fractional langevin model of memory in financial markets the separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after decades of statistical analysis of financial time series certain stylized facts have emerged including the statistics of stock price fluctuations having fat tails and their linear correlations in time being exceedingly short lived on the other hand the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law one explanation of this long term memory is that it is a consequence of the time scale separation between microscopic and macroscopic economic variables we propose a fractional langevin equation as a dynamical model of the observed memory in financial time series

multiple solutions of periodic boundary value problems for first order differential equations by applying the well known leggett williams multiple fixed point theorem and fixed point theorem of cone expansion and compression this paper investigates the existence of multiple positive solutions of periodic boundary value problems for first order differential equations meanwhile two examples are worked out to demonstrate the main results

global solutions of initial value problems for nonlinear second order integro differential equations of mixed type in banach spaces in this paper the global solutions of initial value problems for nonlinear second order integro differential equations of mixed type in banach spaces are investigated the existence and uniqueness of solutions and their iterative approximation are obtained by using monch fixed point theorem and new comparison results the results presented here essentially improve generalize and unify many well known results

periodic boundary value problem for integrodifferential equations of hammerstein type a general comparison result is developed for first order integrodifferential inequality of hammerstein type with periodic boundary condition it is then used to construct monotone sequences to give out the maximal and minimal solutions of a given pbvp of a first order integrodifferential equation of hammerstein type in which the increasingness of the right hand side function in the argument containing the integral operator is not assumed

exact boundary conditions for the initial value problem of convex conservation laws the initial value problem of convex conservation laws which includes the famous burgers inviscid equation plays an important rule not only in theoretical analysis for conservation laws but also in numerical computations for various numerical methods for example the initial value problem of the burgers equation is one of the most popular benchmarks in testing various numerical methods but in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations in this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries which contain a finite computational domain by using the lax s exact formulas for the convex conservation laws the well posedness of the initial boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described numerical tests with the proposed artificial boundary conditions are carried out by using the lax friedrichs monotone difference schemes

multiple positive solutions for a class of nonlinear integro differential equations in banach spaces in this paper the author obtains the existence of two positive solutions for a boundary value problem of nth order nonlinear integro differential equations on an infinite interval in a banach space by means of fixed point index theory

optimal l 1 rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks we derive optimal error bounds for the viscosity method and monotone difference schemes to an initial value problem of scalar conservation laws with initial data being a finite number of piecewise constants subject to the initial discontinuities satisfying the entropy conditions it is known that the entropy solution of the problem is piecewise constant with a finite number of interacting shocks satisfying the entropy conditions a rigorous analysis shows that both the viscosity method and monotone schemes to approach the initial value problem have uniform l 1 error bounds of o epsilon and o delta x for the time infty t 0 respectively where epsilon and delta x are their corresponding viscosity coefficient and discrete mesh length the results are improvements over the half order rates of l 1 convergence numerical experiments for the lax friedrichs scheme are presented and numerical results justify the theoretical analysis

fractional correlation functions in simple viscoelastic liquids abstract we develop a hydrodynamic formulation of fractional fluctuations in a viscoelastic liquid whose longitudinal modulus is a scalar and only depends on time the method is based on the introduction of fractional time derivatives in the hydrodynamic equations due to the viscoelastic memory coupled generalized langevin equations for the fluctuations of the state variables are formulated and solved analytically for a power law memory kernel with long correlation noise the associated fractional fokker planck equation ffpe is also derived and it is shown that it exhibits a fluctuation dissipation theorem fdt the explicit analytic diffusion coefficient is calculated for power law viscoelasticity and is shown to be subdiffusive the fractional correlations for longitudinal velocity and density fluctuations are analytically calculated and used to obtain the light scattering spectrum and the intermediate scattering function of the viscoelastic fluid our model calculation predicts that the fractional effects on these properties are not small effects and might be measurable

boundary value problems for second order integro differential equations of fredholm type in this paper we study a two point boundary value problem for a nonlinear second order integro differential equation of fredholm type by using upper and lower solutions then we apply our results to investigate some higher order boundary value problems

positive solutions for nth order nonlinear impulsive singular integro differential equations on infinite intervals in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation this paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions we introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in mathbf r nd and every nonvanishing vector mathbf a of mathbf r nd first we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs afterwards we prove that this norm is indeed bounded by the mesh size in several cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases such as the one with independent refined meshes

non markovian diffusion equations and processes analysis and simulations abstract in this paper we introduce and analyze a class of diffusion type equations related to certain non markovian stochastic processes we start from the forward drift equation which is made non local in time by the introduction of a suitable chosen memory kernel k t the resulting non markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l t we then consider the subordinated process y t x l t where x t is a markovian diffusion the corresponding time evolution of the marginal density function of y t is governed by a non markovian fokker planck equation which involves the memory kernel k t we develop several applications and derive the exact solutions we consider different stochastic models for the given equations providing path simulations

a unique positive solution for nth order nonlinear impulsive singular integro differential equations on unbounded domains in banach spaces in this paper the fixed point theory combined with a monotone iterative technique is used to investigate the unique positive solution of a boundary problem for nth order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in a banach space the conditions for the existence of a unique positive solution are established in addition an explicit iterative sequence for approximating the solution of the boundary value problem is derived together with an error estimate furthermore the conditions of the theorems can be easily verified

the solutions of initial value problems for nonlinear second order integro differential equations of mixed type in banach spaces abstract in this paper the minimal and maximal solutions of initial value problems for nonlinear second order integro differential equations of mixed type in banach spaces are investigated by using new comparison results and the partial order method the results presented here essentially improve generalize and unify many well known results

parameter estimation for the fractional schrodinger equation using bayesian method in this paper the fractional schrodinger equation is studied the bayesian method is put forward to estimate some relevant parameters of the equation results show that the estimated values can fit well with the exact solution the varying initial values and maximum iterations have little effect on the estimated results it indicates that the bayesian method is efficient for the multi parameter estimation for the fractional schrodinger equation this method can also be used to estimate parameters for the fractional schrodinger equation in other potential field

initial value problem for first order integro differential equation of volterra type on time scales abstract in this paper we combine the monotone iterative technique with fixed point theory to investigate the extremal solutions of nonlinear integro differential equations on time scales several important lemmas and existence criteria are obtained

spectral decomposition of nonlinear systems with memory we present an alternative approach to the analysis of nonlinear systems with long term memory that is based on the koopman operator and a levy transformation in time memory effects are considered to be the result of interactions between a system and its surrounding environment the analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale on average the time evolution of a mode follows a mittag leffler function and the system can be described using the fractional calculus the general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation when analyzing data from an ill defined black box system the spectral decomposition in terms of mittag leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods consequently the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators or operators with long term memory in time when model details are unknown

remarks on the periodic boundary value problems for integrodifferential equations of volterra type this paper is concerned with the monotone method on the periodic boundary value problems for integrodifferential equations of volterra type this problem has been studied by several authors 1 6 the purpose of this paper is to relax the restrictions as much as possible on the coefficients m and n that appear in the one sided lipschitz condition

anomalous diffusion in view of einstein s 1905 theory of brownian motion abstract einstein s theory of brownian motion is revisited in order to formulate a generalized kinetic theory of anomalous diffusion it is shown that if the assumptions of analyticity and the existence of the second moment of the displacement distribution are relaxed the fractional derivative naturally appears in the diffusion equation this is the first demonstration of the physical origin of the fractional derivative in marked contrast to the usual phenomenological introduction of it it is also examined if einstein s approach can be generalized to nonlinear kinetic theory to derive a generalization of the porous medium type equation by the appropriate use of the escort distribution

periodic boundary value problem for a second order integrodifferential equation of hammerstein type a comparison result dealing with second order integrodifferential inequality of hammerstein type with periodic boundary conditions is proved and the extremal solutions of a corresponding second order integrodifferential equation without increasingness assumption on the integral operator are obtained as limits of monotone iterates an application is given to a fourth order mixed boundary value problem

existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws when shock speed s times t x is rational the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved a sufficient condition under which the above solutions can be extended to j implying the existence of discrete shock profiles of numerical schemes is provided a class of monotonicity preserving schemes including all monotonicity preserving schemes with c 1 numerical flux functions the second order upwinding flux based muscl scheme the second order flux based muscl scheme with lax friedrichs splitting and the godunov scheme for scalar conservation laws are found to satisfy this condition thus the existence of discrete shock profiles for these schemes is established when s t x is rational

exact solution of averaging procedure over the cantor set abstract using functional equations with self similar properties we have derived the exact analytical result for convolution of a smooth function with the normalized density of the cantor set in the limit n we have proved that the self similar kernel of this convolution cannot be reduced explicitly to any product of a power and a log periodic function as suggested in literature only its asymptotic behaviour can be expressed in terms of such a product this clarifies the relationship between fractals and fractional calculus

vanishing curvature viscosity for front propagation in this paper we study the front propagation with constant speed and small curvature viscosity we first investigate two related problems of conservation laws one of which is on the nonlinear viscosity methods for the conservation laws and the other one is on the structure of solutions to conservation laws with l 1 initial data we show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first order rate of l 1 convergence the solutions of conservation laws with l 1 initial data are shown to be bounded after t 0 if all singular points of initial data are from shocks these results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed as the small parameter goes to zero after the front breaks down the cusps will disappear promptly and corners will be formed 2000 academic press

existence of solutions for nth order integro differential equations in banach spaces abstract this paper discusses the existence of solutions of initial value problems for n th order nonlinear integro differential equations of mixed type on an infinite interval in a banach space

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

on a connection between a class of q deformed algebras and the hausdorff derivative in a medium with fractal metric abstract over the recent decades diverse formalisms have emerged that are adopted to approach complex systems amongst those we may quote the q calculus in tsallis version of non extensive statistics with its undeniable success whenever applied to a wide class of different systems kaniadakis approach based on the compatibility between relativity and thermodynamics fractional calculus fc that deals with the dynamics of anomalous transport and other natural phenomena and also some local versions of fc that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations the question we might ask is whether or not there are common aspects that connect these alternative approaches in this short communication we discuss a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely tsallis framework and kaniadakis scenario with local form of fractional derivative operators defined in fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure this connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics such as porous or granular media possible connections with other alternative definitions of fc are also contemplated insights on complexity connected to concepts like coarse grained space time and physics in general are pointed out

rates of convergence of non extensive statistical distributions to levy distributions in full and half spaces the levy type distributions are derived using the principle of maximum tsallis non extensive entropy both in the full and half spaces the rates of convergence to the exact levy stable distributions are determined by taking the n fold convolutions of these distributions the marked difference between the problems in the full and half spaces is elucidated analytically it is found that the rates of convergence depend on the ranges of the levy indices an important result emerging from the present analysis is deduced if interpreted in terms of random walks implying the dependence of the asymptotic long time behaviours of the walks on the ranges of the levy indices if n is identified with the total time of the walks

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

on well posedness of an initial value problem for nonlinear second order impulsive integro differential equations of volterra type in banach spaces in this paper we develop an existence and uniqueness theorem of the solution to an initial value problems for a class of second order impulsive integro differential equations of volterra type in a real banach space by using the generalized banach fixed point theorem an explicit iterative scheme for the solution and an error estimate of the approximation sequence for the initial value problem are also derived two examples are presented to demonstrate the application of our results

periodic solutions for functional differential equations with infinite lead and delay abstract this paper discusses the existence of periodic solutions for functional differential equations with infinite lead and delay on the basis of fixed point theorems topological degree theories comparison principles and differential inequality methods some general results are proved

fractional calculus as a macroscopic manifestation of randomness we generalize the method of van hove physica amsterdam 21 517 1955 so as to deal with the case of nonordinary statistical mechanics that being phenomena with no time scale separation we show that in the case of ordinary statistical mechanics even if the adoption of the van hove method imposes randomness upon hamiltonian dynamics the resulting statistical process is described using normal calculus techniques on the other hand in the case where there is no time scale separation this generalized version of van hove s method not only imposes randomness upon the microscopic dynamics but it also transmits randomness to the macroscopic level as a result the correct description of macroscopic dynamics has to be expressed in terms of the fractional calculus

transform of riccati equation of constant coefficients through fractional procedure we use a particular fractional generalization of the ordinary differential equations that we apply to the riccati equation of constant coefficients by this means the latter is transformed into a modified riccati equation with the free term expressed as a power of the independent variable which is of the same order as the order of the applied fractional derivative we provide the solutions of the modified equation and employ the results for the case of the cosmological riccati equation of frw barotropic cosmologies that has been recently introduced by faraoni

short note on the emergence of fractional kinetics abstract in the present short note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non markovian master equations particle trajectories are assumed to be solely markovian and described by the continuous time random walk model but as a consequence of the complexity of the medium each trajectory is supposed to scale in time according to a particular random timescale the link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed in particular when a stationary distribution is considered the timescale distribution is uniquely determined as a function related to the fundamental solution of the space time fractional diffusion equation in contrast when the non stationary case is considered the timescale distribution is no longer unique two distributions are here computed one related to the m wright mainardi function which is green s function of the time fractional diffusion equation and another related to the mittag leffler function which is the solution of the fractional relaxation equation

periodic boundary value problems for integro differential equations of volterra type on etudie l existence des solutions pour un probleme aux valeurs limites periodique pour l equation integrodifferentielle u f t u tu u 0 u 2 f c 0 2 r r r tu t 0 t k t s u s ds et k c 0 2 0 2 r

on the piecewise smooth solutions to non homogeneous scalar conservation laws abstract we study the structure and smoothness of non homogeneous convex conservation laws the question regarding the number of smoothness pieces is addressed it is shown that under certain conditions on the initial data the entropy solution has only a finite number of discontinuous curves we also obtain some global estimates on derivatives of the piecewise smooth entropy solution along the generalized characteristics these estimates play important roles in obtaining the optimal rate of convergence for various approximation methods to conservation laws

positive solutions of an infinite boundary value problem for nth order nonlinear impulsive singular integro differential equations in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

modified equation for adaptive monotone difference schemes and its convergent analysis a modified parabolic equation for adaptive monotone difference schemes based on equal arclength mesh applied to the linear convection equation is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous piecewise smooth solution of the linear convection equation at order one rate in the l 1 norm it is well known that solutions of the monotone schemes with uniform meshes and their modified equation approach the same discontinuous solution at a half order rate in the l 1 norm therefore the convergence analysis for the modified equation provided in this work demonstrates theoretically that the monotone schemes with adaptive grids can improve the solution accuracy numerical experiments also confirm the theoretical conclusions