fractional order legendre collocation method for solving fractional initial value problems in this paper we present a numerical algorithm for solving second order fractional initial value problems this numerical algorithm is based on a fractional legendre collocation spectral method the governing fractional differential equation is converted into a nonlinear system of algebraic equations the error analysis of the proposed numerical algorithm is presented comparisons with other numerical methods shows that the proposed algorithm is more accurate and simpler to implement several examples are discussed to illustrate the efficiency and accuracy of the present scheme

fractional extreme value adaptive training method fractional steepest descent approach the application of fractional calculus to signal processing and adaptive learning is an emerging area of research a novel fractional adaptive learning approach that utilizes fractional calculus is presented in this paper in particular a fractional steepest descent approach is proposed a fractional quadratic energy norm is studied and the stability and convergence of our proposed method are analyzed in detail the fractional steepest descent approach is implemented numerically and its stability is analyzed experimentally

lagrangians of physics and the game of fisher information transfer the lagrangians of physics arise out of a mathematical game between a smart measurer and nature personified by a demon each contestant wants to maximize his level of fisher information i the game is zero sum by conservation of information in the closed system the payoff of the game introduces a variational principle extreme physical information epi which fixes both the lagrangian and the physical constant of each scenario the epi approach provides an understanding of the relationship between measurement and physical law epi also defines a prescription for constructing lagrangians the prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which i remains invariant as an example when applied to the smart measurement of the space time coordinate of a particle the symmetry used is that between position time space and momentum energy space then the unitary transformation is the fourier one and epi derives the following the equivalence of energy momentum and mass the constancy of planck s parameter h and the lagrangian that implies both the klein gordon equation and the dirac equation of quantum mechanics

the need for alternative information measures abstract in 1948 shannon advanced information theory it as a new branch of mathematics and a powerful tool for understanding the intricacies of the communication process nine years later jaynes conceived the maximum entropy principle and was able to shed with it much light onto statistical mechanics and thermodynamics shannon s logarithmic information measure the maxent protagonist was successfully connected with the thermodynamic entropy since 1988 a new project called nonextensive thermostatistics net came into being after the pioneering work by tsallis and collaborators it has achieved today a remarkable degree of success amid a not small degree of controversy it is also a maxent construction but uses a different information measure in the leading role we advance here some epistemological considerations on the proper place of any information measure within a theoretical physics construct in order put net into a more solid conceptual framework

a numerical method for solving a fractional partial differential equation through converting it into an nlp problem in this paper we propose a new approach for solving fractional partial differential equations which is very easy to use and can also be applied to equations of other types the main advantage of the method lies in its flexibility for obtaining the approximate solutions of time fractional and space fractional equations using this approach we convert a fractional partial differential equation into a nonlinear programming problem several numerical examples are used to demonstrate the effectiveness and accuracy of the method

renyi entropies and fisher informations as measures of nonextensivity in a tsallis setting abstract we study nonextensive statistical scenarios a la tsallis with reference to fisher s information and renyi s entropy a new way of evaluating tsallis generalized expectation values is examined within such a context and is shown to lead to a much better cramer rao bound than the customary procedure a connection between the information measures of fisher s and renyi s is found we show that fisher s measure for translation families remains additive even in a non extensive tsallis setting

power law distributions and fisher s information measure abstract we show that thermodynamic uncertainties tu preserve their form in passing from boltzmann gibbs statistics to tsallis one provided that we express these tu in terms of the appropriate variable conjugate to the temperature in a nonextensive context

a simple probabilistic construction yielding generalized entropies and divergences escort distributions and q gaussians abstract we give a simple probabilistic description of a transition between two states which leads to a generalized escort distribution when the parameter of the distribution varies it defines a parametric curve that we call an escort path the renyi divergence appears as a natural by product of the setting we study the dynamics of the fisher information on this path and show in particular that the thermodynamic divergence is proportional to jeffreys divergence next we consider the problem of inferring a distribution on the escort path subject to generalized moment constraints we show that our setting naturally induces a rationale for the minimization of the renyi information divergence then we derive the optimum distribution as a generalized q gaussian distribution

meshless rbfs method for numerical solutions of two dimensional high order fractional sobolev equations abstract in this paper meshless rbfs method is proposed to solve two dimensional time fractional sobolev equations the proposed method uses rbfs for approximation of spatial operator finite difference formula of o t 2 0 1 is used for time fractional derivative approximation while rule 0 1 as time stepping scheme for the advancement of solution in time validation of the proposed method is made by considering various test examples from literature simulated results are found in very good agreement with available exact solutions a rigorous comparative analysis made with other methods testifies proposed method s superiority in higher dimensions efficiency and accuracy of rbfs method are examined by varying number of nodes n in the domain of influence time step size t as well as l 2 l and l rms error norms linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis

mild solutions to the time fractional navier stokes equations in r n abstract this paper addresses the existence and uniqueness of mild solutions to the navier stokes equations with time fractional differential operator of order 0 1 several interesting properties about the solution are also highlighted like regularity and decay rate in lebesgue spaces which will depend on the fractional exponent moreover it is shown that the l p exponent range which the solution belongs to is different from the range for the solution of the classical problem with 1

fisher info and thermodynamics first law abstract we show starting from first principles that thermodynamics first law can be microscopically obtained for fisher s information measure without need of invoking the adiabatic theorem further it is proved that enforcing the fisher first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing fisher s information measure subject to appropriate constraints

a computational wavelet method for variable order fractional model of dual phase lag bioheat equation abstract in this study we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue several models with different levels of simplifications have been proposed one of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time is the 2d transient dual phase lag dpl variable order fractional energy equation for finding the numerical solution of this general case we propose an efficient semi discrete method based on the two dimensional legendre wavelets 2d lws precisely the variable order fractional derivatives of the model are discretized in the first stage and then the response of the model is expanded by the 2d lws consequently the main problem is transformed into an equivalent system of algebraic equations which can be simply tackled the stability of the proposed method is examined theoretically and experimentally also the procedure is described for one example to examine the computational efficiency of method the experimental results show the stability and spectral accuracy of the proposed method according to the achieved results increasing the fractional order from 0 1 to 1 0 leads to increment of maximum tissue temperatures by about 29 near the center of the targeted region

tsallis entropy maximization procedure revisited abstract the proper way of averaging is an important question with regards to tsallis thermostatistics three different procedures have been thus far employed in the pertinent literature the third one i e the tsallis mendes plastino tmp physica a 261 1998 534 normalization procedure exhibits clear advantages with respect to earlier ones in this work we advance a distinct from the tmp one way of handling the lagrange multipliers involved in the extremization process that leads to tsallis statistical operator it is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the tsallis mendes plastino formalism

solvability of a fractional boundary value problem with fractional integral condition abstract using banach contraction principle and leray schauder nonlinear alternative we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems for fractional differential equations with fractional integral condition involving the caputo fractional derivative some examples are given to illustrate our results

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

the role of constraints in tsallis nonextensive treatment revisited abstract we revisit the nonextensive treatment advanced by tsallis et al physica a 261 1998 534 in order to perform a rather exhaustive study of its properties and compare them with the earlier formalism devised by curado and tsallis j phys a 24 1991 l69 some interesting features are in this way revealed

on the maximum entropy principle and the minimization of the fisher information in tsallis statistics we give a new proof of the theorems on the maximum entropy principle in tsallis statistics that is we show that the q canonical distribution attains the maximum value of the tsallis entropy subject to the constraint on the q expectation value and the q gaussian distribution attains the maximum value of the tsallis entropy subject to the constraint on the q variance as applications of the non negativity of the tsallis relative entropy without using the lagrange multipliers method in addition we define a q fisher information and then prove a q cramer rao inequality that the q gaussian distribution with special q variances attains the minimum value of the q fisher information

numerical solution of fractional pantograph differential equations by using generalized fractional order bernoulli wavelet in the current study new functions called generalized fractional order bernoulli wavelet functions gfbwfs based on the bernoulli wavelets are defined to obtain the numerical solution of fractional order pantograph differential equations in a large interval for the concept of fractional derivative we will use caputo sense by using riemann liouville fractional integral operator first the generalized fractional order bernoulli wavelets are constructed then these functions and their properties are employed to derive the gfbwfs operational matrices of fractional integration and pantograph the operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations finally some examples are included for demonstrating the validity and applicability of our method

robust stabilization of uncertain descriptor fractional order systems this paper presents sufficient conditions for the robust asymptotical stabilization of uncertain descriptor fractional order systems with the fractional order a satisfying 0 a 2 the results are obtained in terms of linear matrix inequalities the parameter uncertainties are assumed to be time invariant and norm bounded appearing in the state matrix a necessary and sufficient condition for the normalization of uncertain descriptor fractional order systems is given via linear matrix inequality lmi formulation the state feedback control to robustly stabilize such uncertain descriptor fractional order systems with the fractional order a belonging to 0 a 2 is derived two numerical examples are given to demonstrate the applicability of the proposed approach

fractional differential mask a fractional differential based approach for multiscale texture enhancement in this paper we intend to implement a class of fractional differential masks with high precision thanks to two commonly used definitions of fractional differential for what are known as grumwald letnikov and riemann liouville we propose six fractional differential masks and present the structures and parameters of each mask respectively on the direction of negative x coordinate positive x coordinate negative y coordinate positive y coordinate left downward diagonal left upward diagonal right downward diagonal and right upward diagonal moreover by theoretical and experimental analyzing we demonstrate the second is the best performance fractional differential mask of the proposed six ones finally we discuss further the capability of multiscale fractional differential masks for texture enhancement experiments show that for rich grained digital image the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential based approach appears obvious better than by traditional integral based algorithms

collocation method with convergence for generalized fractional integro differential equations abstract in this paper we study a numerical approach for some class of generalized fractional integro differential equations gfides defined in terms of the b operators presented recently we develop collocation method for linear and nonlinear forms of gfides the numerical approach uses the idea of collocation methods for solving integral equations legendre polynomials are used to approximate the solution in finite dimensional space with convergence analysis the obtained approximate solution recovers the solution of the fractional integro differential equation fide defined using caputo derivatives in a special case fides containing convolution type kernels appear in diverse area of science and engineering applications therefore some test examples varying the kernel in the b operator are considered to perform the numerical investigations the numerical results validate the presented scheme and provide good accuracy using few legendre basis functions

estimation of distribution laws and physical laws by a principle of extremized physical information abstract a new method of estimating physical probability laws is given this arises from a new form of information which in turn follows from four physically based axioms extremization of this physical information gives rise to the required probability laws as verifications of the approach the following statistical and non statistical laws of physics are derived the complex schrodinger and helmholtz wave equations relativistic quantum mechanics in the klein gordon dirac and weyl pauli formulations the boltzmann energy and maxwell boltzmann velocity distributions the lorentz transformation group of special relativity and maxwell s equations also derived are inequalities defining a class of uncertainty principles e g heisenberg s finally the einstein equations of motion but not the fields of general relativity are derived along with an equivalence d i d mc 2 between information flow rate and matter energy

electrostatic fractional image methods for perfectly conducting wedges and cones engheta 1996 introduced a definition for the electric charge fractional order multipoles using the concept of fractional derivatives and integrals here we utilize that definition to introduce a detailed image theory for the two dimensional 2 d electrostatic potential distributions in front of a perfectly conducting wedge with arbitrary wedge angles and for the three dimensional potential in front of a perfectly conducting cone with arbitrary cone angles we show that the potentials in the presence of these structures can be described equivalently as the electrostatic potentials of sets of equivalent image charge distributions that effectively behave as fractional order multipoles hence the name fractional image methods the fractional orders of these so called fractional images depend on the wedge angle for the wedge problem and on the cone angle for the cone problem special cases where these fractional images behave like the discrete images are discussed and physical justification and insights into these results are given

non standard finite difference and chebyshev collocation methods for solving fractional diffusion equation abstract in this paper a new numerical technique for solving the fractional order diffusion equation is introduced this technique basically depends on the non standard finite difference method nsfd and chebyshev collocation method where the fractional derivatives are described in terms of the caputo sense the chebyshev collocation method with the nsfd method is used to convert the problem into a system of algebraic equations these equations solved numerically using newton s iteration method the applicability reliability and efficiency of the presented technique are demonstrated through some given numerical examples

publisher s note stationary and dynamical properties of information entropies in nonextensive systems phys rev e 77 031133 2008 we have discussed dynamical properties of the tsallis entropy and the generalized fisher information in nonextensive systems described by the langevin model subjected to additive and multiplicative noise analytical expressions for the timedependent tsallis entropy and generalized fisher information have been obtained with the use of the q moment approach to the fokker planck equation developed in a previous study h hasegawa phys rev e 77 031133 2008 model calculations of the information entropies in response to an applied pulse and sinusoidal inputs have been presented

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

green functions based on tsallis nonextensive statistical mechanics normalized q expectation value formulation in this paper the green function theory of quantum many particle systems recently presented is reworked within the framework of nonextensive statistical mechanics with a new normalized q expectation values this reformulation introduces a renormalization of temperature of the earlier theory and a self consistency condition the importance of these two features is nontrivial and to emphasize this we explicitly contrast the maximum entropy density matrices derived for an exactly solvable model based on the two types of the constraints the linear response theory is also presented along with its two particle green function version in order to emphasize the importance of the new formalism we collect here the results where both the formalisms have been used to examine the same set of problems this reveals clearly that the new formalism is the method of choice because the numerical results are much more physically meaningful than those found in the old version even though the general features or the answers retain the same characteristics in certain cases in the case where thermodynamic entities are to be examined as in the case of the q dependence of bose einstein condensation the self consistent requirement in the new formalism is numerically much more subtle and thus the earlier results are modified as shown in fig 1

nonextensive normalized treatment of clausius equation abstract we revisit the first law of thermodynamics for nonextensive statistics by recourse to the optimal lagrange multipliers olm formalism we show firstly that information theory allows one to obtain thermodynamics first law in a general ensemble by dealing just with variations in the density operator without disturbing the system s hamiltonian after re deriving clausius equation we discuss it in some detail and show that for the tsallis mendes plastino tmp normalization i nonextensive reversible processes connecting off equilibrium states exist ii for any extensive reversible process between equilibrium states one can find some other equivalent nonextensive irreversible one and iii in some circumstances an extensive irreversible process can be assimilated to a nonextensive reversible one

semiclassical information from deformed and escort information measures abstract escort distributions are a well established but for physicists a relatively new concept that is rapidly gaining wide acceptance in world in this work we wish to revisit the concept within the strictures of the celebrated semiclassical husimi distributions hds and thereby investigate the possibility of extracting new semiclassical information contained not in the hd themselves but in their associated escort husimi distributions we will also establish relations for various information measures between their deformed versions j naudts physica a 316 2002 323 and those built up with escort hds bounds on the concomitant power exponents will be determined

traveling salesman problem and tsallis statistics a generalization of the stochastic method of simulated annealing algorithm based on tsallis statistics is proposed this algorithm is considerably faster than the traditional ones in solving the traveling salesman problem acceptable solutions are found in fewer steps and higher temperatures than both the classical and the fast simulated annealings recent developments in solving np complete problems can be incorporated and improve the performance even more

why tsallis statistics abstract non extensive thermostatistics is a new paradigm statistical mechanics sm that has reached the rather mature age of 15 in 2003 it has considerably extended both sm formalism and its range of applicability it constitutes one of physics new frontiers with advances being made almost on a weekly basis this is a good time to pause and reflect on its significance

tsallis nonextensive thermostatistics and fisher s information measure abstract the connection between fisher s ideas concerning information measures and nonextensive thermostatistics net is investigated the cramer rao bound is generalized to a net environment a relationship between fisher s information and tsallis entropy is established

fisher information borges operators and q calculus abstract we discuss applying the increasingly popular q calculus or deformed calculus so as to suitably generalize fisher s information measure and the cramer rao inequality a q deformation can be attained in multiple ways and we show that most of them do not constitute legitimate procedures within such a context the only completely acceptable q deformation is that ensuing from using the so called borges derivative e p borges physica a 340 2004 95

thermodynamics and the tsallis variational problem abstract an appropriate redefinition of the lagrange multipliers entering the q maxent variational treatment neatly reconciles his formalism with classical thermodynamics the non extensive approach is seen to reproduce classical results for all q values

chaotic attractors in incommensurate fractional order systems in this paper based on the stability theorems in fractional differential equations a necessary condition is given to check the existence of 1 scroll 2 scroll or multi scroll chaotic attractors in a fractional order system this condition is proposed for incommensurate order systems in general but in the special case it converts to the condition given in the previous works for the commensurate fractional order systems though the presented condition is only a necessary and not sufficient condition for the existence of chaos it can be used as a powerful tool to distinguish for what parameters and orders of a given fractional order system chaotic attractors can not be observed and for what parameters and orders the system may generate chaos it can also be used as a tool to confirm or reject results of a numerical simulation some of the numerical results reported in the previous literature are confirmed by this tool

synchronization of chaotic fractional order systems via active sliding mode controller abstract in this paper we propose a controller based on active sliding mode theory to synchronize chaotic fractional order systems in master slave structure master and slave systems may be identical or different based on stability theorems in the fractional calculus analysis of stability is performed for the proposed method finally three numerical simulations synchronizing fractional order lu lu systems synchronizing fractional order chen chen systems and synchronizing fractional order lu chen systems are presented to show the effectiveness of the proposed controller the simulations are implemented using two different numerical methods to solve the fractional differential equations

symmetries of the fokker planck equation and the fisher frieden arrow of time we give here a concrete illustration of frieden s arrow of time connected with fisher s information measure it is shown that this arrow of time is related to symmetries of the fokker planck equation textcopyright 1996 the american physical society

spectral 1 f noise derived from extremized physical information we consider the evolution of a temporal signal x t that is an intrinsic random field of order zero in the sense of a certain measurement estimation experiment the state of disorder of x t should increase toward an equilibrium state the disorder of x t is measured by its physical information scri and the equilibrium state is determined by the condition that scri be an extremum the equilibrium state is shown to have a power spectrum s ensuremath omega of the form mathrm ensuremath omega mathrm ensuremath mathrm ensuremath alpha 1 ensuremath le ensuremath alpha ensuremath le 2 that of 1 f noise

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

a fleming viot particle representation of the dirichlet laplacian we consider a model with a large number n of particles which move according to independent brownian motions a particle which leaves a domain d is killed at the same time a different particle splits into two particles for large n the particle distribution density converges to the normalized heat equation solution in d with dirichlet boundary conditions the stationary distributions converge as n to the first eigenfunction of the laplacian in d with the same boundary conditions

stationary and dynamical properties of information entropies in nonextensive systems the tsallis entropy and fisher information entropy matrix are very important quantities expressing information measures in nonextensive systems stationary and dynamical properties of the information entropies have been investigated in the n unit coupled langevin model subjected to additive and multiplicative white noise which is one of typical nonextensive systems we have made detailed analytical and numerical study on the dependence of the stationary state entropies on additive and multiplicative noise external inputs couplings and number of constitutive elements n by solving the fokker planck equation fpe by both the proposed analytical scheme and the partial difference equation method transient responses of the information entropies to an input signal and an external force have been investigated we have calculated the information entropies also with the use of the probability distribution derived by the maximum entropy method whose result is compared to that obtained by the fpe the cramer rao inequality is shown to be expressed by the extended fisher entropy which is different from the generalized fisher entropy obtained from the generalized kullback leibler divergence in conformity with the tsallis entropy the effect of additive and multiplicative colored noise on information entropies is discussed also

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators