Open Access Published by De Gruyter Open Access

Volume 11 Issue 10

October 2013

Issue of Open Physics

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Contents
Journal Overview

Open Access December 19, 2013

Fractional calculus: theory and numerical methods

Luis Vázquez, Hossein Jafari

Page range: 1163-1163

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Open Access December 19, 2013

On the multi-index (3m-parametric) Mittag-Leffler functions, fractional calculus relations and series convergence

Jordanka Paneva-Konovska

Page range: 1164-1177

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Abstract

In this paper we consider a family of 3m-indices generalizations of the classical Mittag-Leffler function, called multi-index (3m-parametric) Mittag-Leffler functions. We survey the basic properties of these entire functions, find their order and type, and new representations by means of Mellin-Barnes type contour integrals, Wright pΨq-functions and Fox H-functions, asymptotic estimates. Formulas for integer and fractional order integration and differentiations are found, and these are extended also for the operators of the generalized fractional calculus (multiple Erdélyi-Kober operators). Some interesting particular cases of the multi-index Mittag-Leffler functions are discussed. The convergence of series of such type functions in the complex plane is considered, and analogues of the Cauchy-Hadamard, Abel, Tauber and Littlewood theorems are provided.

Open Access December 19, 2013

Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

Yufeng Xu, Om Agrawal

Page range: 1178-1193

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Abstract

In this paper we study a class of new Generalized Fractional Advection-Diffusion Equations (GFADEs) with a new Generalized Fractional Derivative (GFD) proposed last year. The new GFD is defined in the Caputo sense using a weight function and a scale function. The GFADE is discussed in a bounded domain, and numerical solutions for two examples consisting of a linear and a nonlinear GFADE are obtained using an implicit finite difference approach. The stability of the numerical scheme is investigated, and the order of convergence is estimated numerically. Numerical results illustrate that the finite difference scheme is simple and effective for solving the GFADEs. We investigate the influence of weight and scale functions on the diffusion of GFADEs. Linear and nonlinear stretching and contracting functions are considered. It is found that an increasing weight function increases the rate of diffusion, and a scale function can stretch or contract the diffusion on the time domain.

Open Access December 19, 2013

Vectorial fractional integral inequalities with convexity

George Anastassiou

Page range: 1194-1211

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Abstract

Here we present vectorial general integral inequalities involving products of multivariate convex and increasing functions applied to vectors of functions. As specific applications we derive a wide range of vectorial fractional inequalities of Hardy type. These involve the left and right: Erdélyi-Kober fractional integrals, mixed Riemann-Liouville fractional multiple integrals. Next we produce multivariate Poincaré type vectorial fractional inequalities involving left fractional radial derivatives of Canavati type, Riemann-Liouville and Caputo types. The exposed inequalities are of L p type, p ≥ 1, and exponential type.

Open Access December 19, 2013

On the origin of space

Richard Herrmann

Page range: 1212-1220

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Abstract

Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a model is presented, which describes space generation as a dynamic process, where the dimension d of space evolves smoothly with time in the range 0 ≤ d(t) ≤ 3, where the lower and upper boundaries of dimension are derived from first principles. It is demonstrated, that a minimum threshold for the space dimension is necessary to establish an interaction with external probe particles. A possible application in cosmology is suggested.

Open Access December 19, 2013

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Fawang Liu, Shiping Chen, Ian Turner, Kevin Burrage, Vo Anh

Page range: 1221-1232

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Abstract

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

Open Access December 19, 2013

Linear discrete systems with memory: a generalization of the Langmuir model

Dumitru Băleanu, Raoul Nigmatullin

Page range: 1233-1237

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Abstract

In this manuscript we analyzed a general solution of the linear nonlocal Langmuir model within time scale calculus. Several generalizations of the Langmuir model are presented together with their exact corresponding solutions. The physical meaning of the proposed models are investigated and their corresponding geometries are reported.

Open Access December 19, 2013

Dynamical process of complex systems and fractional differential equations

Hiroaki Hara, Yoshiyasu Tamura

Page range: 1238-1245

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Abstract

Behavior of dynamical process of complex systems is investigated. Specifically we analyse two types of ideal complex systems. For analysing the ideal complex systems, we define the response functions describing the internal states to an external force. The internal states are obtained as a relaxation process showing a “power law” distribution, such as scale free behaviors observed in actual measurements. By introducing a hybrid system, the logarithmic time, and double logarithmic time, we show how the “slow relaxation” (SR) process and “super slow relaxation” (SSR) process occur. Regarding the irregular variations of the internal states as an activation process, we calculate the response function to the external force. The behaviors are classified into “power”, “exponential”, and “stretched exponential” type. Finally we construct a fractional differential equation (FDE) describing the time evolution of these complex systems. In our theory, the exponent of the FDE or that of the power law distribution is expressed in terms of the parameters characterizing the structure of the system.

Open Access December 19, 2013

A fractional approach to the Sturm-Liouville problem

Margarita Rivero, Juan Trujillo, M. Velasco

Page range: 1246-1254

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Abstract

The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order. For each fractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a particular example that evidences the applicability of the fractional Sturm-Liouville theory.

Open Access December 19, 2013

Diffusion problems on fractional nonlocal media

Alberto Sapora, Pietro Cornetti, Alberto Carpinteri

Page range: 1255-1261

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Abstract

In this paper, the nonlocal diffusion in one-dimensional continua is investigated by means of a fractional calculus approach. The problem is set on finite spatial domains and it is faced numerically by means of fractional finite differences, both for what concerns the transient and the steady-state regimes. Nonlinear deviations from classical solutions are observed. Furthermore, it is shown that fractional operators possess a clear physical-mechanical meaning, representing conductors, whose conductance decays as a power-law of the distance, connecting non-adjacent points of the body.

Open Access December 19, 2013

A discrete time method to the first variation of fractional order variational functionals

Shakoor Pooseh, Ricardo Almeida, Delfim Torres

Page range: 1262-1267

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Abstract

The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grünwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations.

Open Access December 19, 2013

Dynamics of a backlash chain

José Machado

Page range: 1268-1274

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Abstract

This paper studies the dynamical properties of a system with distributed backlash and impact phenomena. This means that it is considered a chain of masses that interact with each other solely by means of backlash and impact phenomena. It is observed the emergence of non-linear phenomena resembling those encountered in the Fermi-Pasta-Ulam problem.

Open Access December 19, 2013

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

Hala Hejazi, Timothy Moroney, Fawang Liu

Page range: 1275-1283

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Abstract

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

Open Access December 19, 2013

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

Yuriy Povstenko

Page range: 1284-1294

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Abstract

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.

Open Access December 19, 2013

Fractional nonlinear systems with sequential operators

Dorota Mozyrska, Ewa Girejko, Małgorzata Wyrwas

Page range: 1295-1303

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Abstract

In the paper possible approximation of solutions to initial value problems stated for fractional nonlinear equations with sequential derivatives of Caputo type is presented. We proved that values of Caputo derivatives in continuous case can be approximated by corresponding values of h-difference operators with h being small enough. Numerical examples are presented.

Open Access December 19, 2013

Exact solution for the fractional cable equation with nonlocal boundary conditions

Emilia Bazhlekova, Ivan Dimovski

Page range: 1304-1313

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Abstract

The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.

Open Access December 19, 2013

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

Virginia Kiryakova, Yuri Luchko

Page range: 1314-1336

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Abstract

In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ 1,…,δ m) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.

Open Access December 19, 2013

Singular fractional evolution differential equations

Mirjana Stojanovic

Page range: 1337-1349

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Abstract

We give an existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions using fractional analog for Duhamel principle. Paper deals with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes which have met great interest among researchers who consider them as a challenge in recent years.

Open Access December 19, 2013

An expansion formula for fractional derivatives of variable order

Teodor Atanackovic, Marko Janev, Stevan Pilipovic, Dusan Zorica

Page range: 1350-1360

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Abstract

In this work we extend our previous results and derive an expansion formula for fractional derivatives of variable order. The formula is used to determine fractional derivatives of variable order of two elementary functions. Also we propose a constitutive equation describing a solidifying material and determine the corresponding stress relaxation function.

Open Access December 19, 2013

RLC electrical circuit of non-integer order

Francisco Gómez, Juan Rosales, Manuel Guía

Page range: 1361-1365

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Abstract

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.

Open Access December 19, 2013

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

Manule Guía, Francisco Gómez, Juan Rosales

Page range: 1366-1371

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Abstract

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.

Open Access December 19, 2013

Numerical solution of fractional differential equations by using fractional B-spline

Hossein Jafari, Chaudry Khalique, Mohammad Ramezani, Haleh Tajadodi

Page range: 1372-1376

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Abstract

In this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work.

Open Access December 19, 2013

Existence and approximation of solutions of fractional order iterative differential equations

JianHua Deng, JinRong Wang

Page range: 1377-1386

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Abstract

In this paper, we investigate existence and approximation of solutions of fractional order iterative differential equations by virtue of nonexpansive mappings, fractional calculus and fixed point methods. Three existence theorems as well as convergence theorems for a fixed point iterative method designed to approximate these solutions are obtained in two different work spaces via Chebyshev’s norm, Bielecki’s norm and β norm. Finally, an example is given to illustrate the obtained results.

Open Access December 19, 2013

Two methods to solve a fractional single phase moving boundary problem

Xicheng Li, Shaowei Wang, Moli Zhao

Page range: 1387-1391

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Abstract

A moving boundary problem of a melting problem is considered in this study. A mathematical model using the Caputo fractional derivative heat equation is proposed in the paper. Since moving boundary problems are difficult to solve for the exact solution, two methods are presented to approximate the evolution of the temperature. To simplify the computation, a similarity variable is adopted in order to reduce the partial differential equations to ordinary ones.

Open Access December 19, 2013

Variational iteration method — a promising technique for constructing equivalent integral equations of fractional order

Yi-Hong Wang, Guo-Cheng Wu, Dumitru Baleanu

Page range: 1392-1398

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Abstract

The variational iteration method is newly used to construct various integral equations of fractional order. Some iterative schemes are proposed which fully use the method and the predictor-corrector approach. The fractional Bagley-Torvik equation is then illustrated as an example of multi-order and the results show the efficiency of the variational iteration method’s new role.

Open Access December 19, 2013

Nonlocal Cauchy problems for fractional order nonlinear differential systems

JinRong Wang, Xuezhu Li, Yong Zhou

Page range: 1399-1413

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Abstract

In this paper, we discuss nonlocal Cauchy problems for fractional order nonlinear differential systems. Firstly, an important matrix associated with fractional order and two functionals are constructed. Further, some sufficient conditions which guarantee such matrix convergent to zero matrix are presented. Secondly, by using three fixed point theorems via the techniques that use convergent to zero matrix and vector norm, some existence results for the solutions of such fractional order nonlinear differential systems are given under different conditions. Finally, some examples are given to illustrate the results.

Open Access December 19, 2013

Fractional-order TV-L2 model for image denoising

Dali Chen, Shenshen Sun, Congrong Zhang, YangQuan Chen, Dingyu Xue

Page range: 1414-1422

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Abstract

This paper proposes a new fractional order total variation (TV) denoising method, which provides a much more elegant and effective way of treating problems of the algorithm implementation, ill-posed inverse, regularization parameter selection and blocky effect. Two fractional order TV-L2 models are constructed for image denoising. The majorization-minimization (MM) algorithm is used to decompose these two complex fractional TV optimization problems into a set of linear optimization problems which can be solved by the conjugate gradient algorithm. The final adaptive numerical procedure is given. Finally, we report experimental results which show that the proposed methodology avoids the blocky effect and achieves state-of-the-art performance. In addition, two medical image processing experiments are presented to demonstrate the validity of the proposed methodology.

Open Access December 19, 2013

Existence of positive solutions for nonlocal boundary value problem of fractional differential equation

Xiping Liu, Legang Lin, Haiqin Fang

Page range: 1423-1432

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Abstract

In this paper, we study a type of nonlinear fractional differential equations multi-point boundary value problem with fractional derivative in the boundary conditions. By using the upper and lower solutions method and fixed point theorems, some results for the existence of positive solutions for the boundary value problem are established. Some examples are also given to illustrate our results.

Open Access December 19, 2013

Numerical approach to the Caputo derivative of the unknown function

Fanhai Zeng, Changpin Li

Page range: 1433-1439

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Abstract

If a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method.

Open Access December 19, 2013

Finite difference scheme for the time-space fractional diffusion equations

Jianxiong Cao, Changpin Li

Page range: 1440-1456

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Abstract

In this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.

Open Access December 19, 2013

On the generating function e xt+yϕ(t) and its fractional calculus

Alireza Ansari, Amirhossein Sheikhani, Sohrab Kordrostami

Page range: 1457-1462

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Abstract

In this article, we derive the coefficient set {H m(x,y)}m=1∞ using the generating function ext+yϕ(t). When the complex function ϕ(t) is entire, using the inverse Mellin transform, and when ϕ(t) has singular points, using the inverse Laplace transform, the coefficient set is obtained. Also, bi-orthogonality of this set with its associated functions and its applications in the explicit solutions of partial fractional differential equations is discussed.

Open Access December 19, 2013

Legendre multiwavelet collocation method for solving the linear fractional time delay systems

Sohrab Yousefi, Ali Lotfi

Page range: 1463-1469

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Abstract

In this article the Legendre multiwavelet basis with aid of collocation method has been applied to give approximate solution for fractional delay systems. The properties of Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of algebraic system. Numerical results and comparison with exact solutions in the cases when we have exact solution are given in test examples in order to demonstrate the applicability and efficiency of the method.

Open Access December 19, 2013

Numerical solution of fractional differential equations via a Volterra integral equation approach

Shahrokh Esmaeili, Mostafa Shamsi, Mehdi Dehghan

Page range: 1470-1481

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Abstract

The main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.

Open Access December 19, 2013

Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation

Hossein Jafari, Haleh Tajadodi, Dumitru Baleanu, Abdulrahim Al-Zahrani, Yahia Alhamed, Adnan Zahid

Page range: 1482-1486

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Abstract

In this paper the fractional sub-equation method is used to construct exact solutions of the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation.The fractional derivative is described in the Jumarie’s modified Riemann-Liouville sense. Two illustrative examples are given, showing the accuracy and convenience of the method.

Open Access December 19, 2013

Existence of solutions for sequential fractional differential equations with four-point nonlocal fractional integral boundary conditions

Bashir Ahmad, Ahmed Alsaedi, Hana Al-Hutami

Page range: 1487-1493

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Abstract

This paper investigates the existence of solutions for a nonlinear boundary value problem of sequential fractional differential equations with four-point nonlocal Riemann-Liouville type fractional integral boundary conditions. We apply Banach’s contraction principle and Krasnoselskii’s fixed point theorem to establish the existence of results. Some illustrative examples are also presented.

Open Access December 19, 2013

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Eid Doha, Ali Bhrawy, Samer Ezz-Eldien

Page range: 1494-1503

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Abstract

In this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.

Open Access December 19, 2013

Reduced-order anti-synchronization of the projections of the fractional order hyperchaotic and chaotic systems

Mayank Srivastava, Saurabh Agrawal, Subir Das

Page range: 1504-1513

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Abstract

The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.

Open Access December 19, 2013

Dynamics analysis of fractional order Yu-Wang system

Sachin Bhalekar

Page range: 1514-1522

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Abstract

Fractional order version of a dynamical system introduced by Yu and Wang (Engineering, Technology & Applied Science Research, 2, (2012) 209–215) is discussed in this article. The basic dynamical properties of the system are studied. Minimum effective dimension 0.942329 for the existence of chaos in the proposed system is obtained using the analytical result. For chaos detection, we have calculated maximum Lyapunov exponents for various values of fractional order. Feedback control method is then used to control chaos in the system. Further, the system is synchronized with itself and with fractional order financial system using active control technique. Modified Adams-Bashforth-Moulton algorithm is used for numerical simulations.

Open Access December 19, 2013

Homotopy analysis method for solving Abel differential equation of fractional order

Hossein Jafari, Khosro Sayevand, Haleh Tajadodi, Dumitru Baleanu

Page range: 1523-1527

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Abstract

In this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.

Open Access December 19, 2013

Existence and uniqueness of a complex fractional system with delay

Rabha Ibrahim, Hamid Jalab

Page range: 1528-1535

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Abstract

Chaotic complex systems are utilized to characterize thermal convection of liquid flows and emulate the physics of lasers. This paper deals with the time-delay of a complex fractional-order Liu system. We have examined its chaos, computed numerical solutions and established the existence and uniqueness of those solutions. Ultimately, we have presented some examples.

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