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