the analytical solution and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation in this paper we consider the analytical and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation we derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate mittag leffler function an implicit difference approximation is constructed stability and convergence analysis of the numerical scheme are proved by the energy method numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis

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

diffusion on multifractals abstract we introduce a class of random fields with variable mean square regularity order defined on multifractal domains the local singularity order of these random fields then depends on the initial variable mean square regularity order and on the variable local singularity exponent of the multifractal measure defining the local dimension of the domain considered the theory is developed in a generalized framework through the covariance factorization using the tools of reproducing kernel hilbert spaces and fractional sobolev spaces of variable order

on a class of random field models which allows long range dependence summary in this note it is shown how a convenient class of random field models with rational spectral densities can be augmented to incorporate the possibility of long range dependence

convection diffusion equations with random initial conditions abstract we consider an evolution equation generalising the viscous burgers equation supplemented by an initial condition which is a homogeneous random field we develop a non linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour

renormalization and homogenization of fractional diffusion equations with random data this paper presents a renormalization and homogenization theory for fractional in space or in time diffusion equations with singular random initial conditions the spectral representations for the solutions of these equations are provided gaussian and non gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple stochastic integral representations

stratified structure of the universe and burgers equation a probabilistic approach the model of the potential turbulence described by the 3 dimensional burgers equation with random initial data was developped by zeldovich and shandarin in order to explain the existing large scale structure of the universe most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results the gaussian scenario under suitable moment assumptions on the initial velocity field these results and some open questions are discussed in sect 2 where we concentrate on the gaussian model and the shot noise model in sect 3 we construct a probabilistic model of strong initial fluctuations a zero range shot noise field with high amplitudes which reveals an intermittent large time behaviour with the velocity vec v t x determined by the position of the largest initial fluctuation discounted by the heat kernelg t x in a neighborhood ofx the asymptoties of such local maximum ast can be analyzed with the help of the theory of records sect 4 finally in sect 5 we introduce a global definition of a point process oft local maxima and show the weak convergence of the suitably rescaled process to a non trivial limit ast

spectral analysis of fractional kinetic equations with random data we present a spectral representation of the mean square solution of the fractional kinetic equation also known as fractional diffusion equation with random initial condition gaussian and non gaussian limiting distributions of the renormalized solution of the fractional in time and in space kinetic equation are described in terms of multiple stochastic integral representations

global attractors and steady state solutions for a class of reaction diffusion systems abstract we show that weak l p dissipativity implies strong l dissipativity and therefore implies the existence of global attractors for a general class of reaction diffusion systems this generalizes the results of alikakos and rothe the results on positive steady states especially for systems of three equations in our earlier work j differential equations 130 1996 59 91 are improved

a universal well calibrated algorithm for on line classification we study the problem of on line classification in which the prediction algorithm for each significance level is required to output as its prediction a range of labels intuitively those labels deemed compatible with the available data at the level rather than just one label as usual the examples are assumed to be generated independently from the same probability distribution p the prediction algorithm is said to be well calibrated for p and if the long run relative frequency of errors does not exceed almost surely w r to p for well calibrated algorithms we take the number of uncertain predictions i e those containing more than one label as the principal measure of predictive performance the main result of this paper is the construction of a prediction algorithm which for any unknown p and any a makes errors independently and with probability at every trial in particular is well calibrated for p and b makes in the long run no more uncertain predictions than any other prediction algorithm that is well calibrated for p and c processes example n in time o log n

scaling limits of solutions of the heat equation for singular non gaussian data limiting distributions of the parabolically rescaled solutions of the heat equation with singular non gaussian initial data with long range dependence are described in terms of their multiple stochastic integral representations

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

the analytical solution and numerical solution of the fractional diffusion wave equation with damping fractional partial differential equations have been applied to many problems in physics finance and engineering numerical methods and error estimates of these equations are currently a very active area of research in this paper we consider a fractional diffusionwave equation with damping we derive the analytical solution for the equation using the method of separation of variables an implicit difference approximation is constructed stability and convergence are proved by the energy method finally two numerical examples are presented to show the effectiveness of this approximation

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

sample path properties of fractional riesz bessel field of variable order in this paper we consider fractional riesz bessel field of variable order which is also known as multifractional riesz bessel field sample path properties of this random field such as local regularity locally self similar property hausdorff dimension of the graph and long short range dependent property are studied the relationship between the multifractional riesz bessel field and the multifractional brownian field is also established

a fast semi implicit difference method for a nonlinear two sided space fractional diffusion equation with variable diffusivity coefficients in this paper we derive a new nonlinear two sided space fractional diffusion equation with variable coefficients from the fractional fick s law a semi implicit difference method sidm for this equation is proposed the stability and convergence of the sidm are discussed for the implementation we develop a fast accurate iterative method for the sidm by decomposing the dense coefficient matrix into a combination of toeplitz like matrices this fast iterative method significantly reduces the storage requirement of o n 2 and computational cost of o n 3 down to n and o n log n where n is the number of grid points the method retains the same accuracy as the underlying sidm solved with gaussian elimination finally some numerical results are shown to verify the accuracy and efficiency of the new method

the statistics of burgers turbulence initialized with fractional brownian noise data the statistics of the solution to the inviscid burgers equation are investigated when the initial velocity potential is fractional brownian motion using the theory of large deviations for gaussian processes we characterize the tails of the probability distribution functions pdfs of the velocity the distance between shocks and the shock strength these pdfs are shown to decay like stretched exponentials of the form our method of proof can also be used to extend these results to a much larger class of gaussian potentials this work generalizes the results of avellaneda and e 2 3 on the inviscid burgers equation with white noise initial data

unique continuation property for multi terms time fractional diffusion equations this paper concens about a carleman estimate which can give the unique continuation property of solutions for a multi terms time fractional diffusion equation up to order alpha 0 alpha 2 with general time dependent second order strongly elliptic operator for the diffusion by using a special holmgren type transformation which is linear with respect to time the estimate giving a local unique continuation of solutions is derived via some subelliptic estimate for an operator associated to this transformed equation using calculus of pseudo differential operators after that we have given a new argument to derive the global unique continuation of solutions here the global unique continuation means as follows if u is a solution of the multi terms time fractional diffusion equation in a domain over the time interval 0 t and it is supported on t ge 0 then a zero set of solution over a subdomain of omega can be continued to 0 t times omega

the isoperimetric problem on surfaces of revolution of decreasing gauss curvature we prove that the least perimeter way to enclose prescribed area in the plane with smooth rotationally symmetric complete metric of nonincreasing gauss curvature consists of one or two circles bounding a disc the complement of a disc or an annulus we also provide a new isoperimetric inequality in general surfaces with boundary

generalization bounds for averaged classifiers we study a simple learning algorithm for binary classification instead of predicting with the best hypothesis in the hypothesis class that is the hypothesis that minimizes the training error our algorithm predicts with a weighted average of all hypotheses weighted exponentially with respect to their training error we show that the prediction of this algorithm is much more stable than the prediction of an algorithm that predicts with the best hypothesis by allowing the algorithm to abstain from predicting on some examples we show that the predictions it makes when it does not abstain are very reliable finally we show that the probability that the algorithm abstains is comparable to the generalization error of the best hypothesis in the class

n dimensional fractional diffusion equation and green function approach spatially dependent diffusion coefficient and external force we investigate an n dimensional fractional diffusion equation with radial symmetry by using the green function approach we consider in our analysis the spatial dependence on the diffusion coefficient and the presence of an external force in particular we employ boundary conditions in a finite interval and after we extend it to a semi infinite interval we also show that a rich class of diffusive processes including normal and anomalous ones can be obtained from the solutions found here

harmonic analysis of random fractional diffusion wave equations this paper presents the green functions and spectral representations of the mean square solutions of the fractional diffusion wave equations with random initial conditions

prescribing curvature on open surfaces in this article we study the problem sometimes called the berger nirenberg problem of prescribing the curvature on a riemann surface that is on an oriented surface equipped with a conformal class of riemannian metrics

a computationally effective alternating direction method for the space and time fractional bloch torrey equation in 3 d the space and time fractional bloch torrey equation st fbte has been used to study anomalous diffusion in the human brain numerical methods for solving st fbte in three dimensions are computationally demanding in this paper we propose a computationally effective fractional alternating direction method fadm to overcome this problem we consider st fbte on a finite domain where the time and space derivatives are replaced by the caputo djrbashian and the sequential riesz fractional derivatives respectively the stability and convergence properties of the fadm are discussed finally some numerical results for st fbte are given to confirm our theoretical findings

generalized brownian functionals and the solution to a stochastic partial differential equation abstract a class of generalized wiener functionals related to those of hida and watanabe is introduced and the malliavin calculs associated with these functionals is developed these notions are applied to the derivation of a solution to the stochastic partial differential equation y t ly y where l is a second order partial differential operator in the n space variables denotes a white noise in the n 1 space time variables y denotes a skorohod type stochastic integral and is a non random function of the n 1 independent variables

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

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

a coverage theory for least squares summary a sensible use of an estimation method requires that assessment criteria for the quality of the estimate be available we present a coverage theory for the least squares estimate by suitably modifying the empirical costs one constructs statistics that are guaranteed to cover with known probability the cost associated with a next still unseen member of the population all results of this paper are distribution free and can be applied to least squares problems in use across a variety of fields

on a class of retarded integro differential equations with nonlocal initial conditions of concern is the following cauchy problem for fractional integro differential equations with time delay and nonlocal initial condition u t email protected 0 t t s m email protected m 1 au s ds f t u t u k t t 0 u t h t u email protected t email protected email protected 0 where 1 0 a d a x x is a generator of a solution operator on a complex banach space x the convolution integral in the equation is known as the riemann liouville fractional integral k t 0 email protected representing the delay property is a function and h t is an operator defined from email protected 0 xc email protected 0 x into x for some t 0 which constitutes a nonlocal condition the local existence and uniqueness of mild solutions for the cauchy problem under various criteria are proved moreover we present an existence result of the global solution also an example is given to illustrate the applications of the abstract results

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

nonlinear stochastic wave and heat equations we study nonlinear wave and heat equations on d driven by a spatially homogeneous wiener process for the wave equation we consider the cases of d 1 2 3 the heat equation is considered on an arbitrary d space we give necessary and sufficient conditions for the existence of a function valued solution in terms of the covariance kernel of the noise

on the solutions of linear odd order heat type equations with random initial conditions in this paper odd order heat type equations with different random initial conditions are examined in particular we give rigorous conditions for the existence of the solutions in the case where the initial condition is represented by a strictly subgaussian harmonizable process x also the case where is represented by a stochastic integral with respect to a process with independent increment is studied

the inviscid burgers equation with brownian initial velocity the law of the hopf cole solution of the inviscid burgers equation with brownian initial velocity is made explicit as examples of applications we investigate the smoothness of the solution the statistical distribution of the shocks we determine the exact hausdorff function of the lagrangian regular points and investigate the existence of lagrangian regular points in a fixed borel set

a meshless method based on point interpolation method pim for the space fractional diffusion equation this paper aims to develop a meshless approach based on the point interpolation method pim for numerical simulation of a space fractional diffusion equation two fully discrete schemes for the one dimensional space fractional diffusion equation are obtained by using the pim and the strong forms of the space diffusion equation numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach

fractional diffusion equation and green function approach exact solutions abstract we investigate the solutions of a fractional diffusion equation with radial symmetry by using the green function approach and by taking the n dimensional case into account in our analysis a spatial time dependent diffusion coefficient is considered i e d r t d t 1 r the presence of external forces f r k r e with e 1 and f r kr k r e is also taken into account in particular we discuss the results obtained by employing boundary conditions defined on a finite interval and afterwards the analysis is extended to a semi infinite interval finally we also discuss a rich class of diffusive processes that can be obtained from the results presented in this work

gaussian limiting behavior of the rescaled solution to the linear korteweg de vries equation with random initial conditions we analyze the asymptotic behavior of the rescaled solution to the linear korteweg de vries equation when the initial conditions are supposed to be random and weakly dependent by means of the method of moments we prove the gaussianity of the limiting process and we present its correlation function the same technique is applied to the analysis of another third order heat type equation

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

models of anomalous diffusion the subdiffusive case abstract the paper discusses a model for anomalous diffusion processes their one point probability density functions p d f are exact solutions of fractional diffusion equations the model reflects the asymptotic behavior of a jump anomalous random walk process with random jump sizes and random inter jump time intervals with infinite means and variances which do not satisfy the law of large numbers in the case when these intervals have a fractional exponential p d f the fractional komogorov feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed finally some statistical properties of solutions of the related langevin equation are studied the subdiffusive case is explored in detail the emphasis is on a rigorous presentation which however would be accessible to the physical sciences audience

estimates for functionals of solutions to higher order heat type equations with random initial conditions in the present paper we continue the investigation of solutions to higher order heat type equations with random initial conditions which play the important role in many applied areas we consider the random initial conditions given by harmonizable varphi sub gaussian processes the main results are the bounds for the distributions of the suprema over bounded and unbounded domains for solutions of such equations the results obtained in the paper hold in particular for the case of gaussian initial condition

on the strong universal consistency of nearest neighbor regression function estimates two results are presented concerning the consistency of the k nearest neighbor regression estimate we show that all modes of convergence in l 1 in probability almost sure complete are equivalent if the regression variable is bounded under the additional condition k log n we also obtain the strong universal consistency of the estimate

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

unique continuation property for the anomalous diffusion and its application abstract the unique continuation for anomalous diffusion operators with fractional time derivative of order 1 2 is proved by deriving carleman estimates for the operators by applying operators which have a negative sign in front of the diffusion terms of anomalous diffusion operators the anomalous diffusion operators are transformed to parabolic operators of order 4 in the space variables the carleman estimates are derived for these transformed operators considered as semi elliptic operators the usual calderon uniqueness argument for semi elliptic operators does not work due to the singularities in factors of the factorization of their principal parts in order to have smooth factorizations of the operators we took into account the lower order terms of the operators