Search results

1

Fractional thermal diffusion and the heat equation

100%

Gómez F., Morales L., González M., Alvarado V., López G.

Open Physics

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2015

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vol. 13

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issue 1

EN

Fractional calculus is the branch of mathematical analysis that deals with operators interpreted as derivatives and integrals of non-integer order. This mathematical representation is used in the description of non-local behaviors and anomalous complex processes. Fourier’s lawfor the conduction of heat exhibit anomalous behaviors when the order of the derivative is considered as 0 < β,ϒ ≤ 1 for the space-time domain respectively. In this paper we proposed an alternative representation of the fractional Fourier’s law equation, three cases are presented; with fractional spatial derivative, fractional temporal derivative and fractional space-time derivative (both derivatives in simultaneous form). In this analysis we introduce fractional dimensional parameters σx and σt with dimensions of meters and seconds respectively. The fractional derivative of Caputo type is considered and the analytical solutions are given in terms of the Mittag-Leffler function. The generalization of the equations in spacetime exhibit different cases of anomalous behavior and Non-Fourier heat conduction processes. An illustrative example is presented.

2

Posicast control of a class of fractional-order processes

100%

Gonzalez E., Hung J., Dorcak L., Terpak J., Petras I.

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EN

The number of studies on the control of fractional-order processes-processes having dynamics described by differential equations of arbitrary order-has been increasing in the past two decades and it is now ubiquitous. Various methods have emerged and have been proven to effectively control such processes-usually resulting in fractional-order controllers similar to their conventional integer-order counterparts, which include, but are not limited to fractional PID and fractional lead-lag controllers. However, such methods require a lot of computational effort and fractional-order controllers could be challenging when it comes to their synthesis and implementation. In this paper, we propose a simple yet effective delay-based controller with the use of the Posicast control methodology in controlling the overshoot of a fractional-order process of the class $$\mathcal{P}:\left\{ {P\left( s \right) = {1 \mathord{\left/ {\vphantom {1 {\left( {as^\alpha + b} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {as^\alpha + b} \right)}}} \right\}$$ having orders 1 < α < 2. Such controllers have proven to be easy to implement because they only require delays and summers. In this paper, the Posicast control methodology introduced in the past few years is modified to minimize the overshoot of the processes step response to a level that is acceptable in control engineering and automation practices. Furthermore, proof of the existence of overshoot for such class of processes, as well as the determination of the peak-time of the open-loop response of a fractional-order process of the class P is presented. Validation through numerical simulations for a class of fractional-order processes are presented in this paper.

3

Analytic solutions of the Helmholtz and Laplace equations by using local fractional derivative operators

100%

Ahmad J., Mohyud-Din S. T., Srivastava H. M., Yang X.-J.

Waves, Wavelets and Fractals

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2015

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vol. 1

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issue 1

EN

In this paper we develop analytical solutions for the Helmholtz and Laplace equations involving local fractional derivative operators. We implement the local fractional decomposition method (LFDM) for finding the exact solutions. The iteration procedure is based upon the local fractional derivative sense. The numerical results, whichwe present in this paper, show that the methodology used provides an efficient and simple tool for solving fractal phenomena arising in mathematical physics and engineering. Several illustrative examples are also provided.

4

RLC electrical circuit of non-integer order

100%

Gómez F., Rosales J., Guía M.

Open Physics

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2013

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vol. 11

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issue 10

1361-1365

EN

In this work a fractional differential equation for the electrical RLC circuit is studied. The order of the derivative being considered is 0 < γ ≤ 1. To keep the dimensionality of the physical quantities R, L and C an auxiliary parameter γ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between and σ through the physical parameters RLC of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order of the fractional differential equation.

5

Two dimensional fractional projectile motion in a resisting medium

100%

Rosales J., Guía M., Gómez F., Aguilar F., Martínez J.

Open Physics

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2014

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vol. 12

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issue 7

517-520

EN

In this paper we propose a fractional differential equation describing the behavior of a two dimensional projectile in a resisting medium. In order to maintain the dimensionality of the physical quantities in the system, an auxiliary parameter k was introduced in the derivative operator. This parameter has a dimension of inverse of seconds (sec)−1 and characterizes the existence of fractional time components in the given system. It will be shown that the trajectories of the projectile at different values of γ and different fixed values of velocity v 0 and angle θ, in the fractional approach, are always less than the classical one, unlike the results obtained in other studies. All the results obtained in the ordinary case may be obtained from the fractional case when γ = 1.

6

Analysis on the time and frequency domain for the RC electric circuit of fractional order

100%

Guía M., Gómez F., Rosales J.

Open Physics

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2013

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vol. 11

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issue 10

1366-1371

EN

This paper provides an analysis in the time and frequency domain of an RC electrical circuit described by a fractional differential equation of the order 0 < α≤ 1. We use the Laplace transform of the fractional derivative in the Caputo sense. In the time domain we emphasize on the delay, rise and settling times, while in the frequency domain the interest is in the cutoff frequency, the bandwidth and the asymptotes in low and high frequencies. All these quantities depend on the order of differential equation.

7

Fractional diffusion equation in half-space with Robin boundary condition

88%

Duan J.-S., Wang Z., Fu S.-Z.

Open Physics

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2013

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vol. 11

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issue 6

799-805

EN

The initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.

8

Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

88%

Povstenko Y.

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EN

Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.

9

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

88%

Luchko Y., Mainardi F.

Open Physics

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2013

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vol. 11

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issue 6

666-675

EN

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.

10

On LQ optimization problem subject to fractional order irregular singular systems

75%

Muhafzan, Nazra A., Yulianti L., Zulakmal, Revina R., Muhafzan, Nazra A., Yulianti L., Zulakmal, Revina R.

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2020

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vol. 30

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issue 4

Refine search results

Fractional thermal diffusion and the heat equation

Posicast control of a class of fractional-order processes

Analytic solutions of the Helmholtz and Laplace equations by using local fractional derivative operators

RLC electrical circuit of non-integer order

Two dimensional fractional projectile motion in a resisting medium

Analysis on the time and frequency domain for the RC electric circuit of fractional order

Fractional diffusion equation in half-space with Robin boundary condition

Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

On LQ optimization problem subject to fractional order irregular singular systems