linear approximation of transfer function with a pole of fractional power two methods for the linear approximation of a transfer function with a pole of fractional power are presented analog circuit models are developed and their frequency response curves and step response curves are compared it was found that the pade method gives a better approximation than wang and hsia s method within the frequency limit as specified

homotopy perturbation method for quadratic riccati differential equation and comparison with adomian s decomposition method abstract in this paper a homotopy perturbation method is proposed to solve quadratic riccati differential equation comparisons are made between adomian s decomposition method adm and the exact solution and the proposed method the results reveal that the proposed method is very effective and simple

brief state space representation for fractional order controllers this article proposes an infinite dimensional state space realization for linear filters with transfer function c d s c 0 1 s w b 1 s w h d where 0 w b w h and 0

solution of the prey and predator problem by homotopy perturbation method in this article the problem of prey and predator is presented and the homotopy perturbation method is employed to compute an approximation to the solution of the system of nonlinear volterra differential equations governing on the problem the results are compared with the results using the adomian decomposition and the power series methods some plots are presented to show the populations of the prey and the predator versus time for illustrating the reliability and simplicity of the method

short memory principle and a predictor corrector approach for fractional differential equations fractional differential equations are increasingly used to model problems in acoustics and thermal systems rheology and modelling of materials and mechanical systems signal processing and systems identification control and robotics and other areas of application this paper further analyses the underlying structure of fractional differential equations from a new point of view we apprehend the short memory principle of fractional calculus and farther apply a adams type predictor corrector approach for the numerical solution of fractional differential equation and the detailed error analysis is presented combining the short memory principle and the predictor corrector approach we gain a good numerical approximation of the true solution of fractional differential equation at reasonable computational cost a numerical example is provided and compared with the exact analytical solution for illustrating the effectiveness of the short memory principle

solution of the epidemic model by homotopy perturbation method in this article the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic is considered he s homotopy perturbation method is employed to compute an approximation to the solution of the system of nonlinear ordinary differential equations governing on the problem the results are compared with the results obtained by adomian decomposition method some plots are presented to show the reliability and simplicity of the method

improved bounds for the symmetric rendezvous value on the line a notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line when the initial distance between the two players is two we show that the symmetric rendezvous value is within the interval 4 1520 4 2574 which considerably improves the previous best known result 3 9546 4 3931 to achieve the improved bounds we call upon results from absorbing markov chain theory and mathematical programming theory particularly fractional quadratic programming and semidefinite programming moreover we also establish some important properties of this problem which could be of independent interest and useful for resolving this problem completely finally we conjecture that the symmetric rendezvous value is asymptotically equal to 4 25 based on our numerical calculations

a new approach to the solution of the prey and predator problem and comparison of the results with the adomian method in this article the problem of prey and predator is presented and the power series method is employed to compute an approximation to the solution of the system of nonlinear volterra differential equations governing the problem the results are compared with the results using the adomian decomposition method some plots are presented to show the populations of the prey and the predator versus time for illustrating the reliability and simplicity of the method

rational chebyshev series for the thomas fermi function endpoint singularities and spectral methods we solve the thomas fermi problem for neutral atoms u y y 1 y u 3 2 0 on y 0 with u 0 1 and u 0 using rational chebyshev functions tl n y l to illustrate some themes in solving differential equations on a semi infinite interval l is a user choosable numerical parameter the thomas fermi equation is singular at the origin giving a tl convergence rate of only fourth order but this can be removed by the change of variables z y with v z u y z the function v z decays as z with a term in z 3 which is consistent with a geometric rate of convergence however the asymptotic series has additional terms with irrational fractional powers beginning with z 4 5 4 4 in spite of the faster spatial decay the irrational powers degrade the convergence rate to slightly larger than tenth order this vividly illustrates the subtle connection between the spatial decay of u x and the decay with degree of its rational chebyshev series the tl coefficients a n l are hostages to a tug of war between a singularity on the negative real axis which gives a geometric rate of convergence that slows with increasingl and the slow inverse power decay for large z which gives quasi tenth order convergence with a proportionality constant that decreasesinversely as a power of l for l 2 we can approximate u y 0 v z z 0 to 1 part in a million with a truncation n of only 20 l 64 and n 600 gives u y 0 1 5880710226113753127186845 correct to 25 decimal places

solving riccati differential equation using adomian s decomposition method in this paper we suggest a method to solve the matrix riccati differential equation the suggested method which we called multistage adomian s decomposition method madm can be considered as an extension of the adomian s decomposition method adm which is very efficient in solving a variety of differential and algebraic equations the solution is introduced in a recursive form which can be used to obtain the solution for the whole time horizon comparisons are made between madm and the exact solution and further between madm and different numerical methods

on the multichannel rendezvous problem fundamental limits optimal hopping sequences and bounded time to rendezvous one of the fundamental problems in a cognitive radio network known as the multichannel rendezvous problem is for two secondary users to find a common channel that is not blocked by primary users the basic idea for solving such a problem in most works in the literature is for the two users to select their own channel hopping sequences and then rendezvous when they both hop to a common unblocked channel at the same time in this paper we focus on the fundamental limits of the multichannel rendezvous problem and formulate such a problem as a constrained optimization problem where the selection of the random hopping sequences of the two secondary users must satisfy certain constraints we derive various lower bounds for the expected respectively maximum time to rendezvous under certain constraints for some of these lower bounds we are also able to construct optimal channel hopping sequences that achieve the lower bounds inspired by the constructions of quorum systems and relative difference sets our constructions of the channel hopping sequences are based on the mathematical theories of finite projective planes orthogonal latin squares and sawtooth sequences the use of such theories in the constructions of channel hopping sequences appear to be new and better than other existing schemes in terms of minimizing the expected respectively maximum time to rendezvous

stability preservation analysis for frequency based methods in numerical simulation of fractional order systems in this paper the frequency domain based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation first the stability boundary curve is exactly determined for these methods then this boundary is analyzed and compared with an accurate ideal boundary in different frequency ranges also the critical regions in which the stability does not preserve are determined finally the analytical achievements are confirmed via some numerical illustrations

projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control abstract this paper investigates the projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control the pinning control method is employed in designing controllers for guaranteeing projective cluster synchronization based on the lyapunov stability theory and well known barbalat s lemma some new sufficient conditions are derived to guarantee the projective cluster synchronization of the fractional order complex network a numerical simulation is given to verify the effectiveness of the theoretical results

optimal symmetric rendezvous search on three locations in the symmetric rendezvous search game played on n locations two players are initially placed at two distinct locations the game is played in discrete steps at each of which each player can either stay where he is or move to a different location the players share no common labelling of the locations we wish to find a strategy such that if both players follow it independently then the expected number of steps until they are in the same location is minimized informal versions of the rendezvous problem have received considerable attention in the popular press the problem was proposed by alpern in 1976 alpern s 1976 hide and seek games seminar at the institut fur hohere studien vienna and it has proved notoriously difficult to analyse in this paper we prove a 20 year old conjecture that the following strategy is optimal for the game on three locations in each block of two steps stay where you are or search the other two locations in random order doing these with probabilities 1 3 and 2 3 respectively this is now the first nontrivial symmetric rendezvous game of any type to be fully solved

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

on some approximate methods for nonlinear models we discuss some recent inadequate applications of the homotopy perturbation method the adomian decomposition method and the variational iteration method to nonlinear problems

fractional market dynamics a new extension of a fractality concept in nancial mathematics has been developed we have introduced a new fractional langevin type stochastic dierential equation that diers from the standard langevin equation i by replacing the rst order derivative with respect to time by the fractional derivative of order and ii by replacing white noise gaussian stochastic force by the generalized shot noise each pulse of which has a random amplitude with the stable levy distribution as an application of the developed fractional non gaussian dynamical approach the expression for the probability distribution function pdf of the returns has been established it is shown that the obtained fractional pdf ts well the central part and the tails of the empirical distribution of s p 500 returns c 2000 elsevier science b v all rights reserved

variational iteration method for solving the epidemic model and the prey and predator problem in this article two problems are considered the first one is the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic and the other one is the problem of prey and predator the variational iteration method is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations governing on these problems the results are compared with the results obtained by adomian decomposition and power series methods some plots are presented to show the reliability and simplicity of the method

brief paper rational approximations in the simulation and implementation of fractional order dynamics a descriptor system approach this paper deals with issues related to the use of rational approximations in the simulation of fractional order systems and practical implementations of fractional order dynamics and controllers based on the mathematical formulation of the problem a descriptor model is found to describe the rational approximating model this model is analyzed and compared with the original fractional order system under the aspects which are important in their simulation and implementation from the results achieved one can determine in what applications the use of rational approximations would be unproblematic and in what applications it would lead to fallacious results in order to clarify this point some examples are presented in which the effects of using rational approximations are investigated

fractal system as represented by singularity function a fractional slope on the log log bode plot has been observed in characterizing a certain type of physical phenomenon and has been called the fractal system or the fractional power pole in order to represent and study its dynamical behavior a singularity function method is presented which consists of cascaded branches of a number of pole zero negative real pairs or simple rc section the distribution spectrum of the system can also be easily calculated and its accuracy depends on a prescribed error specified in the beginning the method is then extended to a multiple fractal system which consists of a number of fractional power poles the result can be simulated by a combination of singularity functions each representing a single fractal system

the rendezvous search problem the author considers the problem faced by two people who are placed randomly in a known search region and move about at unit speed to find each other in the least expected time this time is called the rendezvous value of the region it is shown how symmetries in the search region may hinder the process by preventing coordination based on concepts such as north or clockwise a general formulation of the rendezvous search problem is given for a compact metric space endowed with a group of isometrics which represents the spatial uncertainties of the players these concepts are illustrated by considering upper bounds for various rendezvous values for the circle and an arbitrary metric network the discrete rendezvous problem on a cycle graph for players restricted to symmetric markovian strategies is then solved finally the author considers the problem faced by two people on an infinite line who each know the distribution of the distance but not the direction to each other

brief paper a proof for non existence of periodic solutions in time invariant fractional order systems the aim of this note is to highlight one of the basic differences between fractional order and integer order systems it is analytically shown that a time invariant fractional order system contrary to its integer order counterpart cannot generate exactly periodic signals as a result a limit cycle cannot be expected in the solution of these systems our investigation is based on caputo s definition of the fractional order derivative and includes both the commensurate or incommensurate fractional order systems

solution of the epidemic model by adomian decomposition method in this article the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic is considered mathematical modeling of the problem leads to a system of non linear ordinary differential equations adomian decomposition method is employed to solve this system of non linear ordinary differential equations the problem of convergence has been addressed

rendezvous search on the line with distinguishable players two players are placed on the real line at a distance d with a distribution f known to both neither knows the direction of the other nor do they have a common notion of a positive direction on the line we seek the least expected rendezvous time r r left f right in which they can meet given maximum speeds of one we consider the cases where f is a bounded point discrete or finite mean distribution we obtain upper bounds or exact values for r and in one case an optimality condition for search strategies a connection with beck s linear search problem is established

a note on time domain simulation of feedback fractional order systems the study of feedback fractional order systems has been receiving considerable attention due to the facts that many physical systems are well characterized by fractional order models and that fractional order controllers are used in feedback systems with the intention of breaking through the performance limitation of integer order controllers owing to the lack of effective analytic methods for the time domain analysis and simulation of linear feedback fractional order systems we suggest in this paper two reliable and accurate numerical methods for inverting fractional order laplace transforms one is based on computing bromwich s integral with a numerical integration scheme capable of accuracy control and the other is based on expanding the time response function in a b spline series in order to demonstrate the superiority in solution accuracy and computational complexity of these two numerical methods over the grunwald letniknov approximation method and podlubny s analytic formulas which are in a form of double infinite series the time domain simulations of the feedback control of a fractional order process with a pd sup spl mu controller and a fractional order band limited lead compensator are worked out the simulation results indicate that a convergence problem indeed occurs in using podlubny s infinite series expressions and that the problem could not be overcome by a series acceleration scheme

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

chaotic behavior in noninteger order cellular neural networks in this paper a simple system showing chaotic behavior is introduced it is based on the well known concept of cellular neural networks cnns which have already given good results in generating complex dynamics the peculiarity of the cnn model consists in the fact that it replaces the traditional first order cell with a noninteger order one the introduction of the fractional cell with a suitable choice of the coupling parameters leads to the onset of chaos in a simple two cell system a theoretical approach based on the harmonic balance theory has been used to investigate the existence of chaos a circuit realization of the proposed fractional two cell chaotic cnn is reported and the corresponding strange attractor is also shown

nyquist based stability analysis of non commensurate fractional order delay systems abstract as a generalization of the first and second order models the elementary fractional order models have been widely used in various engineering fields however most of the previous studies only focus on commensurate fractional order models in this paper a general non commensurate elementary fractional order delay system is investigated first the stability of the studied fractional order delay system is analyzed based on nyquist theorem then a series of sufficient stability conditions are presented for different combinations of parameters including the fractional orders time delay pseudo damping factor and natural frequency 0 finally three examples are given to show the effectiveness of the presented results

on fractional calculus and fractional multipoles in electromagnetism using the concept and tools of fractional calculus we introduce a definition for fractional order multipoles of electric charge densities and we show that as far as their scalar potential distributions are concerned such fractional order multipoles effectively behave as intermediate sources bridging the gap between the cases of integer order point multipoles such as point monopoles point dipoles point quadrupoles etc this technique which involves fractional differentiation or integration of the dirac delta function provides a tool for formulating an electric source distribution whose potential functions can be obtained by using fractional differentiation or integration of potentials of integer order point multipoles of lower or higher orders as illustrative examples the cases of three dimensional point source and two dimensional line source problems in electrostatics are treated in detail and an extension to the time harmonic case is also addressed in the three dimensional electrostatic example we suggest an electric charge distribution which can be regarded as an intermediate case between cases of the electric point monopole point charge and the electric point dipole point dipole and we present its electrostatic potential which behaves as r sup 1 spl alpha p sub spl alpha cos spl theta where 0 spl alpha 1 and p sub spl alpha spl middot is the legendre function of noninteger degree spl alpha thus denoting this charge distribution as a fractional 2 sup spl alpha pole at the two limiting cases of spl alpha 0 and spl alpha 1 this fractional 2 sup spl alpha pole becomes the standard point monopole and point dipole respectively a corresponding intermediate fractional order multipole is also given for the two dimensional electrostatic case potential applications of this treatment to the image method in electrostatic problems are briefly mentioned physical insights and interpretation for such fractional order 2 sup spl alpha poles are also given

optimal trade off between speed and acuity when searching for a small object a searcher seeks to find a stationary hider located at some point h not necessarily a node on a given network q the searcher can move along the network from a given starting point at unit speed but to actually find the hider she must pass it while moving at a fixed slower speed which may depend on the arc in this bimodal search game the payoff is the first time the searcher passes the hider while moving at her slow speed this game models the search for a small or well hidden object e g a contact lens improvised explosive device predator search for camouflaged prey we define a bimodal chinese postman tour as a tour of minimum time i which traverses every point of every arc at least once in the slow mode for trees and weakly eulerian networks networks containing a number of disjoint eulerian cycles connected in a tree like fashion the value of the bimodal search game is i 2 for trees the optimal hider strategy has full support on the network this differs from traditional search games where it is optimal for him to hide only at leaf nodes we then consider the notion of a lucky searcher who can also detect the hider with a positive probability q even when passing him at her fast speed this paper has particular importance for demining problems

convergence of the variational iteration method for solving multi order fractional differential equations in this paper the variational iteration method vim is applied to obtain approximate solutions of multi order fractional differential equations m fdes we can easily obtain the satisfying solution just by using a few simple transformations and applying the vim a theorem for convergence and error estimates of the vim for solving m fdes is given moreover numerical results show that our theoretical analysis are accurate and the vim is a powerful method for solving m fdes

a computational method for solution of the prey and predator problem in this article a mathematical model of the problem of prey and predator being presented and adomian decomposition method is employed to compute an approximation to the solution of the system of nonlinear volterra differential equations governing on the problem some plots for the population of the prey and predator versus time are presented to illustrate the solution

chaos and hyperchaos in the fractional order rossler equations abstract the dynamics of fractional order systems have attracted increasing attentions in recent years in this paper we numerically study the chaotic behaviors in the fractional order rossler equations we found that chaotic behaviors exist in the fractional order rossler equation with orders less than 3 and hyperchaos exists in the fractional order rossler hyperchaotic equation with order less than 4 the lowest orders we found for chaos and hyperchaos to exist in such systems are 2 4 and 3 8 respectively period doubling routes to chaos in the fractional order rossler equation are also found

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

chaos in fractional ordered liu system abstract present paper deals with fractional version of a dynamical system introduced by c liu l liu and t liu c liu l liu t liu a novel three dimensional autonomous chaos system chaos solitons fractals 39 4 2009 1950 1958 numerical investigations on the dynamics of this system have been carried out properties of the system have been analyzed by means of lyapunov exponents furthermore the minimum effective dimensions have been identified for chaos to exist in commensurate and incommensurate orders it is noteworthy that the results obtained are consistent with the analytical conditions given in the literature

hybrid projective synchronization of chaotic fractional order systems with different dimensions abstract the hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper it is shown that the slave system can be synchronized with the projection of the master system generated through state transformation based on the stability theorem of linear fractional order systems a suitable controller for achieving the synchronization is given the hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order the corresponding numerical results verify the effectiveness of the proposed method