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 A local fractional derivative with applications to fractal relaxation and diffusion phenomena

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 723-727 ◽  
Author(s):  
Yang Zhao ◽  
Yan-Guang Cai ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Cantor Sets ◽  
Diffusion Equations ◽  
Exponential Functions ◽  
Local Fractional Derivative ◽  
Diffusion Phenomena ◽  
And Diffusion

In this paper, a new application of the fractal complex transform via a local fractional derivative is presented. The solution for the fractal relaxation and time-fractal diffusion equations are obtained based on the sup-exponential functions defined on Cantor sets.

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 Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ai-Min Yang ◽  
Xiao-Jun Yang ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Series Expansion ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Expansion Method ◽  
Cantor Sets ◽  
Diffusion Equations ◽  
And Diffusion ◽  
Series Expansion Method

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

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 A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets

2019 ◽  
Vol 3 (2) ◽  
pp. 30 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Kamil Jassim
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Cantor Sets ◽  
New Technique ◽  
Helmholtz Equations ◽  
Homotopy Perturbation ◽  
Fractional Derivative Operators ◽  
A New Technique

In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

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.

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