Feng Gao

Local Fractional Newton's Method Derived from Modified Local Fractional Calculus

A local fractional Newtons method, which is derived from the modified local fractional calculus, is proposed in the present paper. Its iterative function is obtained and the convergence of the iterative function is discussed. The comparison between the classical Newton iteration and the local fractional Newton iteration has been carried out. It is shown that the iterative value of the local fractional Newton method better approximates the real-value than that of the classical one.

Fundamentals of Local Fractional Iteration of the Continuously Nondifferentiable Functions Derived from Local Fractional Calculus

A new possible modeling for the local fractional iteration process is proposed in this paper. Based on the local fractional Taylor’s series, the fundamentals of local fractional iteration of the continuously non-differentiable functions are derived from local fractional calculus in fractional space.

A new fractional derivative involving the normalized sinc function without singular kernel

AbstractnIn this paper, a new fractional derivative involving the normalized sinc function without singular kernel is proposed. The Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems. The comparative results between classical and fractional-order operators are presented. The results are significant in the analysis of one-dimensional anomalous heat-transfer problems.n

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

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.

Exact Travelling Wave Solutions for Local Fractional Partial Differential Equations in Mathematical Physics

In the article, we investigate the exact travelling wave solutions for the linear and nonlinear local fractional partial differential equations. The non-differential exact solutions of the fractal diffusion, Korteweg-de Vries, and Boussinesq equations via local fractional derivative are discussed in detail. The local fractional calculus formulations are efficient in description of fractal and complex behaviors of the linear and nonlinear mathematical physics.

Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions

In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to demonstrate the efficiency of the proposed formulations to describe the anomalous advection dispersion processes.