like derivatives and integrals on fractal sets, then compares their behavior with standard non-local operators through examples.
- The section introduces non-local derivatives for functions with fractal support.
- When β = α, the fractal integral has order equal to the fractal dimension.
- Analogous left- and right-sided Riemann-Liouville fractal integrals are defined for orders satisfying n−α ≤ β < n.
- Analogous left- and right-sided Caputo fractal derivatives are then defined for functions in the stated fractal differentiability class.
- Figures 4 and 5 compare standard and fractal non-local derivatives and show a generalized fractal integral for example functions.
4 Generalized functions in the non-local calculus on the fractal subset of real-line
This section introduces generalized special functions and a non-local Laplace transformation for calculus and differential equations on fractal sets. The transformation is then developed for fractal integrals and Riemann-Liouville and Caputo derivatives.
- The section proposes mathematical tools for solving non-local fractal differential equations.
- The Gamma function is defined for fractal calculus and used in the non-local calculus on fractals.
- The generalized Mittag-Leffler function is introduced for fractal fractional differential equations.
- The generalized Laplace transformation is defined for functions with fractal support and applied to non-local differential equations on fractal sets.
- Fractal convolution and Laplace-transform formulas are developed for non-local fractal Riemann-Liouville integrals and derivatives.
- A corresponding Laplace-transform formula is also given for the non-local fractal Caputo derivative of order β ∈[0, 1).
5 Non-local fractal differential equations
The paper applies the proposed fractal Laplace methods to illustrative linear and non-local fractal differential equations. Solutions are plotted on the real-line and Cantor set, and the examples recover standard non-local fractional cases when α = 1.
- The section solves illustrative non-local fractal differential equations, including several linear equations with initial conditions.
- The first two examples formulate linear fractal equations with initial conditions and use the proposed solution procedure.
- Figures 6 and 7 present solutions of Eqs. (51) and (54) on the real-line and Cantor set.
- Figure 8 plots the solution of Eq. (57) on the Cantor set and real-line.
- The solution procedure for Eq. (60) applies the fractal Laplace transformation, performs algebraic calculations, and then uses the inverse transform.
- The solutions of Eqs. (51), (54), and (57) lead to standard non-local fractional cases when α = 1.
6 Conclusion
The work defines new non-local derivatives on fractal sets for describing complex systems with memory effects. It also reports illustrative solutions and recovery of standard non-local fractional cases when α = 1.
- The study defines new non-local derivatives on fractal sets.
- Four illustrative examples were solved in detail.
- When α = 1, the standard non-local fractional cases can be recovered.