a note on the markov property of stochastic processes described by nonlinear fokker planck equations abstract we study the markov property of processes described by generalized fokker planck equations that are nonlinear with respect to probability densities such as mean field fokker planck equations and fokker planck equations related to generalized thermostatistics we show that their transient solutions describe non markov processes in contrast stationary solutions can describe markov processes as a result nonlinear fokker planck equations can be used to model transient non markov processes that converge to stationary markov processes

a numerical approach to the generalized nonlinear fractional fokker planck equation in this paper we propose a fully discrete galerkin finite element method to solve the generalized nonlinear fractional fokker planck equation which has a multi fractional spatial operator characteristic that describes the levy flight in the time direction we use the finite difference method and in the spatial direction we use the fractional finite element method in the framework of the fractional sobolev spaces we derive a fully discrete scheme for the considered equation we prove the existence and uniqueness of the discrete solution and give the error estimates the numerical examples are also included which support the theoretical analysis

a unified view of transport equations abstract distribution functions of many static transport equations are found using the maximum entropy principle the equations of constraint which contain the relevant dynamical information are simply the low lying moments of the distributions systems subject to conservative forces have also been considered in this approach determining the solutions to the transport equations no longer requires solving a partial differential equation but instead experimentally determining the low lying moments and potentials

exact solutions to nonlinear nonautonomous space fractional diffusion equations with absorption we analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions in particular the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation we also discuss some implications by considering a diffusion coefficient d x t d t x theta theta in r and a drift force f k 1 t x k alpha x x alpha 1 in both situations we relate our solutions to those obtained within the maximum entropy principle by using the tsallis entropy

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

the q gamma and q q polygamma functions of tsallis statistics abstract an axiomatic definition is given for the q gamma function q x q r q 0 x r of tsallis non extensive statistical physics the continuous analogue of the q factorial of suyari h suyari physica a 368 1 2006 63 and the q analogue of the gamma function x of euler and gauss a working definition in closed form based on the hurwitz and riemann zeta functions including their analytic continuations is shown to satisfy this definition several relations involving the q gamma and other functions are obtained the q q polygamma functions q q m x m n defined by successive derivatives of ln q q x where ln q a 1 q 1 a 1 q 1 a 0 is the q logarithmic function are also reported the new functions are used to calculate the inferred probabilities and multipliers for tsallis systems with finite numbers of particles n

a procedure for obtaining general nonlinear fokker planck equations abstract a procedure for deriving general nonlinear fokker planck equations fpes directly from the master equation is presented the nonlinear effects are introduced in the transition probabilities which present a dependence on the probabilities for finding the system in a given state it is shown that the fpes obtained from master equations describing transitions among discrete and continuous sets of states are identical within such a procedure we construct nonlinear fpes that appear to be very general our general fpes recover as particular cases nonlinear fpes investigated previously by many authors introduced on a purely phenomenological basis and they lead to the possibility of more complete and complex diffusive equations

a coupled multipoint stress multipoint flux mixed finite element method for the biot system of poroelasticity abstract we present a mixed finite element method for a five field formulation of the biot system of poroelasticity that reduces to a cell centered pressure displacement system on simplicial and quadrilateral grids a mixed stress displacement rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity pressure darcy formulation the spatial discretization is based on combining the multipoint stress mixed finite element msmfe method for elasticity and the multipoint flux mixed finite element mfmfe method for darcy flow it uses the lowest order brezzi douglas marini mixed finite element spaces for the poroelastic stress and darcy velocity piecewise constant displacement and pressure and continuous piecewise linear or bilinear rotation a vertex quadrature rule is applied to the velocity stress and stress rotation bilinear forms which block diagonalizes the corresponding matrices and allows for local velocity stress and rotation elimination this leads to a cell centered positive definite system for pressure and displacement at each time step we perform error analysis for the semidiscrete and fully discrete formulations establishing first order convergence for all variables in their natural norms the numerical tests confirm the theoretical convergence rates and illustrate the locking free property of the method

conditional lie backlund symmetries and functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations abstract the functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations are studied by applying the conditional lie backlund symmetry method a complete list of canonical forms for such equations are presented exact solutions to the resulting equations are constructed the asymptotic behaviors and blow up properties of some solutions are also discussed

deriving partition functions and entropic functionals from thermodynamics abstract given an arbitrary probability distribution and a nondegenerate energy spectrum so that a mean energy u can be computed we derive the partition function z and the entropic functional s that satisfy the basic relation d u t d s the procedure is illustrated by considering examples of typical distributions found currently in nature in particular the power law spectrum is shown to correspond to a critical state associated with tsallis entropy

nonlinear diffusion effects on biological population spatial patterns motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations we investigate its implications in a paradigmatic model for the evolution of a single species density u x t the standard model includes growth and competition in a logistic expression and spreading is modeled through normal diffusion moreover the competition term is nonlocal which has been shown to give rise to spatial patterns we generalize the diffusion term through the nonlinear form tu x t d xxu x t with d 0 encompassing the cases where the state dependent diffusion coefficient either increases 1 or decreases 1 with the density yielding subdiffusion or superdiffusion respectively by means of numerical simulations and analytical considerations we display how that nonlinearity alters the phase diagram the type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape inducing fragmentation in the subdiffusive case the detection of the main persistent mode allows analytical prediction of the critical thresholds

reaction diffusion equations in one dimension particular solutions and relaxation abstract reaction diffusion equations for the concentration of one species in one spatial dimension are considered where the diffusion coefficient as well as creation and annihilation terms are monomials when the exponent of the annihilation term is smaller than the one of the creation term unstable equilibrium solutions may exist in the opposite case travelling wave solutions are found in phase space yielding the value of speed even when the wave form cannot be written in closed form the stability of some solutions is studied numerically showing their robustness in the case of nonlinear diffusion it is found that localized initial conditions evolve to the stable steady state via the travelling waves the invasion of the stable state into the unstable one is via a travelling front with the largest possible gradient this front can be interpreted as the final result of the growth of the most unstable mode the exact solutions allow a modulation approach to study the problem with slowly varying coefficients it is shown that when the creation term is of compact support in space equilibria of compact support are also obtained

a mixed finite element method for nearly incompressible multiple network poroelasticity in this paper we present and analyze a new mixed finite element formulation of a general family of quasi static multiple network poroelasticity mpet equations the mpet equations describe flow a

variational methods in nonextensive tsallis statistics a comparative study abstract in the context of nonextensive tsallis statistics sensible examples are employed in order to compare the variational method proposed in phys rev lett 80 1998 218 with the one in j phys a 26 1993 l893 these examples are such that they can be solved exactly thus a comparison of the exact result with the approximated ones is possible the analysis of these examples indicates that the calculation related to the first variational method is easier to work with than the other method in particular when one is trying to apply the last variational method the calculation of the approximated case can be more cumbersome than solving the exact one

a nonextensive statistical approach to the kinetics of phase transformation abstract some limitations of johnson mehl avrami kolmogorov jmak equation which is used widely for describing kinetics of phase transformation were demonstrated using probabilistic analysis and monte carlo simulations acta mater 48 2000 4217 as well known jmak equation predicts correctly the real transformed fraction only if the number of the growing nuclei in the controlled volume is large however jmak equation seems to fail if the number of growing nuclei is small no matter how large the volume of the controlled volume is in this study we propose a different equation for describing kinetics of phase transformation using a nonextensive formalism namely tsallis thermostatistics which has been commonly employed to study the various physical systems for a decade

nonlinear fokker planck equations and generalized entropies abstract we consider maximum entropy solutions to nonlinear fokker planck equations within general thermostatistical formalisms we show that the family of generalized nonextensive entropies introduced by tsallis is the only one that has an associated family of nonlinear fokker planck equations endowed with time dependent maxent solutions of a generalized gaussian type consequently the natural association that arises between tsallis thermostatistical formalism and special families of nonlinear fokker planck equations is not a universal feature of general thermostatistical formalisms but a particular feature of the tsallis one

on a general link between anomalous diffusion and nonextensivity stochastic processes of systems with composable entropy measures are studied the processes are described by fokker planck equations which are nonlinear with respect to their probability densities and whose diffusion coefficients are derived from the entropy measures in questions in nonextensive extensive systems anomalous normal diffusion is found

stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations abstract we discuss two fundamental aspects of fokker planck equations that are nonlinear with respect to probability densities first we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions second we demonstrate that nonlinear fokker planck equations can be interpreted as linear fokker planck equations describing nonlinear families of markov diffusion processes we exploit this finding in order to derive complete hierarchies of probability densities from nonlinear fokker planck equations

a locking free solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of lagrangian elements abstract this paper presents a new finite element solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of the classical bilinear or trilinear lagrangian elements it solves the primal variable displacement in the strain div formulation and can handle both displacement and traction boundary conditions it is a locking free solver based on conforming finite elements the solver has second order accuracy in displacement and first order accuracy in stress and dilation divergence of displacement as validated by theoretical analysis and illustrated by numerical experiments on benchmarks deal ii implementation is also discussed

hurst exponents for interacting random walkers obeying nonlinear fokker planck equations abstract anomalous diffusion of random walks has been extensively studied for the case of non interacting particles here we study the evolution of nonlinear partial differential equations by interpreting them as fokker planck equations arising from interactions among random walkers we extend the formalism of generalized hurst exponents to the study of nonlinear evolution equations and apply it to several illustrative examples they include an analytically solvable case of a nonlinear diffusion constant and three nonlinear equations which are not analytically solvable the usual fisher equation which contains a quadratic nonlinearity a generalization of the fisher equation with density dependent diffusion constant and the nagumo equation which incorporates a cubic rather than a quadratic nonlinearity we estimate the generalized hurst exponents

general pseudoadditivity of composable entropy prescribed by the existence of equilibrium the concept of composability states that entropy of the total system composed of independent subsystems is a function of entropies of the subsystems here the most general pseudoadditivity rule for composable entropy is derived based only on the existence of equilibrium

nonlinear diffusion equation and nonlinear external force exact solution the solutions of the nonlinear diffusion equation t r1 nd r rn 1 r r r1 n r rn 1f are investigated by considering the presence of an external force f which exhibits an explicit dependence on the distribution first the stationary case is considered after that the dynamical case i e the case dependent on time the stationary solution is obtained by considering the external force f r f1 r f2 r r 1 and the result found is related to the distributions which emerge from the tsallis statistics or the boltzmann gibbs statistics the dynamical solution is investigated by considering the external force f r t k t r k r1 r t 1 and related to the levy distributions in the asymptotic limit in both cases the solutions are expressed in terms of the q exponentials and the q logarithmics functions which emerge from the tsallis formalism

n dimensional nonlinear fokker planck equation with time dependent coefficients an n dimensional nonlinear fokker planck equation is investigated here by considering the time dependence of the coefficients where drift controlled and source terms are present we exhibit the exact solution based on the generalized gaussian function related to the tsallis statistics furthermore we show that a rich class of diffusive processes including normal and anomalous ones can be obtained by changing the time dependence of the coefficients

toward a relative q entropy abstract we address the question and related controversy of the formulation of the q entropy and its relative entropy counterpart for models described by continuous non discrete sets of variables we notice that an l p normalized functional proposed by lutwak yang zhang lyz which is essentially a variation of a properly normalized relative renyi entropy up to a logarithm has extremal properties that make it an attractive candidate which can be used to construct such a relative q entropy we comment on the extremizing probability distributions of this lyz functional its relation to the escort distributions a generalized fisher information and the corresponding cramer rao inequality we point out potential physical implications of the lyz entropic functional and of its extremal distributions

lie symmetry solutions for anomalous diffusion an anomalous diffusion equation is solved by the method of lie symmetries the nonlinear diffusion equation includes fractal dimensions and power law dependence on the radial variable and the diffusion function the lie symmetries appropriate boundary and initial conditions and a first integral replace an ansatz that was previously used to find a class of analytic solutions the mean squared displacement of the diffusion varies in time t by an exponent that is easily found by the lie symmetries and boundary conditions

perturbation and variational methods in nonextensive tsallis statistics a unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on tsallis entropy is given here in the case of the variational method the bogoliubov inequality is generalized in a very natural way following the feynman proof for the usual statistical mechanics the inequality turns out to be form invariant with respect to the entropic index q the method is illustrated with a simple example in classical mechanics the formalisms developed here are expected to be useful in the discussion of nonextensive systems

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

a langevin approach for the microscopic dynamics of nonlinear fokker planck equations abstract for stochastic processes defined by nonlinear fokker planck equations a microscopic dynamics is proposed in terms of generalized langevin equations these langevin equations can for example be used to model stochastic processes with mean field interactions and random walks related to the generalized thermostatistics suggested by tsallis

anisotropic fractional diffusion equation abstract we analyze an anisotropic fractional diffusion equation that extends some known diffusion equations by considering a diffusion coefficient with spatial and time dependence the presence of external forces and time fractional derivatives we obtain new exact classes of solutions for a linear anisotropic fractional diffusion equation and investigate the time scaling behavior and an asymptotic solution for a nonlinear anisotropic fractional diffusion equation we connect the asymptotic solution obtained with the distribution that emerges from the nonextensive statistics to the nonlinear case we also verify different diffusive behavior for instance subdiffusion and superdiffusion in each direction

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

entropy production and nonlinear fokker planck equations the entropy time rate of systems described by nonlinear fokker planck equations which are directly related to generalized entropic forms is analyzed both entropy production associated with irreversible processes and entropy flux from the system to its surroundings are studied some examples of known generalized entropic forms are considered and particularly the flux and production of the boltzmann gibbs entropy obtained from the linear fokker planck equation are recovered as particular cases since nonlinear fokker planck equations are appropriate for the dynamical behavior of several physical phenomena in nature like many within the realm of complex systems the present analysis should be applicable to irreversible processes in a large class of nonlinear systems such as those described by tsallis and kaniadakis entropies

a high order hdg method for the biot s consolidation model abstract we propose a novel high order hdg method for the biot s consolidation model in poroelasticity we present optimal h version error analysis for both the semi discrete and full discrete combined with temporal backward differentiation formula schemes numerical tests are provided to demonstrate the performance of the method

a lattice based approach to option and bond valuation under mean reverting regime switching diffusion processes abstract nowadays the pricing of financial instruments under continuous time markov switching models have received a widespread attention from researchers and practitioners in the finance industry lattice based approaches are amongst the most widely used approaches to solve the pricing problem in this context recently yuen and yang 2010 have proposed a simple and fast recombining trinomial tree method to handle the case of regime switching geometric brownian motion processes in this paper we generalize their approach to the regime switching exponential mean reverting case with state dependent switching rates and derive the necessary conditions for the positivity of conditional branching probabilities we employ the hull and white s tree building procedure to limit tree growth away from the long run mean of the process we use the proposed lattice framework to price contingent claims of european american and barrier type and demonstrate its applicability in pricing default free zero coupon bonds in the proposed tree structure the number of nodes is substantially decreased and the computational cost is effectively reduced compared to the usual approaches extensive numerical experiments illustrate the efficiency and flexibility of the proposed scheme

on one parameter dependent generalizations of boltzmann gibbs statistical mechanics based on a previously postulated entropy that now becomes a particular case we show that there exists an infinite set of entropies with similar properties that reduce in a common limit to the boltzmann shannon form the probabilities for the microcanonical ensemble and for the canonical ensemble are obtained the method used to construct the set is quite simple and quite general and can be applied to generalizations of physical quantities and to other generalized entropies the existence of an infinite set of entropies with in principle similar properties could be a serious drawback for the actual utility of any of them and points to their utter uselessness unless some reason can be given for a special choice of one of them

a nonextensive maximum entropy approach to a family of nonlinear reaction diffusion equations we consider a monoparametric family of reaction diffusion equations endowed with both a nonlinear diffusion term and a nonlinear reaction one that possess exact time dependent particular solutions of the tsallis maximum entropy maxent form the evolution of these solutions is governed by a system of three coupled nonlinear ordinary differential equations that are integrated numerically a simple population dynamics interpretation provides a qualitative understanding of the behaviour of the q maxent solutions when the reaction term vanishes the time dependent distributions studied here reduce to the previously known tsallis maxent solutions for the nonlinear diffusion equation

nonlinear fractional diffusion equation exact results the nonlinear fractional diffusion equation t r1 d r rd 1d r t r r1 d r rd 1f r t t is studied by considering the diffusion coefficient d r t d t r and the external force f r t k1 t r k r in addition a rich class of diffusive processes including normal and anomalous ones is obtained from the study present in this work

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

nonlinear equation for anomalous diffusion unified power law and stretched exponential exact solution the nonlinear diffusion equation ensuremath partial ensuremath rho ensuremath partial t d stackrel ifmmode tilde else fi ensuremath delta ensuremath rho ensuremath nu is analyzed here where stackrel ifmmode tilde else fi ensuremath delta ensuremath equiv 1 r d ensuremath 1 ensuremath partial ensuremath partial r r d ensuremath 1 ensuremath ensuremath theta ensuremath partial ensuremath partial r and d ensuremath theta and ensuremath nu are real parameters this equation unifies the anomalous diffusion equation on fractals ensuremath nu 1 and the spherical anomalous diffusion for porous media ensuremath theta 0 an exact point source solution is obtained enabling us to describe a large class of subdiffusion ensuremath theta g 1 ensuremath ensuremath nu d normal diffusion ensuremath theta 1 ensuremath ensuremath nu d and superdiffusion ensuremath theta l 1 ensuremath ensuremath nu d furthermore a thermostatistical basis for this solution is given from the maximum entropic principle applied to the tsallis entropy

comparison of least squares monte carlo methods with applications to energy real options abstract least squares monte carlo lsm is a state of the art approximate dynamic programming approach used in financial engineering and real options to value and manage options with early or multiple exercise opportunities it is also applicable to capacity investment and inventory production management problems with demand supply forecast updates arising in operations and hydropower reservoir management lsm has two variants referred to as regress now later lsmn l which compute continuation value function approximations c vfas we provide novel numerical evidence for the relative performance of these methods applied to energy swing and storage options two typical real options using a common price evolution model lsmn l estimate c vfas that yield equally accurate near optimal and precise lower and dual upper bounds on the value of these real options estimating the lsmn l c vfas and their associated lower bounds takes similar computational effort in contrast the estimation of a dual bound using the lsml vfa instead of the lsmn cfa takes seconds rather than minutes or hours this finding suggests the use of lsml in lieu of lsmn when estimating dual bounds on the value of early or multiple exercise options as well as of related capacity investment and inventory production policies

a family of evolution equations with nonlinear diffusion verhulst growth and global regulation exact time dependent solutions abstract a family of evolution equations describing a power law nonlinear diffusion process coupled with a local verhulst like growth dynamics and incorporating a global regulation mechanism is considered these equations admit an interpretation in terms of population dynamics and are related to the so called conserved fisher equation exact time dependent solutions exhibiting a maximum nonextensive q entropy shape are obtained

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

solutions for a fractional nonlinear diffusion equation spatial time dependent diffusion coefficient and external forces we analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation we start our investigation by considering the linear case and the nonlinear case afterward the linear case is discussed taking fractional time and spatial derivatives into account in a unified approach we also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by d x t d t x for the nonlinear case we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation

q exponential weibull and q weibull distributions an empirical analysis abstract in a comparative study the q exponential and weibull distributions are employed to investigate frequency distributions of basketball baskets cyclone victims brand name drugs by retail sales and highway length in order to analyze the intermediate cases a distribution the q weibull one which interpolates the q exponential and weibull ones is introduced it is verified that the basketball baskets distribution is well described by a q exponential whereas the cyclone victims and brand name drugs by retail sales ones are better adjusted by a weibull distribution on the other hand for highway length the q exponential and weibull distributions do not give satisfactory adjustment being necessary to employ the q weibull distribution furthermore the introduction of this interpolating distribution gives an illumination from the point of view of the stretched exponential against inverse power law q exponential with q 1 controversy

a study of two modes of locking in poroelasticity in this paper we study two modes of locking phenomena in poroelasticity possion locking and pressure oscillations we first study the regularity of the solution of the biot model to gain some insight into the cause of poisson locking and show that the displacement gets into a divergence free state as the lame constant lambda to infty we also examine the cause of pressure oscillations from an algebraic point of view when a three field mixed finite element method is used based on the results of our study on the causes of the two modes of locking we propose a new family of mixed finite elements that are free of both pressure oscillations and poisson locking some numerical results are presented to validate our theoretical studies

derivation of nonlinear fokker planck equations by means of approximations to the master equation nonlinear fokker planck equations fpes are derived as approximations to the master equation in cases of transitions among both discrete and continuous sets of states the nonlinear effects introduced through the transition probabilities are argued to be relevant for many real phenomena within the class of anomalous diffusion problems the nonlinear fpes obtained appear to be more general than some previously proposed on a purely phenomenological basis ones in spite of this the same kind of solution applies i e it is shown that the time dependent tsallis s probability distribution is a solution of both equations obtained either from discrete or continuous sets of states and that the corresponding stationary solution is in the infinite time limit a stable solution

discrete time modeling of mean reverting stochastic processes for real option valuation abstract in this paper the recombining binomial lattice approach for modeling real options and valuing managerial flexibility is generalized to address a common issue in many practical applications underlying stochastic processes that are mean reverting binomial lattices were first introduced to approximate stochastic processes for valuation of financial options and they provide a convenient framework for numerical analysis unfortunately the standard approach to constructing binomial lattices can result in invalid probabilities of up and down moves in the lattice when a mean reverting stochastic process is to be approximated there have been several alternative methods introduced for modeling mean reverting processes including simulation based approaches and trinomial trees however they unfortunately complicate the numerical analysis of valuation problems the approach developed in this paper utilizes a more general binomial approximation methodology from the existing literature to model simple homoskedastic mean reverting stochastic processes as recombining lattices this approach is then extended to model dual correlated one factor mean reverting processes these models facilitate the evaluation of options with early exercise characteristics as well as multiple concurrent options the models we develop in this paper are tested by implementing the lattice in binomial decision tree format and applying to a real application by solving for the value of an oil and gas switching option which requires a binomial model of two correlated one factor commodity price models for cases where the number of discrete time periods becomes too large to be solved using common decision tree software we describe how recursive dynamic programming algorithms can be developed to generate solutions

quasi stationary states of the nrt nonlinear schrodinger equation abstract with regards to the nonlinear schrodinger equation recently advanced by nobre rego monteiro and tsallis nrt based on tsallis q thermo statistical formalism we investigate the existence and properties of its quasi stationary solutions which have the time and space dependences separated in a q deformed fashion one recovers the normal factorization into purely spatial and purely temporal factors corresponding to the standard linear schrodinger equation when the deformation vanishes q 1 we discuss various specific examples of exact quasi stationary solutions of the nrt equation in particular we obtain a quasi stationary solution for the moshinsky model providing the first example of an exact solution of the nrt equation for a system of interacting particles

nonlinear diffusion equation tsallis formalism and exact solutions we address this work to analyze a nonlinear diffusion equation in the presence of an absorption term taking external forces and spatial time dependent diffusion coefficient into account the nonlinear terms present in this equation are due to a nonlinear generalization of the darcy law and the presence of an absorbent source term we obtain new exact solutions and investigate nonlinear effects produced on the solutions by these terms we also connect the results found here within the tsallis formalism

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

fractional and nonlinear diffusion equation additional results abstract we investigate the solutions of a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation we start our study by considering the linear case and the nonlinear case afterward the linear case is analyzed taking fractional time and spatial derivatives into account in this context we also discuss the modifications that emerge by considering a diffusion coefficient given by d x x for the nonlinear case accomplishing the fractional time derivative we discuss scaling behavior of the time and the asymptotic for the solution of the nonlinear fractional diffusion equation in this case the connection between the asymptotic solution found here and the nonextensive tsallis statistics is performed

fractional nonlinear diffusion equation solutions and anomalous diffusion abstract we devote this work to investigate the solutions of a generalized diffusion equation which contains spatial fractional derivatives and nonlinear terms the presence of external forces and absorbent terms is also considered the solutions found here can have a compact or long tail behavior and in particular for the last case in the asymptotic limit we relate these solutions to the levy or tsallis distributions in addition from the results presented here a rich class of diffusive processes including normal and anomalous ones can be obtained

maximum entropy approach to stretched exponential probability distributions we introduce a nonextensive entropy functionals whose optimization under simple constraints mean values of some standard quantities yields stretched exponential probability distributions which occur in many complex systems the new entropy functional is characterized by a parameter the stretching exponent such that for d 1 the standard logarithmic entropy is recovered we study its mathematical properties showing that the basic requirements for a well behaved entropy functional are verified i e s possesses the usual properties of positivity equiprobability concavity and irreversibility and verifies khinchin axioms except the one related to additivity since s is nonextensive the entropys is shown to be superadditive for 1

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

extensions and solutions for nonlinear diffusion equations and random walks we investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by einstein to explain the brownian motion we show here how to properly modify that framework in order to handle different physical scenarios we obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism finally we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations depending on the choice of distribution functions related to the spreading of systems

multidimensional nonlinear diffusion equation spatial time dependent diffusion coefficient and external forces abstract we analyze a multidimensional nonlinear diffusion equation taking a spatial time dependent diffusion coefficient and external forces into account we obtain new exact classes of solutions and investigate the transverse effects induced by an external force applied in the system we also connect the solutions found here with the distribution that emerges from the nonextensive statistics

numerical approach of fokker planck equation with caputo fabrizio fractional derivative using ritz approximation abstract in this manuscript a type of fokker planck equation fpe with caputo fabrizio fractional derivative is considered we present a numerical approach which is based on the ritz method with known basis functions to transform this equation into an optimization problem it leads to a nonlinear algebraic system then we obtain the coefficients of basis functions by solving the algebraic system the convergence of this technique is discussed extensively three examples are included to show the applicability and validity of this method

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function