Local Fractional Derivative Research Articles

Reduced differential transform method for partial differential equations within local fractional derivative operators

The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in this article. The reduced differential transform method is considered in the local fractional operator sense. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique to solve local fractional partial differential equations.

Analytical Approximate Solutions of Fractional Convection-Diffusion Equation by Means of Local Fractional Derivative Operators

In this article, the local fractional decomposition method ( LFDM) is applied to obtain approximate the analytical solution of nonlinear fractional convection-dif fusion. Numerical solutions obtained by local fractional decomposition method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. A new application of loca l fractional decomposition method (LFDM) was extended to reproduce the analytical solutions to this equati on in the form of a series. It is shown that the solutions obtained by the LFDM are reliable, simple and t hat LFDM is an effective method for strongly nonlinear partial equations.

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

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.

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

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.

About local fractional three-dimensional compressible Navier-Stokes equations in Cantor-type cylindrical co-ordinate system

In this article, we investigate the local fractional 3-D compressible Navier-Stokes equation via local fractional derivative. We use the Cantor-type cylindrical co-ordinate method to transfer 3-D compressible Navier-Stokes equation from the Cantorian co-ordinate system to the Cantor-type cylindrical co-ordinate system.

The local fractional iteration solution for the diffusion problem in fractal media

In this paper, we address the coupling method for the local fractional variational iteration algorithm III and local fractional Laplace transform for the first time, which is called as the local fractional Laplace transform variational iteration algorithm III. The proposed technology is used to find the local fractional iteration solution for the diffusion problem in fractal media via local fractional derivative.

The non-differentiable solution for local fractional Laplace equation in steady heat-conduction problem

In this paper, we investigate the local fractional Laplace equation in the steady heat-conduction problem. The solutions involving the non-differentiable graph are obtained by using the characteristic equation method (CEM) via local fractional derivative. The obtained results are given to present the accuracy of the technology to solve the steady heat-conduction in fractal media.

A remark on local fractional calculus and ordinary derivatives

Abstract In 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.

On three-dimensional incompressible Navier-Stokes fluid on cantor sets in spherical Cantor type co-ordinate system

This paper addresses the systems of the incompressible Navier-Stokes equations on Cantor sets without the external force involving the fractal heat-conduction problem vial local fractional derivative. The spherical Cantor type co-ordinate method is used to transfer the incompressible Navier-Stokes equation from the Cantorian co-ordinate system into the spherical Cantor type co-ordinate system.

The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator

We used the local fractional variational iteration transform method (LFVITM) coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.

On local fractional operators View of computational complexity: Diffusion and relaxation defined on cantor sets

This paper treats the description of non-differentiable dynamics occurring in complex systems governed by local fractional partial differential equations. The exact solutions of diffusion and relaxation equations with Mittag-Leffler and exponential decay defined on Cantor sets are calculated. Comparative results with other versions of the local fractional derivatives are discussed.

Local fractional functional decomposition method for solving local fractional Poisson equation in steady heat-conduction problem

The steady heat-conduction problem via local fractional derivative is investigated in this paper. The analytical solution of the local fractional Poisson equation is obtained. The local fractional functional decomposition method is proposed to find the analytical solution of the partial differential equation in the steady heat-conduction problem.

A new computational method for fractal heat-diffusion via local fractional derivative

The fractal heat-conduction problem via local fractional derivative is investigated in this paper. The solution of the fractal heat-diffusion equation is obtained. The characteristic equation method is proposed to find the analytical solution of the partial differential equation in fractal heat-conduction problem.

A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach

In this article, we first propose a new numerical technique based upon a certain two-dimensional extended differential transform via local fractional derivatives and derive its associated basic theorems and properties. One example of testing the local fractional diffusion equation is then considered. The numerical result presented here illustrates the efficiency and accuracy of the proposed computational technique in order to solve the partial differential equations involving local fractional derivatives.

Local fractional differential equations by the Exp-function method

Purpose – The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation. The exact solutions for the equation are successfully established. The paper aims to discuss these issues. Design/methodology/approach – Use the chain rule of the local fractional derivative and the exp-function method. Findings – Some new exact solutions for the fractional differential equation are successfully established, and the process of the solution is extremely simple and remarkably accessible. Originality/value – The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation.

An efficient analytical method for solving local fractional nonlinear PDEs arising in mathematical physics

In this paper, an efficient analytical technique is developed to investigate and derive the non-differentiable solutions of nonlinear PDEs with local fractional derivatives. Two illustrative examples for the transport and Fokker–Planck equations are given in order to demonstrate the efficiency and reliability of the methodology presented here.

An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives

The non-differentiable asymptotic approximation theory is first proposed in this article. One illustrative example for testing the free damped vibrations problem involving local fractional differential operator is given. The obtained result is remarkably simple and efficient to show the fractal behavior of a linear oscillator in fractal medium. Some graphical evidences are also provided.

A Local Fractional Taylor Expansion and Its Computation for Insufficiently Smooth Functions

AbstractA general fractional Taylor formula and its computation for insufficiently smooth functions are discussed. The Aitken delta square method and epsilon algorithm are implemented to compute the critical orders of the local fractional derivatives, from which more critical orders are recovered by analysing the regular pattern of the fractional Taylor formula. The Richardson extrapolation method is used to calculate the local fractional derivatives with critical orders. Numerical examples are provided to verify the theoretical analysis and the effectiveness of our approach.

Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero

In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional derivatives. The Caputo derivative of fractional order alpha of function f(x) is defined as a fractional integration of order (n-alpha) of the derivative f^(n)(x) of integer order n. The requirement of the existence of integer-order derivatives allows us to conclude that the local fractional derivative cannot be considered as the best method to describe nowhere differentiable functions and fractal objects. We also prove that unviolated Leibniz rule cannot hold for derivatives of orders alpha, which are not equal to one.

Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow

The local fractional Burgers’ equation (LFBE) is investigated from the point of view of local fractional conservation laws envisaging a nonlinear local fractional transport equation with a linear non-differentiable diffusion term. The local fractional derivative transformations and the LFBE conversion to a linear local fractional diffusion equation are analyzed.