the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

variable order fractional differential operators in anomalous diffusion modeling abstract the purpose of this paper is to offer a unified discussion of variable order differential operators in anomalous diffusion modeling the characteristics of the new models in contrast to constant order fractional diffusion models change with time space concentration or other independent quantities we introduced a classification of variable order fractional diffusion models based on the possible physical origins which prompt the variable order some potential applications of the variable order fractional diffusion models are also discussed

non isospectral scattering problems and truncation for hierarchies burgers and dispersive water waves abstract non isospectral scattering problems have been proven useful for several reasons amongst them the information that they provide about painleve truncation for entire hierarchies of integrable partial differential equations pdes we show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1 1 dimensions namely burgers hierarchy and the dispersive water wave hierarchy burgers equation is well known as a model for turbulence and the dispersive water wave equations as a system governing shallow water waves we also see how these results are easily extended to related hierarchies in 2 1 dimensions our results for the 2 1 dimensional burgers hierarchy and for the 1 1 and 2 1 dimensional dispersive water wave hierarchies are all new

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

binary models for marginal independence summary log linear models are a classical tool for the analysis of contingency tables in particular the subclass of graphical log linear models provides a general framework for modelling conditional independences however with the exception of special structures marginal independence hypotheses cannot be accommodated by these traditional models focusing on binary variables we present a model class that provides a framework for modelling marginal independences in contingency tables the approach that is taken is graphical and draws on analogies with multivariate gaussian models for marginal independence for the graphical model representation we use bidirected graphs which are in the tradition of path diagrams we show how the models can be parameterized in a simple fashion and how maximum likelihood estimation can be performed by using a version of the iterated conditional fitting algorithm finally we consider combining these models with symmetry restrictions

wellposedness and regularity of the variable order time fractional diffusion equations abstract it was proved that the solution to a time fractional partial differential equation lacks regularity at the initial time t 0 which is often inconsistent with the physical problem being modeled even though the nature of the physical problem away from t 0 is well captured by a time fractional partial differential equation we argue that the fundamental reason why this phenomenon occurs lies in the incompatibility between the nonlocality of the time fractional partial differential equations and the locality of the classical initial condition at t 0 we show that the variable order time fractional partial differential equations in which the order varies in time as t goes to 0 to accommodate the impact of the local initial condition at time t 0 turn out to be a natural candidate to eliminate the nonphysical singularity of the solutions to the constant order time fractional partial differential equations and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa we prove the wellposedness of a variable order linear time fractional diffusion partial differential equation in multiple space dimensions we also prove that the regularity of its solution depends on the behavior of the variable order and its derivatives at time t 0 in addition to the usual smoothness assumptions more precisely we prove that its solution has full regularity like its integer order analogue if the variable order has an integer limit at t 0 or exhibits singular behaviors at t 0 like in the case of the constant order time fractional partial differential equations if the variable order has a non integer value at time t 0

random networks graphical models and exchangeability we study conditional independence relationships for random networks and their interplay with exchangeability we show that for finitely exchangeable network models the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts we then characterize all possible markov structures for finitely exchangeable random graphs thereby identifying a new class of markov network models corresponding to bidirected kneser graphs in particular we demonstrate that the fundamental property of dissociatedness corresponds to a markov property for exchangeable networks described by bidirected line graphs finally we study those exchangeable models that are also summarized in the sense that the probability of a network depends only on the degree distribution and we identify a class of models that is dual to the markov graphs of frank and strauss particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of markov properties correspond to the empty graph the bidirected line graph of the complete graph and the complete graph

a generalized spectral collocation method with tunable accuracy for variable order fractional differential equations we generalize existing jacobi gauss lobatto collocation methods for variable order fractional differential equations using a singular approximation basis in terms of weighted jacobi polynomials of the form 1 pm x mu p j a b x where mu 1 in order to derive the differentiation matrices of the variable order fractional derivatives we develop a three term recurrence relation for both integrals and derivatives of these weighted jacobi polynomials hence extending the three term recurrence relationship of jacobi polynomials the new spectral collocation method is applied to solve fractional ordinary and partial differential equations with endpoint singularities we demonstrate that the singular basis enhances greatly the accuracy of the numerical solution by properly tuning the parameter mu even for cases where we do not know explicitly the form of singularity in the solution at the boundaries

modified extended tanh function method and its applications to nonlinear equations new exact travelling wave solutions for the generalized shallow water wave equation the improved boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh function method the obtained results include rational periodic singular and solitary wave solutions

using reproducing kernel for solving a class of fractional partial differential equation with non classical conditions today most of the real physical world problems can be best modeled with fractional differential equations besides modeling the solution techniques and their reliabilities are most important therefore high accuracy solutions are always needed in this paper a new method is provided to solve fractional partial differential equation in a very favorable reproducing kernel space it s reproducing kernel function is discussed in detail from the examples considered here it can be easily seen that our method has small computational work fast convergence speed and high precision

time stepping discontinuous galerkin methods for fractional diffusion problems time stepping hp hp versions discontinuous galerkin dg methods for the numerical solution of fractional subdiffusion problems of order alpha with 1 alpha 0 1 0 will be proposed and analyzed generic hp hp version error estimates are derived after proving the stability of the approximate solution for h h version dg approximations on appropriate graded meshes near t 0 t 0 we prove that the error is of order o k max 2 p frac alpha 2 o kmax 2 p 2 where k k is the maximum time step size and p ge 1 p 1 is the uniform degree of the dg solution for hp hp version dg approximations by employing geometrically refined time steps and linearly increasing approximation orders exponential rates of convergence in the number of temporal degrees of freedom are shown finally some numerical tests are given

fractional cable equation models for anomalous electrodiffusion in nerve cells finite domain solutions in recent work we introduced fractional nernst planck equations and related fractional cable equations to model electrodiffusion of ions in nerve cells with anomalous subdiffusion along and across the nerve cells this work was motivated by many computational and experimental studies showing that anomalous diffusion is ubiquitous in biological systems with binding crowding or trapping for example recent experiments have shown that anomalous subdiffusion occurs along the axial direction in spiny dendrites due to trapping by the spines we modeled the subdiffusion in two ways leading to two fractional cable equations and presented fundamental solutions on infinite and semi infinite domains here we present solutions on finite domains for mixed robin boundary conditions the finite domain solutions model passive electrotonic properties of spiny dendritic branch segments with ends that are voltage clamped sealed or killed the behavior of the finite domain solutions is similar for both fractional cable

sharp error estimate of the nonuniform l1 formula for linear reaction subdiffusion equations stability and convergence of the l1 formula on nonuniform time grids are studied for solving linear reaction subdiffusion equations with the caputo derivative a discrete fractional gronwall inequality is developed for the nonuniform l1 formula by introducing a discrete convolution kernel of riemann liouville fractional integral to simplify the consistency analysis of the nonuniform l1 formula we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete riemann liouville integral kernel with the help of discrete fractional gronwall inequality and global consistency error analysis a sharp error estimate reflecting the regularity of solution is obtained for a simple l1 scheme numerical examples are provided to verify the sharpness of the error analysis

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

discovering variable fractional orders of advection dispersion equations from field data using multi fidelity bayesian optimization abstract the fractional advection dispersion equation fade can describe accurately the solute transport in groundwater but its fractional order has to be determined a priori here we employ multi fidelity bayesian optimization to obtain the fractional order under various conditions and we obtain more accurate results compared to previously published data moreover the present method is very efficient as we use different levels of resolution to construct a stochastic surrogate model and quantify its uncertainty we consider two different problem set ups in the first set up we obtain variable fractional orders of one dimensional fade considering both synthetic and field data in the second set up we identify constant fractional orders of two dimensional fade using synthetic data we employ multi resolution simulations using two level and three level gaussian process regression models to construct the surrogates

a new general algebraic method with symbolic computation to construct new doubly periodic solutions of the 2 1 dimensional dispersive long wave equation for constructing more new exact doubly periodic solutions in terms of rational form jacobi elliptic function of nonlinear evolution equations a new direct and unified algebraic method named jacobi elliptic function rational expansion method is presented and implemented in a computer algebraic system compared with most of the existing jacobi elliptic function expansion methods the proposed method can be expected to obtain new and more general formal solutions we choose a 2 1 dimensional dispersive long wave equation to illustrate the method

classical liouville completely integrable systems associated with the solutions of boussinesq burgers hierarchy two new finite dimensional completely integrable systems in the liouville sense are obtained the solutions of boussinesq burgers hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

solving a system of linear volterra integral equations using the new reproducing kernel method in this paper we present a new numerical technique to obtain the approximation solution for linear volterra integral equations of the second kind based on reproducing kernel theory the approximation solution is expressed by n term summation of reproducing kernel functions the merit of the new method includes 1 it is easy to implement this method 2 high accuracy the numerical examples compared with other methods show that the new method is more efficient

error analysis of a finite difference method on graded meshes for a time fractional diffusion equation a reaction diffusion problem with a caputo time derivative of order alpha in 0 1 is considered the solution of such a problem is shown in general to have a weak singularity near the initial time t 0 and sharp pointwise bounds on certain derivatives of this solution are derived a new analysis of a standard finite difference method for the problem is given taking into account this initial singularity this analysis encompasses both uniform meshes and meshes that are graded in time and includes new stability and consistency bounds the final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme so one can choose an optimal mesh grading numerical results are presented that confirm the sharpness of the error analysis

association models for a multivariate binary response models for a multivariate binary response are parameterized by univariate marginal probabilities and dependence ratios of all orders the w order dependence ratio is the joint success probability of w binary responses divided by the joint success probability assuming independence this parameterization supports likelihood based inference for both regression parameters relating marginal probabilities to explanatory variables and association model parameters relating dependence ratios to simple and meaningful mechanisms five types of association models are proposed where responses are 1 independent given a necessary factor for the possibility of a success 2 independent given a latent binary factor 3 independent given a latent beta distributed variable 4 follow a markov chain and 5 follow one of two first order markov chains depending on the realization of a binary latent factor these models are illustrated by reanalyzing three data sets foremost a set of binary time series on auranofin therapy against arthritis likelihood based approaches are contrasted with approaches based on generalized estimating equations association models specified by dependence ratios are contrasted with other models for a multivariate binary response that are specified by odds ratios or correlation coefficients

rational solutions of nonlinear evolution equations vertex operators and bispectrality abstract we prove that if q x t 2 t m and r x t 2 t m are certain rational solutions of the akns hierarchy then there are eigenfunctions x k of the akns zs operator l x q r x satisfying a differential equation in the spectral parameter k of the form b k k x where b k k is a matrix differential operator independent of x and is a nonconstant function of x we also discuss the relation between this result and a similar one for the rational solutions of the schrodinger operator with potentials in the manifold of rational solutions of the kdv hierarchy

numerical solution of the time fractional fokker planck equation with general forcing we study two schemes for a time fractional fokker planck equation with space and time dependent forcing in one space dimension the first scheme is continuous in time and discretized in space using a piecewise linear galerkin finite element method the second is continuous in space and employs a time stepping procedure similar to the classical implicit euler method we show that the space discretization is second order accurate in the spatial l 2 norm uniformly in time whereas the corresponding error for the time stepping scheme is o k alpha for a uniform time step k where alpha in 1 2 1 is the fractional diffusion parameter in numerical experiments using a combined fully discrete method we observe convergence behavior consistent with these results

on soliton type solutions of equations associated with n component systems the algebraic geometric approach to n component systems of nonlinear integrable pde s is used to obtain and analyze explicit solutions of the coupled kdv and dym equations detailed analysis of soliton fission kink to anti kink transitions and multi peaked soliton solutions is carried out transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics

regularity theory for time fractional advection diffusion reaction equations abstract we investigate the behavior of the time derivatives of the solution to a linear time fractional advection diffusion reaction equation allowing space and time dependent coefficients as well as initial data that may have low regularity our focus is on proving estimates that are needed for the error analysis of numerical methods the nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic pde in our analysis we rely on novel energy methods in combination with a fractional gronwall inequality and certain properties of fractional integrals

multivariate logistic models summary when data composed of several categorical responses together with categorical or continuous predictors are observed the multivariate logistic transform introduced by mccullagh and nelder can be used to define a class of regression models that is in many applications particularly suitable for relating the joint distribution of the responses to predictors in this paper we give a general definition of this class of models and study their properties a computational scheme for performing maximum likelihood estimation for data sets of moderate size is described and a system of model formulae that succinctly define particular models is introduced applications of these models to longitudinal problems are illustrated by numerical examples

factorization of the classical boussinesq system the scalar equation equivalent to the classical boussinesq or broer kaup system is shown to factorize into a differential operator acting on burgers equation the linearizability of this latter equation provides a very simple explanation for the recently found property of fusion and fission of a class of solitons for this system

method for deriving rational solutions of some nonlinear evolution equations maps for nonlinear evolution equations are discussed an algorithmic method for deriving rational solutions is presented this approach is illustrated by examples which have solutions in the form of two truncated expansions

log mean linear models for binary data this paper is devoted to the theory and application of a novel class of models for binary data which we call log mean linear lml models the characterizing feature of these models is that they are specied by linear constraints on the

marginal regression analysis of a multivariate binary response we propose the use of the mean parameter for regression analysis of a multivariate binary response we model the association using dependence ratios defined in terms of the mean parameter the components of which are the joint success probabilities of all orders this permits flexible modelling of higher order associations using maximum likelihood estimation we reanalyse two data sets one with variable cluster size and the other a longitudinal data set with constant cluster size

sample size for logistic regression with small response probability abstract the fisher information matrix for the estimated parameters in a multiple logistic regression can be approximated by the augmented hessian matrix of the moment generating function for the covariates the approximation is valid when the probability of response is small with its use one can obtain a simple closed form estimate of the asymptotic covariance matrix of the maximum likelihood parameter estimates and thus approximate sample sizes needed to test hypotheses about the parameters the method is developed for selected distributions of a single covariate and for a class of exponential type distributions of several covariates it is illustrated with an example concerning risk factors for coronary heart disease

on the integrable variant of the boussinesq system painleve property rational solutions a related many body system and equivalence with the akns hierarchy abstract an infinite family of rational solutions of the integrable version of the boussinesq system u t w x uu x 0 w t u xxx uw x 0 and the associated higher order flows is constructed the differential equations governing the motion of the poles are derived and their complete integrability demonstrated this system embeds into two disjoint calogero moser systems coupled only through a constraint a novel treatment of this constraint is given using the equivalence of to the real version of the nonlinear schrodinger system it is seen that rational solutions of the akns hierarchy have been constructed as well finally it is shown that all solutions of these hierarchies including the rational ones also solve the first modified kp equation

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

finite band solutions of the classical boussinesq burgers equations the classical boussinesq burgers hierarchy is decomposed into two systems of solvable ordinary differential equations with the help of the lax representations of stationary evolution equations from which the finite band solutions of the higher order classical boussinesq burgers equations are obtained

on a direct bilinearization method kaup s higher order water wave equation as a modified nonlocal boussinesq equation a systematic procedure for the bilinearization of classes of soliton equations is developed with the help of a generalization of bell s exponential polynomials 1934 application of this procedure to kaup s higher order wave equation 1975 discloses several links with other soliton systems in particular it is found that the kaup equation is the modified version of a sech square soliton system which constitutes an alternative to the good boussinesq equation

solitonic solutions for a variable coefficient variant boussinesq system in the long gravity waves variable coefficient variant boussinesq vcvb system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth in this paper with symbolic computation a lax pair associated with the vcvb system under some constraints for variable coefficients is derived and based on the lax pair two sorts of basic darboux transformations are presented by applying the darboux transformations some solitonic solutions are obtained with the relevant constraints given in the text in addition the vcvb system is transformed to a variable coefficient broer kaup system solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics

crum transformation and rational solutions of the non focusing nonlinear schrodinger equation a crum transformation for the linear problem of the non focusing nonlinear schrodinger nls equation is used to construct the sequence of rational solutions given in terms of tau functions expressed as double wronskian determinants the rational solutions of the adler kaup newell segur akns hierarchy have already been obtained in the work of sachs using the equivalence with the classical boussinesq hierarchy however we find that sachs description of the restrictions that must be placed on the akns tau functions in order to obtain nls rational solutions is not entirely correct we briefly comment on the constrained calogero moser systems associated with the nls rational solutions and indicate how the calogero moser equations can be obtained from a trilinear equation for the tau function which arises by reduction of the kadomtsev petviashvili kp hierarchy

kp constraints from reduced multi component hierarchies the m vector k constrained kadomtsev petviashvili kp hierarchy is shown to be a pseudo reduction of the m 1 component kp hierarchy to facilitate the implementation of this reduction on the level of the solutions the typical multi component kp solutions are mapped onto solutions of a toda molecule type equation from which wronskian and grammian solutions for the constrained kp hierarchy follow the reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

modified singular manifold expansion application to the boussinesq and mikhailov shabat systems we present a modified treatment of the singular manifold method as an improved variant of the painleve analysis for partial differential equations with two branches in the painleve expansion we illustrate the method by fully applying it to the classical boussinesq system and to the mikhailov shabat system

maximum likelihood methods for a generalized class of log linear models we discuss maximum likelihood methods for fitting a broad class of multivariate categorical response data models in particular we derive the large sample distributions for maximum likelihood estimators of parameters of product multinomial generalized log linear models the large sample behavior of other relevant likelihood based statistics such as goodness of fit statistics and adjusted residuals is also described the asymptotic results are derived within the framework of the constraint specification rather than the more common freedom specification of the model we also outline an improved fitting algorithm for computing parameter maximum likelihood estimates and other relevant statistics the broad class of multivariate categorical response data models which are referred to as generalized log linear models can imply structure on several response configuration distributions e g joint and marginal distributions these models which include as special cases log linear logit and cumulative logit models enjoy a wide breadth of application including longitudinal rater agreement and crossover data analyses

conslaw a maple package to construct the conservation laws for nonlinear evolution equations an improved algorithm to explicitly compute the polynomial type conservation laws for nonlinear evolution equations either uniform in rank or not is introduced and a software package conslaw written in maple to automate the computation is developed conslaw can construct the polynomial type conservation laws for polynomial partial differential equations automatically furthermore some new integrable systems can be filtered out by analyzing the compatibility conditions which guarantee the existence of the conservation laws for a given parameterized nonlinear evolution equations the explicit forms of the conserved densities play an important role in studying the integrability such as explicit analytical solutions bi hamiltonian form one parameter family of backlund transformations lax pairs and the checking of the accuracy of numerical integration algorithm the effectiveness of conslaw is illustrated by applying it to a variety of equations

transformation formulae of the solutions of the two dimensional toda lattice two transformation formulae of the solutions of the two dimensional toda lattice are presented the first formula which can be obtained by the zakharov shabat dressing method is proved by applying the hirota bilinear transformation method 1988 and gives the basis for the second new formula

fitted reproducing kernel hilbert space method for the solutions of some certain classes of time fractional partial differential equations subject to initial and neumann boundary conditions latterly many problems arising in different fields of science and engineering can be reduced by applying some appropriate discretization to a series of time fractional partial differential equations unlike the normal case derivative the differential order in such equations is with a fractional order which will lead to new challenges for numerical simulation the purpose of this analysis is to introduce the reproducing kernel hilbert space method for treating classes of time fractional partial differential equations subject to neumann boundary conditions with parameters derivative arising in fluid mechanics chemical reactions elasticity anomalous diffusion and population growth models the method provides appropriate representation of the solutions in convergent series formula with accurately computable components numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n term of the exact solutions finally the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time fractional neumann problems

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

tempered fractional feynman kac equation theory and examples functionals of brownian and non brownian motions have diverse applications and attracted a lot of interest among scientists this paper focuses on deriving the forward and backward fractional feynman kac equations describing the distribution of the functionals of the space and time tempered anomalous diffusion belonging to the continuous time random walk class several examples of the functionals are explicitly treated including the occupation time in half space the first passage time the maximal displacement the fluctuations of the occupation fraction and the fluctuations of the time averaged position

local well posedness for the super korteweg de vries equation abstract we establish some qualitative properties for the cauchy problem associated with the super korteweg de vries equation super kdv super kdv t u x 3 u 1 2 x u 2 1 2 x 2 v 2 0 t v x 3 v x u v 0 x t r we prove local well posedness in weighted sobolev spaces x s 3 h s r h 3 x 2 d x h s r h 3 x 2 d x s 5 integer provided that the initial data is small enough and also in x s 11 h s r h 11 x 2 d x h s r h 11 x 2 d x with s 15 integer for arbitrary initial data the main ingredients of the proof for the first case are new estimates describing the smoothing effect of kato type for the kdv group w t t r that for the second case is via a change of variable performed and a deduction of new smoothing effects related to the kdv c kenig g staffilani local well posedness for higher order nonlinear dispersive systems j fourier anal appl 3 4 1997

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

huygens principle for hyperbolic operators and integrable hierarchies abstract we show that the stationary solutions of the canonical akns hierarchy of nonlinear evolution equations yield perturbations of dirac operators that satisfy a strict form of huygens principle namely the domain of dependence of such dirac operators at any given point y is contained in the light cone s hypersurface issued from y by the canonical akns hierarchy we mean that the differential polynomials defining the flows are isobaric with respect to certain weights the method we employ is of interest by itself indeed we consider the riesz kernels associated to a given hyperbolic differential operator and expand the fundamental solution of perturbations of this operator in a series in such riesz kernels using the coefficients of this hadamard type expansion we introduce a family of vector fields for the d alembertian such vector field family corresponds to the kdv hierarchy and for the dirac operators they include the akns one

the method of approximate particular solutions for the time fractional diffusion equation with a non local boundary condition in this paper we consider the numerical solution of the time fractional diffusion equation with a non local boundary condition the method of approximate particular solutions maps using multiquadric radial basis function mq rbf is employed for this equation due to the accuracy of the mq based meshless methods is severely influenced by the shape parameter we adopt a leave one out cross validation loocv algorithm proposed by rippa 34 to enhance the performance of the maps the numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time fractional diffusion equation with a non local boundary condition

bilinearization of the non local boussinesq equation a single field bilinear system generating the so called non local boussinesq equation is constructed from the bilinearization procedure it can be seen that the associated hierarchy of soliton systems which we construct shares part of the solution set of a hierarchy related to the kadomtsev petviashvili equation a bilinear representation of the recursion operator for the kaup hierarchy is essential in the construction and a systematic way of obtaining such a representation from just two soliton considerations is presented

cumulative distribution networks and the derivative sum product algorithm models and inference for cumulative distribution functions on graphs we present a class of graphical models for directly representing the joint cumulative distribution function cdf of many random variables called cumulative distribution networks cdns unlike graphs for probability density and mass functions for cdfs the marginal probabilities for any subset of variables are obtained by computing limits of functions in the model and conditional probabilities correspond to computing mixed derivatives we will show that the conditional independence properties in a cdn are distinct from the conditional independence properties of directed undirected and factor graphs but include the conditional independence properties of bi directed graphs in order to perform inference in such models we describe the derivative sum product dsp message passing algorithm in which messages correspond to derivatives of the joint cdf we will then apply cdns to the problem of learning to rank players in multiplayer team based games and suggest several future directions for research

discrete maximal regularity of time stepping schemes for fractional evolution equations in this work we establish the maximal ell p regularity for several time stepping schemes for a fractional evolution model which involves a fractional derivative of order alpha in 0 2 alpha ne 1 in time these schemes include convolution quadratures generated by backward euler method and second order backward difference formula the l1 scheme explicit euler method and a fractional variant of the crank nicolson method the main tools for the analysis include operator valued fourier multiplier theorem due to weis math ann 319 735 758 2001 doi 10 1007 pl00004457 and its discrete analogue due to blunck stud math 146 157 176 2001 doi 10 4064 sm146 2 3 these results generalize the corresponding results for parabolic problems

the bilinear negative kadomtsev petviashvili system and its kac moody virasoro symmetry group the symmetry transformation group of the bilinear negative kadomtsev petviashvili system is studied by means of a direct method the kac moody virasoro type lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

rational solutions of the classical boussinesq hierarchy the classical boussinesq system and its associated higher order flows in the bilinear form are considered an infinite family of homogeneous rational solutions of this hierarchy is verified by the use of the wronskian technique developed by freeman and nimmo 1983

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

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

marginal log linear parameterization of conditional independence models models defined by a set of conditional independence restrictions play an important role in statistical theory and applications especially but not only in graphical modelling in this paper we identify a subclass of these consisting of hierarchical marginal log linear models as defined by bergsma rudas 2002a such models are smooth which implies the applicability of standard asymptotic theory and simplifies interpretation furthermore we give a marginal log linear parameterization and a minimal specification of the models in the subclass which implies the applicability of standard methods to compute maximum likelihood estimates and simplifies the calculation of the degrees of freedom of chi squared statistics to test goodness of fit the utility of the results is illustrated by applying them to block recursive markov models associated with chain graphs

a two grid mmoc finite element method for nonlinear variable order time fractional mobile immobile advection diffusion equations abstract a fully discrete two grid modified method of characteristics mmoc scheme is proposed for nonlinear variable order time fractional advection diffusion equations in two space dimensions the mmoc is used to handle the advection dominated transport and the two grid method is designed for efficiently solving the resulting nonlinear system optimal l 2 error estimates are derived for both the mmoc scheme and the corresponding two grid mmoc scheme numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method

the two singular manifold method ii classical boussinesq system for pt i see musette et al ibid vol 24 p 3895 1994 the two singular manifold method a generalization of the singular manifold method of weiss 1983 is applied to the classical boussinesq system also known as the broer kaup system from the point of view of its singularity analysis the important feature of this system is the existence of two principal families with opposite principal parts the usual singular manifold method takes into account only one of these families at a time our generalization takes into account both families and in this way we are able to derive the lax pair and darboux transformation and hence the auto backlund transformation for the classical boussinesq system from its painleve analysis

existence and regularity of solutions to time fractional diffusion equations we investigate some initialboundary value problems for time fractional diffusion equations of order 0 1 such equations model anomalous diffusion on fractals the existence of solution irrelevant to is established only if the external force function f is weighted hlder continuous which is weaker than hlder continuous some interesting versions of maximal and spatial regularity criteria depending on the fractional exponent are also discussed