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- Fractals - Mathematical complexity book - Local fractional calculus- Local fractional partial derivatives - Fractal Lagrange multipliers method - Multiple local fractional integrals - Local fractional gradient-Local fractional divergence theorem- Local fractional Stokes theorem- Green's first theorem in fractal domain- Green's second theorem in fractal domain - Fractal Riemann space - Geometry in fractal Riemann space - Fractal orthogonal coordinate tensors- Local fractional Euler–Lagrange equations - Principle of minimum potential energy in fractal medium- Principle of minimum potential energy in fractal medium - Principle of minimum complementary energy in fractal medium - J-integral formula in fractal fracture mechanics - Local fractional heat conduction equations (1)
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Articles 1 - 30 of 52
Full-Text Articles in Physics
Maxwell’S Equations On Cantor Sets: A Local Fractional Approach, Yang Xiaojun
Maxwell’S Equations On Cantor Sets: A Local Fractional Approach, Yang Xiaojun
Xiao-Jun Yang
Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell’s equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.
Approximation Solutions For Local Fractional Schrödinger Equation In The One-Dimensional Cantorian System, Xiao-Jun Yang
Approximation Solutions For Local Fractional Schrödinger Equation In The One-Dimensional Cantorian System, Xiao-Jun Yang
Xiao-Jun Yang
The local fractional Schr¨odinger equations in the one-dimensional Cantorian systemare investigated.The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative.
Helmholtz And Diffusion Equations Associated With Local Fractional Derivative Operators Involving The Cantorian And Cantor-Type Cylindrical Coordinates, Yang Xiaojun
Xiao-Jun Yang
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.
Analysis Of Fractal Wave Equations By Local Fractional Fourier Series Method, Xiao-Jun Yang
Analysis Of Fractal Wave Equations By Local Fractional Fourier Series Method, Xiao-Jun Yang
Xiao-Jun Yang
The fractal wave equations with local fractional derivatives are investigated in this paper.The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.
Systems Of Navier-Stokes Equations On Cantor Sets
Systems Of Navier-Stokes Equations On Cantor Sets
Xiao-Jun Yang
We present systems of Navier-Stokes equations on Cantor sets, which are described by the local fractional vector calculus. It is shown that the results for Navier-Stokes equations in a fractal bounded domain are efficient and accurate for describing fluid flow in fractal media.
Local Fractional Series Expansion Method For Solving Wave And Diffusion Equations On Cantor Sets, Yang Xiaojun
Local Fractional Series Expansion Method For Solving Wave And Diffusion Equations On Cantor Sets, Yang Xiaojun
Xiao-Jun Yang
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.
Mathematical Aspects Of Heisenberg Uncertainty Principle Within Local Fractional Fourier Analysis, Yang Xiaojun
Mathematical Aspects Of Heisenberg Uncertainty Principle Within Local Fractional Fourier Analysis, Yang Xiaojun
Xiao-Jun Yang
In this paper, we discuss the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. The Schrödinger equation and Heisenberg uncertainty principles are structured within local fractional operators.
Approximate Solutions For Diffusion Equations On Cantor Space-Time, Xiao-Jun Yang
Approximate Solutions For Diffusion Equations On Cantor Space-Time, Xiao-Jun Yang
Xiao-Jun Yang
In this paper we investigate diffusion equations on Cantor space-time and we obtain approximate solutions by using the local fractional Adomian decomposition method derived from the local fractional operators. Analytical solutions are given in terms of the Mittag-Leffler functions defined on Cantor sets.
Fractional Complex Transform Method For Wave Equations On Cantor Sets Within Local Fractional Differential Operator, Xiao-Jun Yang
Fractional Complex Transform Method For Wave Equations On Cantor Sets Within Local Fractional Differential Operator, Xiao-Jun Yang
Xiao-Jun Yang
This paper points out the fractional complex transform method for wave equations on Cantor sets within the local differential fractional operators. The proposed method is efficient to handle differential equations on Cantor sets.
Damped Wave Equation And Dissipative Wave Equation In Fractal Strings Within The Local Fractional Variational Iteration Method, Xiao-Jun Yang
Damped Wave Equation And Dissipative Wave Equation In Fractal Strings Within The Local Fractional Variational Iteration Method, Xiao-Jun Yang
Xiao-Jun Yang
No abstract provided.
Cantor-Type Cylindrical-Coordinate Method For Differential Equations With Local Fractional Derivatives, Xiao-Jun Yang
Cantor-Type Cylindrical-Coordinate Method For Differential Equations With Local Fractional Derivatives, Xiao-Jun Yang
Xiao-Jun Yang
In this Letter, we propose to use the Cantor-type cylindrical-coordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantorian-coordinate systems to Cantor-type cylindrical-coordinate systems.
Fractal Heat Conduction Problem Solved By Local Fractional Variation Iteration Method, Yang Xiaojun
Fractal Heat Conduction Problem Solved By Local Fractional Variation Iteration Method, Yang Xiaojun
Xiao-Jun Yang
This paper points out a novel local fractional variational iteration method for processing the local fractional heat conduction equation arising in fractal heat transfer.
A Cauchy Problem For Some Local Fractional Abstract Differential Equation With Fractal Conditions, Yang Xiaojun, Zhong Weiping, Gao Feng
A Cauchy Problem For Some Local Fractional Abstract Differential Equation With Fractal Conditions, Yang Xiaojun, Zhong Weiping, Gao Feng
Xiao-Jun Yang
Fractional calculus is an important method for mathematics and engineering [1-24]. In this paper, we review the existence and uniqueness of solutions to the Cauchy problem for the local fractional differential equation with fractal conditions \[ D^\alpha x\left( t \right)=f\left( {t,x\left( t \right)} \right),t\in \left[ {0,T} \right], x\left( {t_0 } \right)=x_0 , \] where $0<\alpha \le 1$ in a generalized Banach space. We use some new tools from Local Fractional Functional Analysis [25, 26] to obtain the results.
One-Phase Problems For Discontinuous Heat Transfer In Fractal Media, Yang Xiaojun
One-Phase Problems For Discontinuous Heat Transfer In Fractal Media, Yang Xiaojun
Xiao-Jun Yang
We first propose the fractal models for the one-phase problems of discontinuous transient heat transfer.The models are taken in sense of local fractional differential operator and used to describe the (dimensionless)melting of fractal solid semi-infinite materials initially at their melt temperatures.
Local Fractional Fourier Series With Application To Wave Equation In Fractal Vibrating String, Yang Xiaojun
Local Fractional Fourier Series With Application To Wave Equation In Fractal Vibrating String, Yang Xiaojun
Xiao-Jun Yang
We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag- Leffler function.
The Discrete Yang-Fourier Transforms In Fractal Space, Yang Xiao-Jun
The Discrete Yang-Fourier Transforms In Fractal Space, Yang Xiao-Jun
Xiao-Jun Yang
The Yang-Fourier transform (YFT) in fractal space is a generation of Fourier transform based on the local fractional calculus. The discrete Yang-Fourier transform (DYFT) is a specific kind of the approximation of discrete transform, used in Yang-Fourier transform in fractal space. This paper points out new standard forms of discrete Yang-Fourier transforms (DYFT) of fractal signals, and both properties and theorems are investigated in detail.
Expression Of Generalized Newton Iteration Method Via Generalized Local Fractional Taylor Series, Yang Xiao-Jun
Expression Of Generalized Newton Iteration Method Via Generalized Local Fractional Taylor Series, Yang Xiao-Jun
Xiao-Jun Yang
Local fractional derivative and integrals are revealed as one of useful tools to deal with everywhere continuous but nowhere differentiable functions in fractal areas ranging from fundamental science to engineering. In this paper, a generalized Newton iteration method derived from the generalized local fractional Taylor series with the local fractional derivatives is reviewed. Operators on real line numbers on a fractal space are induced from Cantor set to fractional set. Existence for a generalized fixed point on generalized metric spaces may take place.
The Zero-Mass Renormalization Group Differential Equations And Limit Cycles In Non-Smooth Initial Value Problems, Yang Xiaojun
The Zero-Mass Renormalization Group Differential Equations And Limit Cycles In Non-Smooth Initial Value Problems, Yang Xiaojun
Xiao-Jun Yang
In the present paper, using the equation transform in fractal space, we point out the zero-mass renormalization group equations. Under limit cycles in the non-smooth initial value, we devote to the analytical technique of the local fractional Fourier series for treating zero-mass renormalization group equations, and investigate local fractional Fourier series solutions.
A Novel Approach To Processing Fractal Dynamical Systems Using The Yang-Fourier Transforms, Yang Xiaojun
A Novel Approach To Processing Fractal Dynamical Systems Using The Yang-Fourier Transforms, Yang Xiaojun
Xiao-Jun Yang
In the present paper, local fractional continuous non-differentiable functions in fractal space are investigated, and the control method for processing dynamic systems in fractal space are proposed using the Yang-Fourier transform based on the local fractional calculus. Two illustrative paradigms for control problems in fractal space are given to elaborate the accuracy and reliable results.
Theory And Applications Of Local Fractional Fourier Analysis, Yang Xiaojun
Theory And Applications Of Local Fractional Fourier Analysis, Yang Xiaojun
Xiao-Jun Yang
Local fractional Fourier analysis is a generalized Fourier analysis in fractal space. The local fractional calculus is one of useful tools to process the local fractional continuously non-differentiable functions (fractal functions). Based on the local fractional derivative and integration, the present work is devoted to the theory and applications of local fractional Fourier analysis in generalized Hilbert space. We investigate the local fractional Fourier series, the Yang-Fourier transform, the generalized Yang-Fourier transform, the discrete Yang-Fourier transform and fast Yang-Fourier transform.
Heat Transfer In Discontinuous Media, Yang Xiaojun
Heat Transfer In Discontinuous Media, Yang Xiaojun
Xiao-Jun Yang
From the fractal geometry point of view, the interpretations of local fractional derivative and local fractional integration are pointed out in this paper. It is devoted to heat transfer in discontinuous media derived from local fractional derivative. We investigate the Fourier law and heat conduction equation (also local fractional instantaneous heat conduct equation) in fractal orthogonal system based on cantor set, and extent them. These fractional differential equations are described in local fractional derivative sense. The results are efficiently developed in discontinuous media.
A Short Note On Local Fractional Calculus Of Function Of One Variable, Yang Xiaojun
A Short Note On Local Fractional Calculus Of Function Of One Variable, Yang Xiaojun
Xiao-Jun Yang
Local fractional calculus (LFC) handles everywhere continuous but nowhere differentiable functions in fractal space. This note investigates the theory of local fractional derivative and integral of function of one variable. We first introduce the theory of local fractional continuity of function and history of local fractional calculus. We then consider the basic theory of local fractional derivative and integral, containing the local fractional Rolle’s theorem, L’Hospital’s rule, mean value theorem, anti-differentiation and related theorems, integration by parts and Taylor’ theorem. Finally, we study the efficient application of local fractional derivative to local fractional extreme value of non-differentiable functions, and give …
A New Successive Approximation To Non-Homogeneous Local Fractional Volterra Equation, Yang Xiaojun
A New Successive Approximation To Non-Homogeneous Local Fractional Volterra Equation, Yang Xiaojun
Xiao-Jun Yang
A new successive approximation approach to the non-homogeneous local fractional Valterra equation derived from local fractional calculus is proposed in this paper. The Valterra equation is described in local fractional integral operator. The theory of local fractional derivative and integration is one of useful tools to handle the fractal and continuously non-differentiable functions, was successfully applied in engineering problem. We investigate an efficient example of handling a non-homogeneous local fractional Valterra equation.
Advanced Local Fractional Calculus And Its Applications, Yang Xiaojun
Advanced Local Fractional Calculus And Its Applications, Yang Xiaojun
Xiao-Jun Yang
This book is the first international book to study theory and applications of local fractional calculus (LFC). It is an invitation both to the interested scientists and the engineers. It presents a thorough introduction to the recent results of local fractional calculus. It is also devoted to the application of advanced local fractional calculus on the mathematics science and engineering problems. The author focuses on multivariable local fractional calculus providing the general framework. It leads to new challenging insights and surprising correlations between fractal and fractional calculus. Keywords: Fractals - Mathematical complexity book - Local fractional calculus- Local fractional partial …
A Short Introduction To Yang-Laplace Transforms In Fractal Space, Yang Xiaojun
A Short Introduction To Yang-Laplace Transforms In Fractal Space, Yang Xiaojun
Xiao-Jun Yang
The Yang-Laplace transforms [W. P. Zhong, F. Gao, In: Proc. of the 2011 3rd International Conference on Computer Technology and Development, 209-213, ASME, 2011] in fractal space is a generalization of Laplace transforms derived from the local fractional calculus. This letter presents a short introduction to Yang-Laplace transforms in fractal space. At first, we present the theory of local fractional derivative and integral of non-differential functions defined on cantor set. Then the properties and theorems for Yang-Laplace transforms are tabled, and both the initial value theorem and the final value theorem are investigated. Finally, some applications to the wave equation …
Local Fractional Integral Equations And Their Applications, Yang Xiaojun
Local Fractional Integral Equations And Their Applications, Yang Xiaojun
Xiao-Jun Yang
This letter outlines the local fractional integral equations carried out by the local fractional calculus (LFC). We first introduce the local fractional calculus and its fractal geometrical explanation. We then investigate the local fractional Volterra/ Fredholm integral equations, local fractional nonlinear integral equations, local fractional singular integral equations and local fractional integro-differential equations. Finally, their applications of some integral equations to handle some differential equations with local fractional derivative and local fractional integral transforms in fractal space are discussed in detail.
Local Fractional Partial Differential Equations With Fractal Boundary Problems, Yang Xiaojun
Local Fractional Partial Differential Equations With Fractal Boundary Problems, Yang Xiaojun
Xiao-Jun Yang
This letter points out the new alternative approaches to processing local fractional partial differential equations with fractal boundary conditions. Applications of the local fractional Fourier series, the Yang-Fourier transforms and the Yang-Laplace transforms to solve of local fractional partial differential equations with fractal boundary conditions are investigated in detail.
Local Fractional Kernel Transform In Fractal Space And Its Applications, Yang Xiaojun
Local Fractional Kernel Transform In Fractal Space And Its Applications, Yang Xiaojun
Xiao-Jun Yang
In the present paper, we point out the local fractional kernel transform based on local fractional calculus (FLC), and its applications to the Yang-Fourier transform, the Yang-Laplace transform, the local fractional Z transform, the local fractional Stieltjes transform, the local fractional volterra/ Fredholm integral equations, the local fractional volterra/ Fredholm integro-differential equations, the local fractional variational iteration algorithms, the local fractional variational iteration algorithms with an auxiliary fractal parameter, the modified local fractional variational iteration algorithms, and the modified local fractional variational iteration algorithms with an auxiliary fractal parameter.
A New Viewpoint To Fourier Analysis In Fractal Space, Yang Xiaojun
A New Viewpoint To Fourier Analysis In Fractal Space, Yang Xiaojun
Xiao-Jun Yang
Fractional analysis is an important method for mathematics and engineering [1-21], and fractional differentiation inequalities are great mathematical topic for research [22-24]. In the present paper we point out a new viewpoint to Fourier analysis in fractal space based on the local fractional calculus [25-58], and propose the local fractional Fourier analysis. Based on the generalized Hilbert space [48, 49], we obtain the generalization of local fractional Fourier series via the local fractional calculus. An example is given to elucidate the signal process and reliable result.
Generalized Sampling Theorem For Fractal Signals, Yang Xiaojun
Generalized Sampling Theorem For Fractal Signals, Yang Xiaojun
Xiao-Jun Yang
Local fractional calculus deals with everywhere continuous but nowhere differentiable functions in fractal space. The local fractional Fourier series is a generalization of Fourier series in fractal space, and the Yang-Fourier transform is a generalization of Fourier transform in fractal space. This letter points out the generalized sampling theorem for fractal signals (local fractional continuous signals) by using the local fractional Fourier series and Yang-Fourier transform techniques based on the local fractional calculus. This result is applied to process the local fractional continuous signals.