State space approach to thermoelectric fluid with fractional order heat transfer

A new mathematical model for electromagnetic thermofluid equation heat transfer with thermoelectric properties using the methodology of fractional calculus is constructed. The governing coupled equations in the frame 11 of the boundary layer model are applied to variety problems. Laplace transforms and state space techniques (Ezzat Can J Phys Rev 86:1241–1250 in 2008) are used to get the solution of a thermal shock problem, a layer problem and a problem for the semi-infinite space in the presence of heat sources. According to the numerical results and its graphs, a parametric study of time-fractional order 0 < α ≤ 1, on temperature and the thermoelectric figure of merit are conducted.     

Approximating fractional derivatives in the perspective of system control

The theory of fractional calculus goes back to the beginning of the theory of differential calculus, but its application received attention only recently. In the area of automatic control some work was developed, but the proposed algorithms are still in a research stage. This paper discusses a novel method, with two degrees of freedom, for the design of fractional discrete-time derivatives. The performance of several approximations of fractional derivatives is investigated in the perspective of nonlinear system control.     

On the MHD flow of fractional generalized Burgers＇ fluid with modified Darcy＇s law

This work is concerned with applying the fractional calculus approach to the magnetohydrodynamic (MHD) pipe flow of a fractional generalized Burgers’ fluid in a porous space by using modified Darcy’s relationship. The fluid is electrically conducting in the presence of a constant applied magnetic field in the transverse direction. Exact solution for the velocity distribution is developed with the help of Fourier transform for fractional calculus. The solutions for a Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the present analysis.     

GENERAL SECOND ORDER FLUID FLOW IN A PIPE

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KNOWLEDGE AND APPLICATION: FURTHER DISCUSSIONS ON THE COURSE OF CALCULUS IN THEORETICAL AND APPLIED MECHANICS CURRICULUN

讨论了力学专业中微积分教学的目的、方法,是计划在北京大学出版社出版的《微积分导引(下册)》的前言。     

Geometric calculus for second-order differential equations and generalizations of the inverse problem of Lagrangian mechanics

We review the main features of the geometric calculus which has been introduced over the past 15 years in the study of second-order ordinary differential equations and then explain how a recently introduced generalization of the inverse problem of Lagrangian mechanics can be very concisely dealt with by this calculus in an intrinsic way. This paper is an account of the lecture with the same title presented at the ICDVC-2010 Conference in Hangzhou, May 12–14 (2010).     

Managements of scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials

Nonlinear Dynamics - This paper proposes a series of fractional-order control methods (FOCMs) based on fractional calculus (FC) for a class of general nonlinear systems. In order to deal with the...     

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.     

Comments on time-varying fractional order

Nonlinear Dynamics - The calculus of arbitrary order, known as the fractional calculus, allows for real- and complex-valued orders. Interest in time-varying order is growing in order to study...     

Chaotic oscillation of a nonlinear power system

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Plasmonic cloaking and scattering cancelation for electromagnetic and acoustic waves

One popular approach to cloaking objects from electromagnetic waves at moderately long wavelengths is the scattering cancelation technique. This mechanism is based on the use of a single homogeneous thin layer to cover an object of interest in order to provide scattering suppression in a given frequency band. This approach has also been recently extended to acoustic waves. This paper provides an investigation of the physical nature of scattering cancelation by a uniform thin layer for both electromagnetic and acoustic waves in inviscid fluids. Two distinct scattering cancelation regions are obtained within the available parameter space: a non-resonant plasmonic cloaking region and an anti-resonant cloaking region, which are identified and compared in both the electromagnetic and acoustic domains. Although both types of operations allow for the suppression of the dominant scattering orders, the resulting internal fields and physical functionality of the cloaks present distinct differences between the two domains. We discuss analogies and differences between these functionalities and their implications in electromagnetic and acoustic cloaking problems, with an insight into their practical implementation.     

Integral Representations of Increments of Stochastic Processes

A formulation is presented in which the increment of a stochastic process is represented as an integral of the derivative of the process. It is shown that this representation is an alternative to the more common approach of writing equations for the differentials of stochastic processes. A possible advantage of the integral formulation is that its reliance on derivatives, rather than differentials, makes the operations of stochastic calculus more closely resemble those of ordinary deterministic calculus. This similarity to well-known mathematics may serve to make stochastic calculus accessible to a broader audience than in the past. The integral formulation is herein shown to be compatible with the Itô differential rule for non-Gaussian processes and is used to describe the increment of the nonstationary response of a system governed by a vector stochastic equation with parametric delta-correlated excitation.     

Generalized Euler–Lagrange equation for nonsmooth calculus of variations

In this paper, we first introduce a novel generalized derivative and obtain the generalized first-order Taylor expansion of the nonsmooth functions. Then we derive the generalized Euler–Lagrange equation for the nonsmooth calculus of variations and solve this equation by using Chebyshev pseudospectral method, approximately. Finally, the optimal solutions of some problems in the nonsmooth calculus of variations are approximated.     

Predefined-time control of distributed-order systems

Nonlinear Dynamics - Distributed-order calculus can be understood as a further generalisation of integer- and fractional-order calculus. Such a general case allows the modelling and understanding...     

Two sample applications of a classical isoperimetric problem of the calculus of variations to fluid mechanical problems in fluidization and spouting

This paper is devoted to outlining precisely the basic mathematics of a classical isoperimetric problem of the calculus of variations and showing how significant fluid mechanical problems in fluidization and spouting can be addressed using this approach.     

An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index

In this paper, we consider the main problem of variational calculus when the derivatives are Riemann?CLiouville-type fractional with incommensurate orders in general. As the most general form of the performance index, we consider a fractional integral form for the functional that is to be extremized. In the light of fractional calculus and fractional integration by parts, we express a generalized problem of the calculus of variations, in which the classical problem is a special case. Considering five cases of the problem (fixed, free, and dependent final time and states), we derive a necessary condition which is an extended version of the classical Euler?CLagrange equation. As another important result, we derive the necessary conditions for an optimization problem with piecewise smooth extremals where the fractional derivatives are not necessarily continuous. The latter result is valid only for the integer order for performance index. Finally, we provide some examples to clarify the effectiveness of the proposed theorems.     

Particle swarm optimization with fractional-order velocity

This paper proposes a novel method for controlling the convergence rate of a particle swarm optimization algorithm using fractional calculus (FC) concepts. The optimization is tested for several well-known functions and the relationship between the fractional order velocity and the convergence of the algorithm is observed. The FC demonstrates a potential for interpreting evolution of the algorithm and to control its convergence.     

Unsteady Rotating Flows of a Viscoelastic Fluid with the Fractional Maxwell Model Between Coaxial Cylinders

The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.The project supported by the National Natural Science Foundation of China (10272067, 10426024), the Doctoral Program Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.