high frequency solutions for the singularly perturbed one dimensional nonlinear schrodinger equation this article is devoted to the nonlinear schrodinger equation when the parameter e approaches zero all possible asymptotic behaviors of bounded solutions can be described by means of envelopes or alternatively by adiabatic profiles we prove that for every envelope there exists a family of solutions reaching that asymptotic behavior in the case of bounded intervals we use a combination of the nehari finite dimensional reduction together with degree theory our main contribution is to compute the degree of each cluster which is a key piece of information in order to glue them

the use of defect correction to refine the eigenelements of compact integral operators we show how the principle of defect correction can be applied to an integral eigenvalue problem we present as particular cases of this principle the two important methods which are known as the two level multigrid and iterative refinement

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

numerical approximation of a time fractional black scholes equation abstract in this paper a time fractional black scholes equation is examined we transform the initial value problem into an equivalent integral differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization a rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first order convergence with respect to the time variable is proved for overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the black scholes differential operator we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization it is proved that the scheme is stable and second order convergent with respect to the spatial variable numerical experiments support these theoretical results

on the steps of convergence of approximate eigenvectors in the rayleigh schrodinger series error bounds are given for the iterative computation of the eigenvectors in the rayleigh schrodinger series these bounds remove the discrepancy in the theoretical behaviour and numerical results noted by redont under the assumption of collectively compact convergence as a particular case it follows that the eigenvector in the iterative galerkin method proposed by sloan improves upon the eigenvector in the galerkin method this is illustrated by numerical experiments

an approximate analytical solution of time fractional telegraph equation abstract in this article the powerful easy to use and effective approximate analytical mathematical tool like homotopy analysis method ham is used to solve the telegraph equation with fractional time derivative 1

numerically pricing american options under the generalized mixed fractional brownian motion model abstract in this paper we introduce a robust numerical method based on the upwind scheme for the pricing of american puts under the generalized mixed fractional brownian motion gmfbm model by using portfolio analysis and applying the wick ito formula a partial differential equation pde governing the prices of vanilla options under the gmfbm is successfully derived for the first time based on this we formulate the pricing of american puts under the current model as a linear complementarity problem lcp unlike the classical black scholes b s model or the generalized b s model discussed in cen and le 2011 the newly obtained lcp under the gmfbm model is difficult to be solved accurately because of the numerical instability which results from the degeneration of the governing pde as time approaches zero to overcome this difficulty a numerical approach based on the upwind scheme is adopted it is shown that the coefficient matrix of the current method is an m matrix which ensures its stability in the maximum norm sense remarkably we have managed to provide a sharp theoretic error estimate for the current method which is further verified numerically the results of various numerical experiments also suggest that this new approach is quite accurate and can be easily extended to price other types of financial derivatives with an american style exercise feature under the gmfbm model

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

a new operator splitting method for american options under fractional black scholes models abstract a new operator splitting method is proposed for american options under time fractional black scholes models the fractional linear complementarity problem is split into two easy sub problems with the leading coefficients separated from the convolution sum and matched through a general correction step the method is implementation friendly in the sense that one can easily modify a fractional european solver to obtain the proposed method since the correction step is decoupled and is trivial to solve the method is validated through numerical experiments and demonstrated to be superior to the traditional approach the paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models

accelerated spectral analysis of compact operators an acceleration technique is developed that significantly improves the accuracy of a wide class of well known numerical procedures for approximating the spectra of linear compact operators a given portion of the spectrum is determined to any desired accuracy by implementing an eigenvalue problem of fixed dimension for a matrix polynomial in the eigenparameter of sufficiently high degree the usual approach employs the eigenvalue problem for a matrix polynomial of degree one and sufficiently large dimension the method is applicable to degenerate as well as nondegenerate portions of the spectrum error expressions are obtained and several numerical examples for integral operators are presented

analytically pricing double barrier options based on a time fractional black scholes equation this paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system in this scenario the option price is governed by a modified black scholes equation with a time fractional derivative in comparison with standard derivatives of integer order the fractional order derivatives are characterized by their globalness i e the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself the existence of the time fractional derivative in conjunction with the presence of two barriers of the double barrier options has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted albeit difficult we have managed to find an explicit closed form analytical solution for double barrier options which has been taken to price the single barrier options and european path independent options under the same framework as a special case of the current solution in addition not only have we provided a theoretical proof for the convergence of the newly found analytic solution in series form which is a vital step to show the closeness of our solution we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice

correlated continuous time random walk with time averaged waiting time abstract in this paper we study the dynamics of a correlated continuous time random walk with time averaged waiting time the mean square displacement msd shows this process is subdiffusive and generalized einstein relation holds we also get the asymptotic behavior of the probability density function pdf of this process is stretched gaussian at last by computing the time averaged msd we find ergodicity breaking occurs in this process

fractional fokker planck equation on heterogeneous fractal structures in external force fields and its solutions abstract we introduce a heterogeneous fractional fokker planck equation hffpe involving external force fields describing systems on heterogeneous fractal structure medium the hffpe is shown to obey generalized einstein relation and its stationary solution is the generalized boltzmann distribution it is proved that the asymptotic shape of its solution is a stretched gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for generic potentials furthermore it is shown that the solution of ffpe with parameter has the same properties as that ones of hffpe

a diffusion model for transport on loopless aggregates the authors propose a diffusion equation for transport on a loopless aggregate the relevant distance variable is the chemical distance and the spatially dependent coefficients of the diffusion equation are found in terms of b l the number of bonds at chemical distance l from the vertex of the tree and bs l the number of bonds on the skeleton of the tree it is shown that the resulting motion can be modelled in terms of a biased one dimensional random walk with a probability of remaining in place that approaches 1 as l to infinity this is interpreted as being due to motion along dead ends an important consequence of the model is a simple derivation of the expressions relating the diffusion exponent and fracton dimension to the fractal dimension and the intrinsic dimensions of the tree and its skeleton the mean first passage time to go from the vertex to an arbitrary shell is also found

a new regularization method for solving a time fractional inverse diffusion problem in this paper we consider an inverse problem for a time fractional diffusion equation in a one dimensional semi infinite domain the temperature and heat flux are sought from a measured temperature history at a fixed location inside the body we show that such problem is severely ill posed and further apply a new regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under the a priori bound assumptions for the exact solution finally numerical examples are given to show that the proposed numerical method is effective

numerical solution of the time fractional black scholes model governing european options when considering the price change of the underlying fractal transmission system a fractional black scholes b s model with an order time fractional derivative is derived in this paper we discuss the numerical simulation of this time fractional black scholes model tfbsm governing european options a discrete implicit numerical scheme with a spatially second order accuracy and a temporally 2 order accuracy is constructed then the stability and convergence of the proposed numerical scheme are analyzed using fourier analysis some numerical examples are chosen in order to demonstrate the accuracy and effectiveness of the proposed method finally as an application we use the tfbsm and the above numerical technique to price several different european options

analysis of an axis symmetric fractional diffusion wave problem this paper presents an axis symmetric fractional diffusion wave problem which is considered in polar coordinates the dynamic characteristics of the system are described with a partial fractional differential equation in terms of the riemann liouville fractional derivative this continuum problem is reduced to a countable infinite problem by using the method of separation of variables in this way the closed form solution of the problem is obtained the grunwald letnikov approach is applied to take a numerical evaluation the compatibility and effectiveness of this approach are realized by some simulation results which are obtained by a matlab program it can be seen that the analytical and numerical solutions overlap

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

numerically pricing double barrier options in a time fractional black scholes model abstract the numerical solution of the time fractional black scholes model tfbsm of order 0 1 governing european options is studied zhang et al 2016 derived a numerical scheme of second order in space we improve their results by constructing a scheme of fourth order in space while keeping 2 in time the solvability stability and convergence of the proposed numerical scheme are proved using a fourier analysis the results are demonstrated on two examples

fractal ladder models and power law wave equations extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power law dependence on frequency to describe this power law behavior in tissue a hierarchical fractal network model is proposed the viscoelastic and self similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite ladder topology involving alternating springs and dashpots in the low frequency limit this ladder network yields a stress strain constitutive equation with a time fractional derivative by combining this constitutive equation with linearized conservation principles and an adiabatic equation of state the fractional partial differential equations previously proposed by m caputo linear models of dissipation whose q is almost frequency independent ii geophys j r astron soc 13 529 539 1967 and mg wismer finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation j acoust soc am 120 3493 3502 2006 are derived the resulting attenuation coefficient is a power law with exponents ranging between 1 and 2 while the phase velocity is in agreement with the kramers kronig relations the power law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry

derivation and solutions of some fractional black scholes equations in coarse grained space and time application to merton s optimal portfolio abstract by using the new fractional taylor s series of fractional order f x h e h d x f x where e denotes the mittag leffler function and d x is the so called modified riemann liouville fractional derivative which we introduced recently to remove the effects of the non zero initial value of the function under consideration one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a usual gaussian white noise one can then derive two new families of fractional black scholes equations and one shows how one can obtain their solutions merton s optimal portfolio is once more considered and some new results are contributed with respect to the modeling on one hand and to the solution on the other hand finally one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics

a shooting approach to layers and chaos in a forced duffing equation abstract we study equilibrium solutions for the problem u t e 2 u xx u 3 u cos x u x 0 t u x 1 t 0 using a shooting method we find solutions for all nonzero e for small e we add to the solutions found by previous authors especially angenent mallet paret and peletier and hale and sakamoto and also give new elementary ode proofs of their results among the new results is the existence of internal layer type solutions considering the ode satisfied by equilibria but on an infinite interval we obtain chaos results for 0 3 2 2 3 and 0 1 4 we also consider the problem of bifurcation of solutions as increases

answer to an open problem proposed by r metzler and j klafter in a positive answer to the open problem proposed by r metzler and j klafter it is proved that the asymptotic shape of the solution for a wide class of fractional fokker planck equations is a stretched gaussian for the initial condition being a pulse function in the homogeneous and heterogeneous fractal structures whose mean square displacement behaves like x 2 t t and x 2 t x t 0 1 respectively

the projection kantorovich method for eigenvalue problems abstract a modified projection method for eigenvalues and eigenvectors of a compact operator t on a banach space is defined and analyzed the method is derived from the kantorovich regularization for second kind equations involving the operator t it is shown that when t is a positive self adjoint operator on a hilbert space and the projections are orthogonal the modified method always gives eigenvalue approximations which are at least as accurate as those obtained from the projection method for self adjoint operators the required computation is essentially the same for both methods numerical computations for two integral operators are presented one has t positive self adjoint while in the other t is not self adjoint in both cases the eigenvalue approximations from the modified method are more accurate than those from the projection method

fractional diffusion equation on fractals one dimensional case and asymptotic behaviour a fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered it represents the simplest generalization of the fractional diffusion equation valid in euclidean systems the solution of the fractional equation in one dimension is discussed and compared with exact results for the fractional brownian motion and the one dimensional version of the standard diffusion equation on fractals in higher dimensions it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality

arbitrage with fractional gaussian processes abstract while the arbitrage opportunity in the black scholes model driven by fractional brownian motion has a long history the arbitrage strategy in the black scholes model driven by general fractional gaussian processes is in its infancy the development of stochastic calculus with respect to fractional gaussian processes allowed us to study such models in this paper following the idea of shiryaev 1998 an arbitrage strategy is constructed for the black scholes model driven by fractional gaussian processes when the stochastic integral is interpreted in the riemann stieltjes sense arbitrage opportunities in some fractional gaussian processes including fractional brownian motion sub fractional brownian motion bi fractional brownian motion weighted fractional brownian motion and tempered fractional brownian motion are also investigated

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

fast numerical simulation of a new time space fractional option pricing model governing european call option abstract when the fluctuation of option price is regarded as a fractal transmission system and the stock price follows a levy distribution a time space fractional option pricing model tsfopm is obtained then we discuss the numerical simulation of the tsfopm a discrete implicit numerical scheme with a second order accuracy in space and a 2 order accuracy in time is constructed where is a transmission exponent the stability and convergence of the obtained numerical scheme are analyzed moreover a fast bi conjugate gradient stabilized method is proposed to solve the numerical scheme in order to reduce the storage space and computational cost then a numerical example with exact solution is presented to demonstrate the accuracy and effectiveness of the proposed numerical method finally the tsfopm and the above numerical technique are applied to price european call option the characteristics of the fractional option pricing model are analyzed in comparison with the classical black scholes b s model

a variational method for the existence of bounded solutions of a sublinear forced oscillator we prove that for every bounded and measurable forcing p t the differential equation ddot u u 1 3 p t has bounded solutions with arbitrarily large amplitude in general it is not possible to say that all solutions are bounded as shown by an example due to littlewood the proof is based on a variational method which can be seen as a dual version of nehari s method for boundary value problems on compact intervals

fractional fokker planck equation and oscillatory behavior of cumulant moments the fokker planck equation is considered which is connected to the birth and death process with immigration by the poisson transform the fractional derivative in time variable is introduced into the fokker planck equation in order to investigate an origin of oscillatory behavior of cumulant moments from its solution the probability density function the generating function gf for the corresponding probability distribution is derived we consider the case when the gf reduces to that of the negative binomial distribution nbd if the fractional derivative is replaced to the ordinary one the h j moment derived from the gf of the nbd decreases monotonically as the rank j increases however the h j moment derived in our approach oscillates which is contrasted with the case of the nbd calculated h j moments are compared with those of charged multiplicities observed in pp e e and e p collisions a phenomenological meaning of introducing the fractional derivative in time variable is discussed

solutions for multidimensional fractional anomalous diffusion equations in this paper we investigate the solutions of a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time dependent diffusion coefficient and n dimensional case into account which subjects to natural boundaries and the general initial condition in our analysis the presence of external force is also taken into account we obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the tsallis entropy

universality of stretched gaussian asymptotic behaviour for the fractional fokker planck equation in external force fields we introduce a heterogeneous fractional fokker planck equation hffpe on heterogeneous fractal structure media describing systems involving external force fields the hffpe is shown to obey the generalized einstein relation and its stationary solution is the boltzmann distribution it is proven that the asymptotic shape of its solution is a stretched gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for constant and generic potentials with polar singularity at origin

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

diffusion on complex cellular automata patterns we studied random walks on two dimensional patterns formed by the sequence of configurations of complex elementary cellular automata ca with random initial configurations the walkers are allowed to jump between nearest neighbours or next nearest neighbours 1 sites on patterns of rules 22 54 90 122 126 150 and 182 the diffusion is normal 1 2 on patterns 18 and 146 the diffusion is anomalous with 0 415 0 005 and the spectral dimension is ds 1 25 0 1 from the analysis of the diffusion in the horizontal and vertical directions space and time directions of the ca problem respectively of patterns 18 and 146 we obtained x 0 22 and y 0 42 because they have branches of 1 sites which are long in the vertical direction but narrow in the horizontal direction due to the anisotropic diffusion the results do not satisfy the relation ds 2df dw with dw 1 and fractal dimension df 2 considering that the fractal dimension of the region visited by the walker is equal to the dimension of the substrate df we suggest the new scaling relation ds 2 x y this relation is supported by our numerical results and may be generalized to other structures with anisotropic diffusion

exact solutions for nonlinear fractional anomalous diffusion equations abstract this work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption we first investigate the nonlinear anomalous diffusion equations with one fractional derivative and then multi fractional ones in both situations we obtain the corresponding exact solution its diffusive behavior and the sufficient and necessary conditions for solutions satisfying the boundary condition w t 0 and the sharp initial condition w x 0 x

the differential equation describing random walks on the koch curve consider a particle which is released at some point on a fractal and which moves about the fractal at random a long standing goal has been to determine a differential equation governing the probability density function which describes this walk as well as being interesting in its own right this problem is thought to provide an insight into the problem of anomalous diffusion many attempts to derive such an equation have been made all with limited success perhaps because of the tension between smoothness required by differential equation tools and the lack of smoothness inherent in fractals here we present for the first time the equation governing the random walk on a simple fractal the koch curve we show that this equation makes computation of the probability density function for this problem a simple matter

determination of diffusion kernel on fractals when the diffusion set is a fractal whether the generating mappings of the diffusion set are linear or nonlinear the concrete expressions for the diffusion kernel are obtained moreover the fractional diffusion exponent of the diffusion kernel must satisfy 0 1 in addition the inverse problem of the above case is also discussed

fractional diffusion equation on fractals three dimensional case and scattering function a fractional equation for diffusion in isotropic and homogeneous fractal structures is discussed it generalizes the fractional diffusion equation valid for d dimensional euclidean systems the asymptotic behaviour of the probability density function is obtained exactly analytical expressions are derived for the scattering and relaxation functions which can be studied by x ray and neutron scattering experiments on fractals

analytic approach to the problem of convergence of truncated levy flights towards the gaussian stochastic process an analytic expression for characteristic function defining a truncated l evy flight is derived it is shown that the characteristic function yields results in agreement with recent simulations of truncated l evy flights by mantegna and stanley phys rev lett 73 2946 1994 with the analytic expression for the characteristic function the convergence of the l evy process towards the gaussian is demonstrated without simulations in the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

off lattice anomalous diffusion in a thin sticks system abstract an off lattice monte carlo simulation of a diffusion process through a random sticks network in two dimensions is presented in this paper the statistical properties of the hindered motion are analysed according to the geometric characteristics and concentration of obstacles here represented by randomly disperse segments parallel to one another in particular short and long time behaviour of the mean square displacements are considered with respect to the type of interactions between the tracer and the sticks the diffusion coefficient is estimated and a power law dependence on the obstacle length is determined finally an application of the present results to experimental data of diffusion processes in inhomogeneous media is discussed

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

high energy solutions for a phase transition problem abstract in this article we study the existence and profile of high energy solutions for the phase transition model 2 u a x w u 0 in 1 1 u 1 u 1 0 with w a double well balanced potential and is a small parameter these solutions will be shown to be highly oscillatory we will characterize their asymptotic behavior in terms of an envelope function

heterogeneous memorized continuous time random walks in an external force fields in this paper we study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated memorized waiting times which involves reimann liouville fractional derivative or reimann liouville fractional integral we show that the mean squared displacement of the test particle x x which is dependent on its location x of the form el wakil and zahran chaos solitons fractals 12 1929 1935 2001 begin aligned langle mathbb x x 2 rangle t langle delta x x t 2 rangle 0 sim x theta t gamma quad 0 gamma 1 quad theta d w 2 end aligned 1 where d w 2 is the anomalous exponent the diffusion exponent gamma is dependent on the model parameters we obtain the fokker planck type dynamic equations and their stationary solutions are of the boltzmann gibbs form these processes obey a generalized einstein stokes smoluchowski relation and the second einstein relation we observe that the asymptotic behavior of waiting times and subordinations are of stretched gaussian distributions we also discuss the time averaged in the case of an harmonic potential and show that the process exhibits aging and ergodicity breaking

solutions for a generalized fractional anomalous diffusion equation in this paper we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time dependent diffusion coefficient and an external force into account which subjects to the natural boundaries and the generic initial condition we obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the tsallis entropy

highly oscillatory solutions of a neumann problem for a p laplacian equation abstract we deal with a boundary value problem of the form 1 p u a x w u 0 u 0 0 u 1 where p s s p 2 s for s r and p 1 and w 1 1 r is a double well potential we study the limit profile of solutions of 1 when 0 and conversely we prove the existence of nodal solutions associated with any admissible limit profile when is small enough

transfer processes in fractal media an irreversible process in fractal media involves coupling relation between the space and the time the present note displays how the fractional derivation has to be introduced to describe this effect as a result the law of the chemical diffusion to a fractal is given

distributions and moments of structural properties for percolation clusters we study numerically and by scaling methods the distributions and moments of several structural properties of percolation clusters in two and three dimensions the clusters are generated at criticality and properties such as the distribution of the mass as a function of linear size or chemical distance are studied our results suggest that the hierarchy of moments can be represented by a single gap exponent using a scaling approach we obtain analytical forms for the different distribution functions which agree very well with the numerical data

multi layered stationary solutions for a spatially inhomogeneous allen cahn equation abstract we consider stationary solutions of a spatially inhomogeneous allen cahn type nonlinear diffusion equation in one space dimension the equation involves a small parameter e and its nonlinearity has the form h x 2 f u where h x represents the spatial inhomogeneity and f u is derived from a double well potential with equal well depth when e is very small stationary solutions develop transition layers we first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h x and that at most a single layer can appear near each local minimum point of h x we then discuss the stability of layered stationary solutions and prove that the morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h x we also show the existence of stationary solutions having clustering layers at the local maximum points of h x

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

superconvergent nystrom and degenerate kernel methods for eigenvalue problems the nystrom and degenerate kernel methods based on projections at gauss points onto the space of discontinuous piecewise polynomials of degree r 1 for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel exhibit order 2r we propose new superconvergent nystrom and degenerate kernel methods that improve this convergence order to 4r for eigenvalue approximation and to 3r for spectral subspace approximation in the case where the kernel is sufficiently smooth moreover for a simple eigenvalue we show that by using an iteration technique an eigenvector approximation of order 4r can be obtained the methods introduced here are similar to that studied by kulkarni in 10 and exhibit the same convergence orders so a comparison with these methods is worked out in detail also the error terms are analyzed and the obtained methods are numerically tested finally these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

first passage time survival probability and propagator on deterministic fractals the first passage time density psi r t defined as the probability density for the time spent by a random walker to travel for the first time the distance r that separates the starting site from its nearest neighbours and the survival probability s r t i e the probability that a random walker who starts at a site has not been absorbed by traps located on its nearest neighbours at distance r in the time interval 0 t were calculated for the class of deterministic fractals in which sites are isolated from the rest of the lattice by their nearest neighbours the large xi identical to r square root 2dt 1dw asymptotic expressions for these quantities are psi r t approximately a xi nu 2 dw exp c xi nu and h r t 1 s r t approximately a c dw 1 xi nu 2 exp c xi nu with v dw dw 1 a and c being characteristic constants for each fractal the asymptotic expression for s r t is used to justify that for this class of deterministic fractals the propagator or green function is given asymptotically by p r t t ds 2 xi alpha exp c xi nu for large xi with alpha nu 2 df this value of alpha differs from others proposed recently

dynamical features of reaction diffusion fronts in fractals the speed of front propagation in fractals is studied by using i the reduction of the reaction transport equation into a hamilton jacobi equation and ii the local equilibrium approach different equations proposed for describing transport in fractal media together with logistic reaction kinetics are considered finally we analyze the main features of wave fronts resulting from this dynamic process i e why they are accelerated and what is the exact form of this acceleration

on the accuracy of the rayleigh schrodinger approximations improved upper bounds are given for the kth terms of the rayleigh schrodinger series for the eigenelements of a perturbed operator these are used to obtain error bounds which are better than the previously known estimates for the iterative refinements of computed eigenvalues and eigenvectors in the case of a collectively compact approximation if the perturbation operator is of a special kind stronger results are available these results show a shift in the convergence pattern of approximate eigenvectors as compared to that of approximate eigenvalues

correlated continuous time random walk and option pricing in this paper we study a correlated continuous time random walk cctrw with averaged waiting time whose probability density function pdf is proved to follow stretched gaussian distribution then we apply this process into option pricing problem supposing the price of the underlying is driven by this cctrw we find this model captures the subdiffusive characteristic of financial markets by using the mean self financing hedging strategy we obtain the closed form pricing formulas for a european option with and without transaction costs respectively at last comparing the obtained model with the classical black scholes model we find the price obtained in this paper is higher than that obtained from the black scholes model a empirical analysis is also introduced to confirm the obtained results can fit the real data well

generalized einstein relation and the metzler klafter conjecture in a composite subdiffusive regime abstract in this paper a generalized diffusion model driven by the composite subdiffusive fractional brownian motion fbm is employed based on this stochastic process we derive a fractional fokker planck equation ffpe and obtain its solution it is proved that the generalized einstein relation ger and the metzler and klafter conjecture on the asymptotic behavior of stretched gaussian hold the ffpe in a composite subdiffusive regime