a numerical approach to the generalized nonlinear fractional fokker planck equation in this paper we propose a fully discrete galerkin finite element method to solve the generalized nonlinear fractional fokker planck equation which has a multi fractional spatial operator characteristic that describes the levy flight in the time direction we use the finite difference method and in the spatial direction we use the fractional finite element method in the framework of the fractional sobolev spaces we derive a fully discrete scheme for the considered equation we prove the existence and uniqueness of the discrete solution and give the error estimates the numerical examples are also included which support the theoretical analysis

pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

fractional derivatives in complex planes abstract many models are formulated in terms of fractional derivatives such as in viscoelasticity electrochemistry electrode electrolyte polarization signal processing diffusion processes control processing etc in this paper we first study important properties of the caputo derivative in real line then we study the recently developed fractional derivative in complex plane by ortigueira which is very useful in signal processing we also generalize the caputo derivative in real line to that in complex plane then study its properties these discussions are helpful in understanding fractional calculus and establishing fractional models in science and engineering

approximations of fractional integrals and caputo fractional derivatives abstract in this paper we propose two algorithms for numerical fractional integration and caputo fractional differentiation we present a modification of trapezoidal rule that is used to approximate finite integrals the new modification extends the application of the rule to approximate integrals of arbitrary order 0 we then using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order 0 where the fractional derivative based on caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points the study is conducted through illustrative examples and error analysis

a rbf based differential quadrature method for solving two dimensional variable order time fractional advection diffusion equation abstract numerical simulation technique of two dimensional variable order time fractional advection diffusion equation is developed in this paper using radial basis function based differential quadrature method rbf dq to the best of the authors knowledge this is the first application of this method to variable order time fractional advection diffusion equations for the general case of irregular geometries the meshless local form of rbf dq is used and the multiquadric type of radial basis functions is selected for the computations this approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the dirichlet and neumann boundary conditions on the irregular boundaries the method is validated by the well documented test examples involving variable order fractional modeling of air pollution the numerical results demonstrate that the proposed method provides accurate solutions for two dimensional variable order time fractional advection diffusion equations

fractional kinetic equations and universal decoupling of a memory function in mesoscale region it is proved that kinetic equations containing non integer integrals and derivatives appear in the result of reduction of a set of micromotions to some averaged collective motion in the mesoscale region in other words it means that after a proper statistical average the microscopic dynamics is converted into a collective motion in the mesoscopic regime a fractal medium containing weakly and strongly correlated relaxation units has been considered it is shown that in most cases the original of the memory function recovers the riemann liouville integral for a strongly correlated fractal medium a generalization of the riemann liouville integral is obtained for the fractal branched processes one can derive the stretched exponential law of relaxation that is widely used for description of relaxation phenomena in disordered media it is shown that the generalized stretched exponential function describes the averaged collective motion in the fractal branched complex systems it is confirmed that the generalized stretched exponential law can be definitely applicable for description of a set of spontaneous evoked postsynaptic signals in the living matter the application of these fractional kinetic equations for the description of the dielectric relaxation phenomena is also discussed these kinetic equations and their generalizations can be applicable for the description of different relaxation or diffusion processes in the intermediate mesoscale region

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

analytical solution for the time fractional telegraph equation by the method of separating variables in this paper a method of separating variables is effectively implemented for solving a time fractional telegraph equation tfte we discuss and derive the analytical solution of the tfte with three kinds of nonhomogeneous boundary conditions namely dirichlet neumann and robin boundary conditions

exact solutions to nonlinear nonautonomous space fractional diffusion equations with absorption we analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions in particular the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation we also discuss some implications by considering a diffusion coefficient d x t d t x theta theta in r and a drift force f k 1 t x k alpha x x alpha 1 in both situations we relate our solutions to those obtained within the maximum entropy principle by using the tsallis entropy

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

spectral analysis of fractional hyperbolic diffusion equations with random data the paper studies the fundamental solutions to fractional in time hyperbolic diffusion equation or telegraph equations and their properties then it derives the exact solutions of the fractional hyperbolic diffusion equation with random data in terms of series expansions of isotropic in space spherical random fields on the unit sphere numerical illustration are presented to illustrate the theoretical results

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

theories of thermal stresses based on space time fractional telegraph equations the generalized telegraph equations with time and space fractional derivatives are considered the corresponding theories of thermal stresses are formulated the proposed theories interpolate the classical thermoelasticity the theory of lord and shulman thermoelasticity without energy dissipation of green and naghdi and theories of fractional thermoelasticity proposed earlier the fundamental solution to the nonhomogeneous space time fractional telegraph equation as well as the corresponding thermal stresses are obtained in the axisymmetric case

the fractional d alembert s formulas abstract in this paper we develop generalized d alembert s formulas for abstract fractional integro differential equations and fractional differential equations on banach spaces some examples are given to illustrate our abstract results and the probability interpretations of these fractional d alembert s formulas are also given moreover we also provide d alembert s formulas for abstract fractional telegraph equations

fractional non linear linear and sublinear death processes this paper is devoted to the study of a fractional version of non linear open image in new window t 0 linear m t t 0 and sublinear mathfrak m nu t t 0 death processes fractionality is introduced by replacing the usual integer order derivative in the difference differential equations governing the state probabilities with the fractional derivative understood in the sense of dzhrbashyan caputo we derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values a useful subordination relation is also proved allowing us to express the death processes as compositions of their classical counterparts with the random time process t2 t t 0 this random time has one dimensional distribution which is the folded solution to a cauchy problem of the fractional diffusion equation

applications of inverse tempered stable subordinators abstract the inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements with an exponential tempering that allows all moments to exist this paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time fractional diffusion equation and its folded density solves a tempered time fractional telegraph equation two explicit formulae for the density function are developed and applied to compute explicit solutions to tempered fractional cauchy problems where a tempered fractional derivative replaces the first derivative in time several examples are given including tempered fractional diffusion equations on bounded or unbounded domains and the probability distribution of a tempered fractional poisson process it is shown that solutions to the tempered fractional diffusion equation have a cusp at the origin

free path lengths in quasicrystals previous studies of kinetic transport in the lorentz gas have been limited to cases where the scatterers are distributed at random e g at the points of a spatial poisson process or at the vertices of a euclidean lattice in the present paper we investigate quasicrystalline scatterer configurations which are non periodic yet strongly correlated a famous example is the vertex set of a penrose tiling our main result proves the existence of a limit distribution for the free path length which answers a question of wennberg the limit distribution is characterised by a certain random variable on the space of higher dimensional lattices and is distinctly different from the exponential distribution observed for random scatterer configurations the key ingredients in the proofs are equidistribution theorems on homogeneous spaces which follow from ratner s measure classification

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

finite difference finite element method for two dimensional space and time fractional bloch torrey equations in this paper a class of two dimensional space and time fractional bloch torrey equations 2d stfbtes are considered some definitions and properties of fractional derivative spaces are presented by finite difference method and galerkin finite element method a semi discrete variational formulation for 2d stfbtes is obtained the stability and convergence of the semi discrete form are discussed then a fully discrete scheme of 2d stfbtes is derived and the convergence is investigated finally some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

existence results for an evolution problem with fractional nonlocal conditions a second order abstract problem involving derivatives of non integer order in the nonlinearity as well as in the nonlocal conditions is considered this problem covers many situations arising in real world phenomena and extends the case of integer order to the case of non integer order we establish the existence of mild solutions under extra appropriate sufficient assumptions on the nonlinearity and on the initial data these mild solutions are shown to be classical

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

solving fractional optimal control problems within a chebyshev legendre operational technique abstractin this manuscript we report a new operational technique for approximating the numerical solution of fractional optimal control foc problems the operational matrix of the caputo fractional derivative of the orthonormal chebyshev polynomial and the legendre gauss quadrature formula are used and then the lagrange multiplier scheme is employed for reducing such problems into those consisting of systems of easily solvable algebraic equations we compare the approximate solutions achieved using our approach with the exact solutions and with those presented in other techniques and we show the accuracy and applicability of the new numerical approach through two numerical examples

the quantitative behaviour of polynomial orbits on nilmanifolds a theorem of leibman asserts that a polynomial orbit g n g n z on a nilmanifold g g is always equidistributed in a union of closed sub nilmanifolds of g g in this paper we give a quantitative version of leibman s result describing the uniform distribution properties of a finite polynomial orbit g n g n n in a nilmanifold more specifically we show that there is a factorisation g eg where e n is smooth n g n z is periodic and rational and g n g n p is uniformly distributed up to a specified error d inside some subnilmanifold g g of g g for all sufficiently dense arithmetic progressions p n our bounds are uniform in n and are polynomial in the error tolerance d in a companion paper we shall use this theorem to establish the mobius and nilsequences conjecture from an earlier paper of ours

high order unconditionally stable difference schemes for the riesz space fractional telegraph equation in this paper a class of unconditionally stable difference schemes based on the pade approximation is presented for the riesz space fractional telegraph equation firstly we introduce a new variable to transform the original differential equation to an equivalent differential equation system then we apply a second order fractional central difference scheme to discretize the riesz space fractional operator finally we use 1 1 2 2 and 3 3 pade approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations matrix analysis is used to show the unconditional stability of the proposed algorithms two examples with known exact solutions are chosen to assess the proposed difference schemes numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space fractional hyperbolic equation

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi type equation with robin boundary condition the traditional element free galerkin efg approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares mls approximation in the current paper we propose a new version of the efg method based on the shape functions of reproducing kernel particle method rkpm in other words based on the developed approach in han and meng 2001 the fractional tricomi type equation will be solved using the new technique the fractional derivative has been introduced in the caputos sense and is approximated by a finite difference plan of order o 3 1 2 we use the efg rkpm to discrete the spatial direction we illustrate some numerical results on non rectangular domains the unconditional stability and convergence of the new technique have been proved numerical examples display the theoretical results and the efficiency of the proposed approach also the numerical results are compared with the finite element method fem and efg mls procedure

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

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

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

differentiation to fractional orders and the fractional telegraph equation using methods of differential and integral calculus we present and discuss the calculation of a fractional green function associated with the one dimensional case of the so called general fractional telegraph equation with one space variable this is a fractional partial differential equation with constant coefficients the equation is solved by means of juxtaposition of transforms i e we introduce the laplace transform in the time variable and the fourier transform in the space variable several particular cases are discussed in terms of the parameters some known results are recovered as a by product of our main result we obtain two new relations involving the two parameter mittag leffler function

error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives abstract in the current investigation an error estimate has been proposed to solve the two dimensional weakly singular integro partial differential equation with space and time fractional derivatives based on the finite element finite difference scheme the time and space derivatives are based on the riemann liouville and riesz fractional derivatives respectively at first the temporal variable has been discretized by a second order difference scheme and then the space variable has been approximated by the finite element method fem the analytical study shows that the presented scheme is unconditionally stable and convergent finally some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique

a mixed finite element method for a time fractional fourth order partial differential equation in this paper a numerical theory based on the mixed finite element method for a time fractional fourth order partial differential equation pde is presented and analyzed an auxiliary variable email protected is introduced then the fourth order equation can be split into the coupled system of two second order equations the time caputo fractional derivative is discretized by a finite difference method and the spatial direction is approximated by the mixed finite element method the stabilities based on a priori analysis for two variables are discussed and some a priori error estimates in l 2 norm for the scalar unknown u and the variable email protected are derived respectively moreover an a priori error result in h 1 norm for the scalar unknown u also is proved for verifying the theoretical analysis a numerical test is made by using matlab procedure

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

fractional telegraph equations we investigate several aspects of the fractional telegraph equations in an effort to better understand the anomalous diffusion processes observed in blood flow experiments in the earlier work eckstein et al electron j differential equations conf 03 1999 39 50 the telegraph equation d 2 u 2adu au 0 was used where d d dt a nd it was shown that as t tends to infinity u is approximated by v w here 2adv av 0 here a d 2 dx 2 on l 2 r o ra can be a more general nonnegative selfadjoint operator in this paper the concern is with the fractional telegraph equation e 2 u 2aeu au 0 where e d and 0 1 after solving this equation it is shown that u is approximated

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

fourth order finite difference schemes for time space fractional sub diffusion equations in this paper we devote to the study of high order finite difference schemes for one and two dimensional time space fractional sub diffusion equations a fourth order finite difference scheme is invoked for the spatial fractional derivatives and the l 1 approximation is applied to the temporal fractional parts for the two dimensional case an alternating direction implicit scheme based on l 1 approximation is proposed the stability and convergence of the proposed methods are studied numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes

fractional difference finite element approximations for the time space fractional telegraph equation abstract in this paper we study the numerical solution of the time space fractional order fractional for simplicity telegraph equation which can be used in signal analysis for transmission and propagation of electrical signals also the modeling of the reaction diffusion and the random walk of suspension flows and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the spatial riemann liouville fractional derivative with order 1 1 2 and the finite difference schemes for the temporal caputo derivatives with orders 1 2 1 and 2 are analyzed respectively results on the existence and uniqueness of the solution the numerical stability and the error estimates are displayed in details numerical examples are included to confirm the theoretical analysis

a non local in time telegraph equation abstract in this work we derive a non local in time telegraph equation our model includes as particular cases the classical telegraph equation and the fractional in time telegraph equation among others further we define the fundamental solution of the problem and we prove that it can be interpreted as a probability density function finally using versions of the karamata feller tauberian theorem we study the temporal behavior of the variance of the distribution process associated with the solution of the equation in large times as well as in short times

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

galerkin finite element method for two dimensional riesz space fractional diffusion equations abstract in this article a class of two dimensional riesz space fractional diffusion equations is considered some fractional spaces are established and some equivalences between fractional derivative spaces and fractional sobolev space are presented by the galerkin finite element method and backward difference method a fully discrete scheme is obtained according to lax milgram theorem the existence and uniqueness of the solution to the fully discrete scheme are investigated the stability and convergence of the scheme are also derived finally some numerical examples are given for verification of our theoretical analysis

a two grid mixed finite element method for a nonlinear fourth order reaction diffusion problem with time fractional derivative in this article we develop a two grid algorithm based on the mixed finite element mfe method for a nonlinear fourth order reaction diffusion equation with the time fractional derivative of caputo type we formulate the problem as a nonlinear fully discrete mfe system where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the mfe method to solve the nonlinear mfe system more efficiently we propose a two grid algorithm which is composed of two steps we first solve a nonlinear mfe system on a coarse grid by nonlinear iterations then solve the linearized mfe system on the fine grid by newton iteration numerical stability and optimal error estimate o k 2 h r 1 h 2 r 2 in l 2 norm are proved for our two grid scheme where k h and h are the time step size coarse grid mesh size and fine grid mesh size respectively we implement the two grid algorithm and present the numerical results justifying our theoretical error estimate the numerical tests also show that the two grid method is much more efficient than solving the nonlinear mfe system directly

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

the existence of mild and classical solutions for a second order abstract fractional problem in this paper we consider a second order abstract problem involving derivatives of non integer order the existence and uniqueness of mild and classical solutions are established in appropriate spaces under weak assumptions

unified fluid flow model for pressure transient analysis in naturally fractured media naturally fractured reservoirs present special challenges for flow modeling with regards to their internal geometrical structure the shape and distribution of matrix porous blocks and the geometry of fractures play key roles in the formulation of transient interporosity flow models although these models have been formulated for several typical geometries of the fracture networks they appeared to be very dissimilar for different shapes of matrix blocks and their analysis presents many technical challenges the aim of this paper is to derive and analyze a unified approach to transient interporosity flow models for slightly compressible fluids that can be used for any matrix geometry and fracture network a unified fractional differential transient interporosity flow model is derived using asymptotic analysis for singularly perturbed problems with small parameters arising from the assumption of a much smaller permeability of the matrix blocks compared to that of the fractures this methodology allowed us to unify existing transient interporosity flow models formulated for different shapes of matrix blocks including bounded matrix blocks unbounded matrix cylinders with any orthogonal crossection and matrix slabs the model is formulated using a fractional order diffusion equation for fluid pressure that involves caputo derivative of order 1 2 with respect to time analysis of the unified fractional derivative model revealed that the surface area to volume ratio is the key parameter in the description of the flow through naturally fractured media expressions of this parameter are presented for matrix blocks of the same geometrical shape as well as combinations of different shapes with constant and random sizes numerical comparisons between the predictions of the unified model and those obtained

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

an effective ratner equidistribution result for operatorname sl 2 mathbb r ltimes mathbb r 2 let g sl 2 r proportional to r 2 be the affine special linear group of the plane and set gamma sl 2 z proportional to z 2 we prove a polynomially effective asymptotic equidistribution result for the orbits of a 1 dimensional nonhorospherical unipotent flow on gamma g

on the uniqueness of the unbounded classical solutions of the navier stokes and associated equations abstract in this note we present a new theorem on the uniqueness of unbounded classical solutions to some class of equations which include the navier stokes equations and the associated leray and foias temam equations as special cases

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

a priori estimates in terms of the maximum norm for the solutions of the navier stokes equations in this paper we consider the cauchy problem for the incompressible navier stokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field for illustrative purposes we first derive corresponding a priori estimates for certain parabolic systems because of the pressure term the case of the navier stokes equations is more difficult however

on the time fractional cattaneo equation of distributed order abstract the present work revisits the time fractional cattaneo equation tfce by considering a generic class of time fractional derivatives of distributed order 0 1 at first we shed light on two main forms of tfce one natural generalization uses the caputo fractional derivative and the other compte metzler generalization is defined in the riemann liouville sense and the equivalence condition between their solutions is provided some special cases of these two forms are addressed hence different versions of distributed order tfce are proposed the fundamental solution of the two general forms of distributed order tfce are derived using integral transform technique the non negativity of solutions of tfce and its distributed order versions is examined using the special type functions technique gorenflo luchko and stojanovic 2013 two classes of order distribution are considered discrete order and continuous order distributions and some random choices of non negative order densities affirm graphically the non negativity of the general forms of distributed order tfce the double order as an example on the discrete order and exponentially exp and sinusoidally sin distributed order as an example on the continuous order are chosen finally connections between some models of distributed order tfce and continuous time random walk theory are established and the corresponding mean squared displacements are discussed

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

trapezoidal scheme for time space fractional diffusion equation with riesz derivative abstract in this paper a finite difference scheme is proposed to solve time space fractional diffusion equation which has second order accuracy in both time and space direction the time and space fractional derivatives are considered in the senses of caputo and riesz respectively first the centered difference approach is used to approximate the riesz fractional derivative in space then the obtained fractional ordinary differential equations are transformed into equivalent volterra integral equations and then the trapezoidal rule is utilized to approximate the volterra integral equations the stability and convergence of our scheme are proved via mathematical induction method finally numerical experiments are performed to confirm the high accuracy and efficiency of our scheme

a fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

finite difference finite element method for a nonlinear time fractional fourth order reaction diffusion problem abstract in this article a finite difference finite element algorithm which is based on a finite difference approximation in time direction and finite element method in spatial direction is presented and discussed to cast about for the numerical solutions of a time fractional fourth order reaction diffusion problem with a nonlinear reaction term to avoid the use of higher order elements the original problem with spatial fourth order derivative need to be changed into a second order coupled system by introducing an intermediate variable u then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction the unconditionally stable result in the norm which just depends on initial value and source item is derived some a priori estimates of l 2 norm with optimal order of convergence o t 2 h m 1 where t and h are time step length and space mesh parameter respectively are obtained to confirm the theoretical analysis some numerical results are provided by our method

analytic and approximate solutions of the space and time fractional telegraph equations the adomian decomposition method is used to obtain analytic and approximate solutions of the space and time fractional telegraph equations the space and time fractional derivatives are considered in the caputo sense the analytic solutions are calculated in the form of series with easily computable terms some examples are given the results reveal that the adomian method is very effective and convenient

orbits of discrete subgroups on a symmetric space and the furstenberg boundary let x be a symmetric space of noncompact type and let be a lattice in the isometry group of x we study the distribution of orbits of acting on the symmetric space x and its geometric boundary x generalizing the main equidistribution result of margulis s thesis m theorem 6 to higher rank symmetric spaces more precisely for any y x and b x we investigate the distribution of the set y b 1 in x x it is proved in particular that the orbits of in the furstenberg boundary are equidistributed and that the orbits of in x are equidistributed in sectors defined with respect to a cartan decomposition our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces which we obtain using shah s result s corollary 1 2 based on ratner s measure classification theorem r1 theorem 1

on the uniform equidistribution of long closed horocycles it is well known that on any given hyperbolic surface of finite area a closed horocycle of length becomes asymptotically equidistributed as in this paper we prove that any subsegment of length greater than 1 2 e of such a closed horocycle also becomes equidistributed as the exponent 1 2 e is the best possible and improves upon a recent result by hejhal he3 we give two proofs of the above result our second proof leads to explicit information on the rate of convergence we also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to

a finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation abstract an efficient numerical technique is proposed to solve one and two dimensional space fractional tempered fractional diffusion wave equations the space fractional is based on the riemann liouville fractional derivative at first the temporal direction is discretized using a second order accurate difference scheme then a classic galerkin finite element is employed to obtain a full discrete scheme furthermore for the time discrete and the full discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method finally two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique

quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms a quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms is proved we also give an application to eigenvalue spacing on flat 2 tori with aharonov bohm flux

gaps in n mod 1 and ergodic theory cut the unit circle s r z at the points 1 2 n where x x mod 1 and let j1 jn denote the complementary intervals or gaps that remain we show that in contrast to the case of random points whose gaps are exponentially distributed the lengths ji n are governed by an explicit piecewise real analytic distribution f t dt with phase transitions at t 1 2 and t 2 the gap distribution is related to the probability p t that a random unimodular lattice translate r meets a fixed triangle st of area t in fact p t f t the proof uses ergodic theory on the universal elliptic curve e sl2 r r 2 sl2 z z and ratner s theorem on unipotent invariant measures

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

a note on the finite element method for the space fractional advection diffusion equation in this paper a note on the finite element method for the space fractional advection diffusion equation with non homogeneous initial boundary condition is given where the fractional derivative is in the sense of caputo the error estimate is derived and the numerical results presented support the theoretical results

expanding translates of curves and dirichlet minkowski theorem on linear forms we show that a multiplicative form of dirichlet s theorem on simulta neous diophantine approximation as formulated by minkowski cannot be improved for almost all points on any analytic curve on r k which is not contained in a proper affine subspace such an investigation was initiated by davenport and schmidt in the late sixties the diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in r k 1 gets equidistributed in the limit we use ratner s theorem on unipotent flows linearization techniques and a new observation about intertwined linear dynamics of various sl m r s contained in sl k 1 r

the uniqueness of nondecaying solutions for the navier stokes equations the uniqueness of solutions of the navier stokes equations in the whole space is established when the velocity field is bounded and the pressure field is a bmo valued locally integrable in time function for bounded initial data here the velocity field may not decay at space infinity although there are a few results concerning uniqueness without the decay assumption our result is new and applicable for solutions constructed by solving the integral equations

numerical approximation of higher order time fractional telegraph equation by using a combination of a geometric approach and method of line we propose a simple and accurate numerical scheme for solving the time fractional telegraph tft equation within caputo type fractional derivative a fictitious coordinate is imposed onto the problem in order to transform the dependent variable u x t into a new variable with an extra dimension in the new space with the added fictitious dimension a combination of method of line and group preserving scheme gps is proposed to find the approximate solutions this method preserves the geometric structure of the problem power and accuracy of this method has been illustrated through some examples of tft equation

galerkin finite element method for two dimensional space and time fractional bloch torrey equation abstract in this paper we consider the galerkin finite element method for the two dimensional space and time fractional bloch torrey equation 2d stfbte utilizing the l 2 1 formula to discretize temporal caputo derivative we obtain the local truncation error with o 3 in the temporal direction where is the order of time fractional derivative furthermore the semi discrete form for the problem is given and the stability and convergence of the semi discrete variational formulation are rigorously proved in l 2 norm and fractional norm respectively then we derive a fully discrete scheme for the 2d stfbte and investigate its convergence theoretically finally extensive numerical results based on linear piecewise polynomials verify the theoretical results and show the effectiveness of the proposed scheme

asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds extending the earlier results for analytic curve segments in this article we describe the asymptotic behaviour of evolution of a finite segment of a c n smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n manifold in particular we show that if the curve satisfies certain natural geometric conditions the pushforward of the parameter measure on the curve under the ge odesic flow converges to the normalized canonical riemannian measure on the tangent bundle in the limit we also study the limits of geodesic evolution of shrinking segments we use ratner s classification of ergodic invariant measures for unipo tent flows on homogeneous spaces of so n 1 and an observation re lating local growth properties of smooth curves and dynamics of linear sl 2 r actions