critical exponents of self avoiding walks on a family of truncated n simplex lattices the authors have studied self avoiding walks saws on a family of truncated n simplex lattices which provide a family of fractals in which the fractal dimension d can be varied to a wide range while the spectral dimension d is held almost fixed by means of exact renormalisation group transformations they have calculated the critical exponents nu alpha and gamma and the connectivity constant mu of the saws for n 5 and 6 they propose an approximate theory for calculating the critical exponents of the saws on fractals which is expected to be accurate at large values of n they show that the theory gives results which are in good agreement with exact values even for small values of n and leads to simple closed relations for the exponents

exact analysis of the self avoiding random walks on two infinite families of fractals abstract we introduce two infinite families of fractals that we name the family and the koch family according to their first members which are the plane filling lattice and the koch fractal curve respectively the fractal dimension d f of family varies from 2 to 1 and from 1 465 to 1 in the case of the koch family when the fractal enumerator b an odd integer varies from 3 to we have calculated exactly the critical exponents of the self avoiding random walks saws on these fractals our results render it possible to perform a complete and exact study of the fractal to euclidean crossover which in this case occurs when b it turns out that all critical exponents when d f 1 b tend to the corresponding euclidean values with a unique correction term of the type constant ln b

on the wave equation with hyperbolic dynamical boundary conditions interior and boundary damping and supercritical sources abstract the aim of this paper is to study the problem u t t u p x u t f x u in 0 u 0 on 0 0 u t t u u q x u t g x u on 0 1 u 0 x u 0 x u t 0 x u 1 x in where is a bounded open subset of r n with c 1 boundary n 2 1 is relatively open on denotes the laplace beltrami operator on is the outward normal to and the terms p and q represent nonlinear damping terms while f and g are nonlinear perturbations in the paper we establish local and global existence uniqueness and hadamard well posedness results when source terms can be supercritical or super supercritical

correction to scaling exponents and amplitudes for the correlation length of linear polymers in two dimensions we consider the scaling behaviour of the radius of gyration for a system of dilute linear polymers we focus on pn the mean square end to end distance of an n step self avoiding walk for which an additional two terms were recently calculated for the close packed triangular lattice combining several extrapolation methods we find that a consistent description of the scaling behaviour exists if and only if the correction to scaling exponent a is roughly half as large as commonly believed we conclude that all data ale consistent with the equation pn an2 1 b na c n where y a a 1 j2 abso 21 and c lo

free energy of magnetic model systems at the fractal to euclidian crossover abstract we prove that the free energy per spin of the nearest neighbour classical n vector spin model situated on the sierpinski gasket sg family of fractals tends to the free energy per spin of the same model situated on the standard triangular lattice when the linear size b of the sg fractal generator tends to infinity this finding can be regarded as the first step in the analytical investigation of the spin system thermodynamics at the fractal to euclidean crossover

existence of a solution of the wave equation with nonlinear damping and source terms abstract we study the nonlinear wave equation involving the nonlinear damping term u t u t m 1 and a source term of type u u p 1 for 1 p m we prove a global existence theorem with large initial data for 1 m p a blow up result is established for sufficiently large initial data

conformal invariance and self avoiding walks in restricted geometries the prediction of conformal invariance for the statistics of self avoiding random walks restricted to both semi infinite and wedge shaped geometries are tested by extrapolating exact enumerations close agreement is found both for the angular distribution of the end to end vector and for the dependence of the critical exponent gamma 2 on the opening angle of the wedge

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

exact enumeration of self avoiding walks on critical percolation clusters in 2 7 dimensions we study self avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method which can handle walks of several thousand steps we had previously presented results for the two and three dimensional cases here we take a wider perspective and vary the system s dimensions up to d 7 beyond the supposed upper critical dimension of these results may serve to check analytical predictions and help understand how the medium s fractal structure impacts on the walks scaling behavior

systems of nonlinear wave equations with damping and supercritical boundary and interior sources utt u g1 ut f1 u v vtt v g2 vt f2 u v in a bounded domain r with robin and dirichlet boundary conditions on u and v respectively the nonlinearities f1 u v and f2 u v are with supercritical exponents representing strong sources while g1 ut and g2 vt act as damping in addition the boundary condition also contains a nonlinear source and a damping term by employing nonlinear semigroups and the theory of monotone operators we obtain several results on the existence of local and global weak solutions and uniqueness of weak solutions moreover we prove that such unique solutions depend continuously on the initial data

phase transitions for polymers on fractal lattices abstract models of linear and branched polymers on finitely ramified fractal lattices are briefly reviewed their foremost interest is to provide exactly solvable systems with phase transitions at finite values of the fugacity or temperature this gives insight into the behaviour of polymers in inhomogeneous media their collapse transition and their adsorption at surfaces some unexpected results are also found such as the possible existence of essential singularities in generating functions or the non convergence of critical exponents towards their eucledean limit

self avoiding walks in quenched random environments the self avoiding walk in a quenched random environment is studied using realspace and field theoretic renormalization and flory arguments these methods indicate that the system is described for d de by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path this fixed point is characterized by scaling laws for the size of the walk l n with n the number of steps and the fluctuations in the free energy af l the bound 1 co pore and suggests that this inequality holds for d 2 and 3 although pure cannot be excluded particularly for d 2 for d d c there is a transition between strong and weak disorder phases at which pure the strong disorder fixed point for saws on percolation clusters is discussed the analogy with directed walks is emphasized

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

regularity theory of hyperbolic equations with non homogeneous neumann boundary conditions ii general boundary data abstract this paper studies the regularity of solutions of general mixed second order time dependent hyperbolic problems of neumann type in a previous paper i lasiecka and r triggiani ann mat pura appl iv clvii 1990 285 367 using pseudo differential calculus we have provided sharp regularity results of the solutions and their traces when the non homogeneous data are in l 2 now we complement this study by providing a regularity when the non homogeneous data are more regular than as well as less regular than l 2 in contrast with our previous paper a functional analytic approach based on the l 2 results of our previous paper is used throughout

local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping abstract we consider the wave equation with interior and boundary nonlinear sources and damping and we are interested in local and global wellposedness of finite energy solutions the main difficulty is represented by the fact that the lopatinski condition fails to hold unless the dim 1 and thus the analysis of the boundary nonlinearities becomes a subtle issue

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

renormalisation on sierpinski type fractals presents a family of deterministic fractals which generalise the d dimensional sierpinski gaskets and establishes their order of ramification and their fractal and spectral dimensions random walks on these fractals are renormalisable and lead to rational not necessarily polynomial mapping

self avoiding walks on fractals studied by the monte carlo renormalization group the authors have applied the monte carlo renormalization group mcrg method to study the problem of self avoiding walks on the sierpinski gasket family of fractals each member of the family is labelled by an integer b 2 or b or infinity and when b to infinity both the fractal df and spectral ds dimension approach their euclidean value 2 they have calculated the critical exponent nu associated with the mean square end to end distance up to b 80 their mcrg results deviate at most 0 03 from the available exact results for 2 or b or 9 the obtained data show clearly that nu monotonically decreases with b and crosses the euclidean value nu 3 4 at b approximately 27 that is before entering the fractal to euclidean crossover region that occurs in the limit b to infinity

theory of self avoiding walks on percolation fractals a phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self avoiding walks saw is used to derive a new expression for the flory exponent describing the average radius of gyration of saws on fractals we focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of saws the mean field version of our new formula corresponds to the aharony and harris expression who used the standard flory approach for its derivation

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

local hadamard well posedness for nonlinear wave equations with supercritical sources and damping abstract we consider the wave equation with supercritical interior and boundary sources and damping terms the main result of the paper is local hadamard well posedness of finite energy weak solutions the results obtained 1 extend the existence results previously obtained in the literature by allowing more singular sources 2 show that the corresponding solutions satisfy hadamard well posedness conditions during the time of existence this result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

critical behavior of self avoiding walks on fractals using a new graph counting technique suitable for self similar fractals exact 18th order series expansions for saws on some sierpinski carpets are generated from them the critical fugacityxc and critical exponents saw and saw are obtained the results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied we find for some carpets with low lacunarity that saw saw d for saws on the fractals which are embedded in ad dimensional euclidean space

a potential well theory for the wave equation with a nonlinear boundary condition soit d un sous ensemble connexe ouvert borne de r n a frontiere de lipschitz d soit d l union de deux sous varietes disjointes a n 1 dimensions et et leur confluence de lipschitz on considere le probleme aux valeurs limites et initiales 2 u t 2 u dans d o t u x o u x u x o t v x dans d u x t o sur o t u n f u x t sur o t

connective constant of saws on the sierpinski gasket family using a graph counting technique suitable for regular fractals an exact evaluation of the total number of embeddings of self avoiding walks on the generalized sierpinski gasket is obtained numerical estimates for the connective constants are quoted for the first time where b is the generation parameter of the gaskets it is shown that the number of distinct n step saws per site converges to the triangular lattice values when our analysis indicates that converges to the euclidean value in the same limit and an asymptotic expression is given

weak solutions and blow up for wave equations of p laplacian type with supercritical sources this paper investigates a quasilinear wave equation with kelvin voigt damping utt pu ut f u in a bounded domain 3 and subject to dirichlet boundary conditions the operator p 2 p 3 denotes the classical p laplacian the nonlinear term f u is a source feedback that is allowed to have a supercritical exponent in the sense that the associated nemytskii operator is not locally lipschitz from w01 p into l2 under suitable assumptions on the parameters we prove existence of local weak solutions which can be extended globally provided the damping term dominates the source in an appropriate sense moreover a blow up result is proved for solutions with negative initial total energy

theta point exponent for polymer chains on percolation fractals we derive a new expression for the flory exponent describing the average radius of gyration of polymer chains at the theta point for this we make use of the appropriate distribution function for the radius of gyration we start from euclidean lattices and extend the results to percolation fractals by taking into account the basic geometry and the topology of such structures we show that such basic features have a very prominent effect on the flory exponent of the chain polymer on fractals at the theta point

critical behaviour at an edge for the saw and ising model the authors have studied the behaviour of two and three dimensional self avoiding walks confined to a wedge of wedge angle alpha series have been obtained and analysed for the angular dependent critical exponents characterising various edge susceptibilities in terms of a general scaling form for the edge free energy fe approximately t d 2 nu psi e ht y0 nu h1t y1 nu h2t y2 nu they find for the two dimensional case the following scaling indices y0 91 48 y1 3 8 y2 alpha 5 pi 8 alpha they argue that these results are exact from which follow all exponents for the bulk surface and edge problem in three dimensions the authors obtain y0 approximately 2 488 y1 0 65 or 0 02 y2 alpha alpha b pi alpha where a 0 51 or 0 04 b 0 847 or 0 017 which for y2 is precisely of the functional form given by mean field theory y2 alpha 1 2 pi 2 they argue that a 1 2 for all three dimensional o n models this simple angular dependence of y2 is different from that suggested by cardy s one loop epsilon expansion 1983 for the square lattice they have also studied the case in which the wedge is rotated through an angle of pi 4 and find that the various exponents are unchanged for the three dimensional ising model in a wedge analogy with the saw results plus mean field results in conjunction with rg and series work yield y0 approximately 2 485 y1 0 71 or 0 02 and y2 a b pi alpha with a 1 2 and b 0 79 or 0 02

universal crossing of the self avoiding walk critical exponent nu at the euclidean value 3 4 for several different fractal families we find that the end to end distance critical exponent nu is close to the euclidean value 3 4 for self avoiding walks saws on fractals that are composed of homogeneous parts of size bh approximately 26 more precisely we find that saws on fractals with smaller homogeneous parts b 3 4 while nu 27 we establish this result in the case of three quite different fractal families the sierpinski gasket sg family the plane filling pf family and the checkerboard cb family of fractals in the case of the first two families sg and pf the relevant values for nu were found previously by applying the monte carte renormalization group mcrg method but the specific value bh was not recognized on the other hand for the cb fractals only the exact renormalization group rg results were known previously for 3 or b or 9 where b is an odd integer that serves as the fractal enumerator in this paper we extend the sequence of the known results for the cb family up to b 81 by generalizing the mcrg method the new results have revealed the occurrence of the aforementioned crossing of the euclidean value 3 4 we discuss the significance of the crossing within the current knowledge of saws on fractals

statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three dimensional space abstract we study the adsorption problem of linear polymers immersed in a good solvent when the container of the polymer solvent system is taken to be a member of the sierpinski gasket sg family of fractals embedded in the three dimensional euclidean space members of the sg family are enumerated by an integer b 2 b and it is assumed that one side of each sg fractal is impenetrable adsorbing boundary we calculate the surface critical exponents 11 1 and s which within the self avoiding walk model saw of polymer chain are associated with the numbers of all possible saws with both one and no ends grafted to the adsorbing surface adsorbing boundary respectively by applying the exact renormalization group method for 2 b 4 we have obtained specific values for these exponents for various types of polymer conformations to extend the obtained sequences of exact values for surface critical exponents we have applied the monte carlo renormalization group method for fractals with 2 b 40 the obtained results show that all studied exponents are monotonically increasing functions of the parameter b for all possible polymer states we discuss mutual relations between the studied critical exponents and compare their values with those found for other types of lattices in order to attain a unified picture of the attacked problem

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved with gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

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

a finite element method for space time directional fractional diffusion partial differential equations in the plane and its error analysis abstract we present and analyze a fast finite element method for space time fractional directional partial differential equations in a bounded domain in the plane the fast solver significantly reduces the computational work of solving the discrete linear algebraic systems from o m n 3 m 2 n by a direct solver to o m n log m n per krylov subspace iteration and a memory requirement from o m n 2 to o n log m an error estimate is proved numerical results are presented to show the utility of the method

hadamard well posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping abstract presented here is a study of a viscoelastic wave equation with supercritical source and damping terms we employ the theory of monotone operators and nonlinear semigroups combined with energy methods to establish the existence of a unique local weak solution in addition it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

global existence for the wave equation with nonlinear boundary damping and source terms abstract the paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms the typical problem studied is u tt u 0 in 0 u 0 on 0 0 u u t m 2 u t u p 2 u on 0 1 u 0 x u 0 x u t 0 x u 1 x on where r n n 1 is a regular and bounded domain 0 1 m 1 2 p r where r 2 n 1 n 2 when n 3 r when n 1 2 and the initial data are in the energy space we prove local existence of the solutions in the energy space when m r r 1 p or n 1 2 and global existence when p m or the initial data are inside the potential well associated to the stationary problem

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

finite size scaling for random walks on fractals random walks are simulated on finite stages of construction of regular fractal lattices it is proved that the mean square displacement rn2 obeys a finite size scaling hypothesis and the critical exponent nu w is estimated the efficiency of the method is proved when applied to finitely ramified fractals in which the problem is exactly solvable nu w is obtained with good accuracy approximately 1 for a class of infinitely ramified fractals the sierpinski carpets the results correct previous estimates based on simulations which did not use finite size scaling it is shown that nu w decreases when dc decreases with very small corrections due to other geometrical properties such as lacunarity the comparison with estimates of the ideal chain exponent nu c shows that the two problems are not equivalent on these fractals and that in general nu w nu c estimates of nu w with the same accuracy are obtained on two sierpinski pastry shells 2 df 3 where anomalous diffusion is also observed

end to end distance of linear polymers in two dimensions a reassessment recent exact enumeration studies of the two dimensional self avoiding walk have produced evidence in support of non analytic scaling corrections newly extended series for the triangular and square lattices to n 19 and 25 respectively are examined in the light of recent monte carlo results suggesting the absence of corrections of this type using a modified approach to series analysis it is concluded that the end to end distance can best be understood in terms of analytic corrections alone in view of the lack of theoretical support there is little reason to introduce additional kinds of correction in attempting to explain the data

on nonlinear wave equations with degenerate damping and source terms in this article we focus on the global well posedness of the differential equation u tt u u k j u t u p 1 u in x 0 t where j is a sub differential of a continuous convex function j under some conditions on j and the parameters in the equations we obtain several results on the existence of global solutions uniqueness nonexistence and propagation of regularity under nominal assumptions on the parameters we establish the existence of global generalized solutions with further restrictions on the parameters we prove the existence and uniqueness of a global weak solution in addition we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k m and the initial energy is negative we also address the issue of propagation of regularity specifically under some restriction on the parameters we prove that solutions that correspond to any regular initial data such that u 0 h 2 h 1 0 u 1 h 1 0 are indeed strong solutions

trace regularity of the solutions of the wave equation with homogeneous neumann boundary conditions and data supported away from the boundary let rn be a smooth domain with boundary let u be the solution of a second order hyperbolic scalar equation with homogeneous neumann boundary conditions defined on due to initial conditions u0 u1 and to a forcing term the present paper studies the regularity of the map u0 u1 u where u0 u1 l2 q h1 l2 q 0 t and u is the dirichlet trace of the solution in particular the case of compactly supported data u0 u1 is considered and contrasted with corresponding results where the data do not have compact support in this latter case an optimal regularity result is given for the wave equation on the half space of dimension 2 it states contrary to expectations from elliptic or parabolic theory that u h34 but u h34 e e 0 where 0 t

asymptotic form of the spectral dimension at the fractal to lattice crossover the authors study the spectral dimension of the sierpinski gasket family of fractals each member of the family is labelled by an integer b 2 or b infinity so that when b to infinity both the fractal and spectral dimension approach the euclidean value 2 the asymptotic law for the spectral dimension was recently suggested to be ds approximately 2 b 1nb beta with b and beta being some constants here the authors demonstrate that this form should be replaced by the asymptotic law ds approximately 2 1n 1nb constant 1nb their analysis is based on the exact calculation of the electric resistances rb for all members of the family up to b 650

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

self avoiding walks attached to triangular and face centred cubic lattice surfaces extended exact enumeration data the analysis of exact enumeration data of self avoiding walks attached to the surfaces of triangular and face centred cubic lattices is consistent with gamma 1 0 718 or 0 008 136 delta pc where delta pc pc 0 0995 in three dimensions and 0 9530 or gamma 1 or 0 9548 in two dimensions where pc for the triangular lattice is assumed to lie in the range 0 240 915 or pc or 0 240 93

inhomogeneous dirichlet boundary value problems of space fractional diffusion equations and their finite element approximations we prove the wellposedness of the galerkin weak formulation and petrov galerkin weak formulation for inhomogeneous dirichlet boundary value problems of constant or variable coefficient conservative caputo space fractional diffusion equations we also show that the weak solutions to their riemann liouville analogues do not exist in general in addition we develop an indirect finite element method for the dirichlet boundary value problems of caputo fractional differential equations which reduces the computational work for the numerical solution of variable coefficient fractional diffusion equations from o n 3 to o n and the memory requirement from o n 2 to o n on any quasiuniform space partition we further prove a nearly sharp error estimate for the method which is expressed in terms of the smoothness of the prescribed data of the problem only we carry out numerical experiments to investigate the performance of the method in comparison with the galerkin finite element method

flory approximant for self avoiding walks on fractals a flory approximant for the exponent describing the end to end distance of a self avoiding walk saw on fractals is derived the approximant involves the fractal dimensionalities of the backbone and of the minimal path and the exponent describing the resistance of the fractal the approximant yields values which are very close to those available from exact and numerical calculations

where two fractals meet the scaling of a self avoiding walk on a percolation cluster the scaling properties of self avoiding walks on a d dimensional diluted lattice at the percolation threshold are analyzed by a field theoretical renormalization group approach to this end we reconsider the model of y meir and a b harris phys rev lett 63 2819 1989 and argue that via renormalization its multifractal properties are directly accessible while the former first order perturbation did not agree with the results of other methods our analytic result gives an accurate description of the available mc and exact enumeration data in a wide range of dimensions 2 ensuremath leqslant d ensuremath leqslant 6

a fast finite element method for space fractional dispersion equations on bounded domains in mathbb r 2 we develop a fast and accurate finite element method for space fractional dispersion equations in two space dimensions which are expressed in terms of fractional directional derivatives in all the directions that are integrated with respect to a probability measure on the unit circle the fast method significantly reduces the computational work of solving the discrete linear algebraic systems from o n 3 by a direct solver to o n log n per iteration and a memory requirement from o n 2 to o n furthermore the fast method reduces the evaluation of the stiffness matrix which often constitutes a large portion of the cpu time from o n 2 to o n the developed preconditioned fast krylov subspace iterative solver significantly reduces the number of iterations in a krylov subspace iterative method and may improve the convergence behavior of the solver numerical results show the utility of the method

true self avoiding walks on a percolation cluster the authors investigate by monte carlo simulations the critical behaviour of true self avoiding walks tsaws on a percolation cluster performed very close to the percolation threshold specifically they generate tsaws on a site percolated incipient infinite cluster for various values of the self avoidance parameter g 0 they found that such walks exhibit critical behaviour different from that of ordinary self avoiding walks and also from that of random walks of no constraint the flory exponent obtained was v 0432 or 0 005 for all g 0 which agrees well with the flory type formula suggested by rammal 1984

superlocalization correlations and random walks on fractals abstract upper and lower bounds are found for the anomalous decay exponent a of localized wave functions and of various two point correlation functions on typical fractal configurations the decay of the superlocalized wave functions is used to evaluate the decay of the probability distribution of a random walk which is then used to obtain a flory like expression for self avoiding walks emphasis is placed on the differences between typical and average quantities

a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of variable coefficient conservative fractional diffusion equations fractional diffusion equations were shown to provide an adequate and accurate description of transport processes exhibiting anomalous diffusion behavior recently spectral galerkin methods were developed for space fractional diffusion equations aiming at achieving exponential convergence an optimal order error estimate in the fractional energy norm was proved under the assumption that the true solution to the fractional diffusion equation has the desired regularity an optimal order error estimate in the l 2 norm was proved via the well known nitsche lifting technique under the assumption that the true solution to the corresponding boundary value problem of the fractional diffusion equation has the required regularity for each right hand side in this paper we show that the true solution to the dirichlet boundary value problem of a conservative fractional diffusion equation of order 2 with 0 1 as well as a constant diffusivity coefficient and a constant source term is not in the fractional sobolev space h 3 2 in general but is still in the besov space b 3 2 l 2 hence the provable convergence rate of a spectral galerkin method in the l 2 norm is at most of the order o n 3 2 where n is the degree of the polynomial space in the numerical method numerical experiments show that the spectral galerkin method exhibits a subquadratic convergence in the l 2 norm for any 0 1 we develop a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of one sided variable coefficient conservative fractional diffusion equations the method has a proved high order convergence rate of arbitrary order i without requiring the smoothness of the true solution u to the given boundary value problem but only assuming that the diffusivity coefficient and the right hand source term have the desired regularity ii for a variable diffusivity coefficient and iii for an inhomogeneous dirichlet boundary condition numerical experiments substantiate the theoretical analysis and show that the method exhibits exponential convergence provided the diffusivity coefficient and the right hand source term have the desired regularity

self avoiding walks on percolation clusters the author studies self avoiding walks saws on percolation clusters a scaling function representation for r the mean end to end distance is proposed which describes a crossover from ordinary saws to saws on fractals he distinguishes between saws on a single cluster for which r approximately nx where x nu 1 and saws on all clusters for which r approximately nz where z nu 2 and n is the number of monomers in the walk he estimates nu 1 d 2 approximately 1 285 and the correction to scaling exponent omega d 2 approximately 1 3 and nu 1 d 3 approximately 1 38 two plausible generalisations of the flory 1953 approximation for nu 1 are investigated and it is argued that none of them provides a satisfactory approximation for nu 1 at all dimensions

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

hamiltonian walks on sierpinski and n simplex fractals we study hamiltonian walks hws on sierpinski and n simplex fractals via numerical analysis of exact recursion relations for the number of hws we calculate the connectivity constant and find the asymptotic behaviour of the number of hws depending on whether or not the polymer collapse transition is possible on a studied lattice different scaling relations for the number of hws are obtained these relations are in general different from the well known form characteristic of homogeneous lattices which has thus far been assumed to also hold for fractal lattices

series studies of self avoiding walks near the theta point on 2d critical percolation clusters the thermally weighted average end to end distance rn of interacting self avoiding walks saws are obtained enumerating all the finite n stepped saw configurations on the infinite percolation cluster of bond diluted square lattice at the percolation threshold averaging over 250 percolation clusters and enumerating all the possible saws on them for n up to 31 rn is fitted to a scaling form r2n approximately n2 nu theta f nphi tau t theta theta is the temperature interval away from the theta point phi is the crossover exponent and nu theta is the tricritical size exponent the best fit is obtained for theta approximately 0 71 compared to theta 0 approximately 1 54 on a pure square lattice nu approximately 0 74 compared to nu theta phi 4 mod 7 approximately 0 57 in two dimensions and phi approximately 0 20 they also obtain an estimate for of saw size exponent nu c for collapsed phase on the percolation cluster

probability densities of random walks in random systems abstract we review recent results for the probability distribution of random walkers in random systems where diffusion is anomalous and the mean square displacement scales with time as r 2 t t 2 w d w 2 the random systems are characterized by structural disorder and by random transition rates in general the mean distribution function p r t of the random walkers is a stretched gaussian and scales as log p r t p r 0 r r t u where u d w d w 1 on random fractals the fluctuations of the density distribution p r t for fixed distance r and time t have a broad logarithmic distribution the average moments p q scale in a multifractal way as p q where q q l space the fluctuations of p are narrow and p q p q

statistics of equally weighted random paths on a class of self similar structures we study the statistics of equally weighted random walk paths on a family of sierpinski gasket lattices whose members are labelled by an integer b 2 b 2 mean path end to end distance grows more slowly than any power of its length n we provide arguments for the emergence of usual power law critical behaviour in the limit b when fractal lattices become almost compact