Temperature Distribution In An Oscillatory Flow With A Sinusoidal Wall Temperature, 2012 University of Texas at Austin

#### Temperature Distribution In An Oscillatory Flow With A Sinusoidal Wall Temperature, Eduardo Ramos, Brian Storey, Fernando Sierra, Raul Zuniga, Andriy Avramenko

*Brian Storey*

The temperature field generated by an oscillatory boundary layer flow in the presence of a wall with a sinusoidal temperature distribution is analyzed. A linear perturbation method is used to find closed form analytical solutions for the temperature field when the amplitude of the velocity oscillation is small. The analytical solutions only consider long-time behavior when the temperature fields oscillate with the frequency of the flow. The structure of the equation that governs the temperature correction due to convection is similar to that of diffusive waves with the solution consisting of traveling or standing waves. The temperature distribution is also ...

'Kinetic Sculptures': A Centerpiece Project Integrated With Mathematics And Physics, 2012 Olin College of Engineering

#### 'Kinetic Sculptures': A Centerpiece Project Integrated With Mathematics And Physics, Yevgeniya Zastavker, Jill Crisman, Mark Jeunnette, Burt Tilley

*Yevgeniya V. Zastavker*

An integrated set of courses, or Integrated Course Block (ICB), developed for incoming first-year students at the Franklin W. Olin College of Engineering, is presented. Bound by a common theme of `Kinetic Sculptures', the individual courses in this ICB are mathematics (single variable calculus and ordinary differential equations), physics (kinetics and dynamics of linear and rotational motion, thermodynamics and fluids), and an open-ended engineering project. The project part of the ICB allows students to explore the motion through the design of kinetic (moving) sculptures while utilizing the mathematics and physics concepts learned in the accompanying courses. This paper considers the ...

Short Period Gravity Waves In The Arctic Atmosphere Over Alaska, 2012 Utah State University

#### Short Period Gravity Waves In The Arctic Atmosphere Over Alaska, Michael Negale, Kim Nielsen, Michael J. Taylor, Britta Irving, Richard Collins

*Physics Student Research*

The propagation nature and sources of short-period gravity waves have been studied extensively at low and mid-latitudes, while their extent and nature at the polar regions are less known. During the last decade, observations from select sites on the Antarctic continent have revealed a significant presence of these waves over the southern Polar Region as well as shown unexpected dynamical behavior. In contrast, observations over the Arctic region are few and the dynamical behavior is unknown. A recent project was initiated in January 2011 to investigate the presence and dynamics of these waves over interior Alaska. This site provides an ...

The Discrete Yang-Fourier Transforms In Fractal Space, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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.

Multiple Equilibrium States In A Micro-Vascular Network, 2012 Franklin W. Olin College of Engineering

#### Multiple Equilibrium States In A Micro-Vascular Network, David Gardner, Yiyang Li, Benjamin Small, John Geddes, Russell Carr

*John B. Geddes*

We use a simple model of micro-vascular blood flow to explore conditions that give rise to multiple equilibrium states in a three-node micro-vascular network. The model accounts for two primary rheological effects: the Fåhræus–Lindqvist effect, which describes the apparent viscosity of blood in a vessel, and the plasma skimming effect, which governs the separation of red blood cells at diverging nodes. We show that multiple equilibrium states are possible, and we use our analytical and computational tools to design an experiment for validation.

The Zero-Mass Renormalization Group Differential Equations And Limit Cycles In Non-Smooth Initial Value Problems, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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 ...

A New Successive Approximation To Non-Homogeneous Local Fractional Volterra Equation, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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 derivatives - Fractal ...

A Short Introduction To Yang-Laplace Transforms In Fractal Space, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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, 2012 China University of Mining & Technology

#### 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.

Picard’S Approximation Method For Solving A Class Of Local Fractional Volterra Integral Equations, 2012 China University of Mining & Technology

#### Picard’S Approximation Method For Solving A Class Of Local Fractional Volterra Integral Equations, Yang Xiaojun

*Xiao-Jun Yang*

In this letter, we fist consider the Picard’s successive approximation method for solving a class of the Volterra integral equations in local fractional integral operator sense. Special attention is devoted to the Picard’s successive approximate methodology for handling local fractional Volterra integral equations. An illustrative paradigm is shown the accuracy and reliable results.