scholarly journals Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Chun-Ying Long ◽  
Yang Zhao ◽  
Hossein Jafari
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Mathematical Models ◽  
Cantor Sets ◽  
Forest Gap ◽  
Local Fractional Derivative ◽  
Local Fractional Calculus ◽  
Growth Equations ◽  
Gap Models ◽  
Individual Trees

The forest new gap models via local fractional calculus are investigated. The JABOWA and FORSKA models are extended to deal with the growth of individual trees defined on Cantor sets. The local fractional growth equations with local fractional derivative and difference are discussed. Our results are first attempted to show the key roles for the nondifferentiable growth of individual trees.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

scholarly journals Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
H. M. Srivastava ◽  
Alireza Khalili Golmankhaneh ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Coupling Method ◽  
Initial Value ◽  
Local Fractional Calculus ◽  
Sumudu Transform

Local fractional derivatives were investigated intensively during the last few years. The coupling method of Sumudu transform and local fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper. The presented method is applied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative. The obtained results are given to show the advantages.

scholarly journals Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Dumitru Baleanu ◽  
J. A. Tenreiro Machado ◽  
Carlo Cattani ◽  
Mihaela Cristina Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Decomposition Methods ◽  
Cantor Set ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Local Fractional Derivative ◽  
Variational Iteration

We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

scholarly journals Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ya-Juan Hao ◽  
H. M. Srivastava ◽  
Hossein Jafari ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Cantor Sets ◽  
Diffusion Equations ◽  
Cylindrical Coordinates ◽  
Coordinate Method ◽  
Local Fractional Derivative ◽  
Cylindrical Coordinate ◽  
Fractional Differential ◽  
Fractional Derivative Operators ◽  
And Diffusion

The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates.

scholarly journals Signal Processing for Nondifferentiable Data Defined on Cantor Sets: A Local Fractional Fourier Series Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi-Yong Chen ◽  
Carlo Cattani ◽  
Wei-Ping Zhong
Keyword(s):  
Signal Processing ◽  
Fourier Series ◽  
Fractional Calculus ◽  
Present Method ◽  
Continuous Functions ◽  
Cantor Sets ◽  
Point Of View ◽  
Local Fractional Calculus ◽  
Processing Point

From the signal processing point of view, the nondifferentiable data defined on the Cantor sets are investigated in this paper. The local fractional Fourier series is used to process the signals, which are the local fractional continuous functions. Our results can be observed as significant extensions of the previously known results for the Fourier series in the framework of the local fractional calculus. Some examples are given to illustrate the efficiency and implementation of the present method.

scholarly journals NON-DIFFERENTIABLE EXACT SOLUTIONS FOR THE NONLINEAR ODES DEFINED ON FRACTAL SETS

Fractals ◽  
2017 ◽  
Vol 25 (04) ◽  
pp. 1740002 ◽  
Author(s):  
XIAO-JUN YANG ◽  
FENG GAO ◽  
H. M. SRIVASTAVA
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Special Functions ◽  
Cantor Sets ◽  
Nonlinear Odes ◽  
Fractal Sets ◽  
Local Fractional Derivative

In the present paper, a family of the special functions via the celebrated Mittag–Leffler function defined on the Cantor sets is investigated. The nonlinear local fractional ODEs (NLFODEs) are presented by following the rules of local fractional derivative (LFD). The exact solutions for these problems are also discussed with the aid of the non-differentiable charts on Cantor sets. The obtained results are important for describing the characteristics of the fractal special functions.

scholarly journals Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Shao-Hong Yan ◽  
Xiao-Hong Chen ◽  
Gong-Nan Xie ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Present Method ◽  
Decomposition Method ◽  
Cantor Set ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Fokker Planck Equations ◽  
Fokker Planck

The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

scholarly journals A periodic solution for the local fractional Boussinesq equation on cantor sets

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3719-3723
Author(s):  
Xiu-Rong Guo ◽  
Gui-Lei Chen ◽  
Mei Guo ◽  
Zheng-Tao Liu
Keyword(s):  
Periodic Solution ◽  
Fractional Derivative ◽  
Boussinesq Equation ◽  
Generalized Functions ◽  
Cantor Sets ◽  
Symmetry Condition ◽  
Direct Integration ◽  
Local Fractional Derivative

In this paper, the periodic solution for the local fractional Boussinesq equation can be obtained in the sense of the local fractional derivative. It is given by applying direct integration with symmetry condition. Meanwhile, the periodic solution of the non-differentiable type with generalized functions defined on Cantor sets is analyzed. As a result, we have a new point to look the local fractional Boussinesq equation through the local fractional derivative theory.

scholarly journals Fractal complex transform technology for fractal Kkorteweg-de Vries equation within a local fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 841-845
Author(s):  
Jinze Xu ◽  
Zeng-Shun Chen ◽  
Jian-Hong Wang ◽  
Ping Cui ◽  
Yunru Bai
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Vries Equation ◽  
Special Functions ◽  
Traveling Wave Solutions ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Linear Partial Differential Equations ◽  
Wave Solutions ◽  
De Vries

In this paper, we present the fractal complex transform via a local fractional derivative. The traveling wave solutions for the fractal Korteweg-de Vries equations within local fractional derivative are obtained based on the special functions defined on Cantor sets. The technology is a powerful tool for solving the local fractional non-linear partial differential equations.

Sign in / Sign up

Export Citation Format

Share Document