pcr algorithm for the parallel computation of the solution of a class of singular linear systems abstract the paper presents a new highly parallel algorithm for computing the solution of a class of singular linear systems wawx b b r wa k 2 x r aw k 1 by the algorithm the solution x ad wb is obtained in t m 7 log 2 n 1 log 2 m 2 log 2 m r 1 log 2 mn steps with p mn processors when n 2 m 1 and with p 2m m 1 processors otherwise

neumann boundary value problems for a time fractional diffusion wave equation in a half plane the time fractional diffusion wave equation with the caputo derivative of the order 0 a 2 is considered in a half plane two types of neumann boundary condition are examined the mathematical condition with the prescribed boundary value of the normal derivative and the physical one with the prescribed boundary value of the matter flux

fractional order legendre collocation method for solving fractional initial value problems in this paper we present a numerical algorithm for solving second order fractional initial value problems this numerical algorithm is based on a fractional legendre collocation spectral method the governing fractional differential equation is converted into a nonlinear system of algebraic equations the error analysis of the proposed numerical algorithm is presented comparisons with other numerical methods shows that the proposed algorithm is more accurate and simpler to implement several examples are discussed to illustrate the efficiency and accuracy of the present scheme

solving constrained matrix equations and cramer rule this paper presents the solution of a general constrained matrix equation using generalized inverses and gives an explicit expression for the elements of the solution matrix using cramer rule

subproper and regular splittings for restricted rectangular linear system iterative schemes for approximating a solution to restricted rectangular but consistent linear system of equations ax b email protected t are considered the methods are based upon so called subproper splitting a m n which is a generalization of the concept of subproper splitting introduced by neumann and studied further by berman and neumann and others we give a necessary and sufficient condition on the splitting such that the iterative sequence converges to a solution of ax b in the case of email protected at for every x 0 where email protected c m x n and t is a subspace of c n monotonicity and the concept of regular subproper splitting are used to determine a necessary and a sufficient condition for the convergence of the iterative scheme finally we present two numerical examples to verify our conclusions

a cramer rule for minimum norm t least squares s solution of inconsistent linear equations abstract in this paper the minimum norm t least squares s solution of inconsistent linear equations ax y a c m n r y r a is given a determinantal form and reduces to common cramer s rule if a is nonsingular

a cramer rule for least norm solutions of consistent linear equations abstract the least euclidean norm solution of a consistent linear equation ax b is given a determinantal form which reduces to cramer s rule if a is nonsingular

adomian decomposition a tool for solving a system of fractional differential equations abstract adomian decomposition method has been employed to obtain solutions of a system of fractional differential equations convergence of the method has been discussed with some illustrative examples in particular for the initial value problem d 1 y 1 d n y n t a y 1 y n t y i 0 c i i 1 n where a a i j is a real square matrix the solution turns out to be y x e 1 n 1 x 1 a 1 x n a n y 0 where e 1 n 1 denotes multivariate mittag leffler function defined for matrix arguments and a i is the matrix having ith row as a i 1 a i n and all other entries are zero fractional oscillation and bagley torvik equations are solved as illustrative examples

a cramer rule for the least norm least squared error solution of inconsistent linear equations abstract a recent cramer rule for obtaining the least norm solution of a consistent system of linear equations ax b is shown to also provide the least squared error solution in the inconsistent case i e the solution a b where a is the moore penrose inverse of a

theory of spike spiral waves in a reaction diffusion system we discovered a type of spiral wave solutions in reaction diffusion systems spike spiral wave which significantly differs from the spiral waves observed in the models of fitzhugh nagumo type we present an asymptotic theory of these waves in the gray scott model chem sci eng 38 29 1983 we derive the kinematic relations describing the shape of this spiral and find the dependence of its main parameters on the control parameters the theory does not rely on the specific features of the gray scott model and thus is expected to be applicable to a broad range of reaction diffusion systems

fox h functions in fractional diffusion the h functions introduced by fox in 1961 are special functions of a very general nature which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework in this paper we express the fundamental solutions of the cauchy problem for the space time fractional diffusion equation in terms of proper fox h functions based on their mellin barnes integral representations we pay attention to the particular cases of space fractional time fractional and neutral fractional diffusion

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

mathematical modeling of different types of instabilities in time fractional reaction diffusion systems abstract in this article we investigate possible scenarios of pattern formations in reaction diffusion systems with time fractional derivatives linear stability analysis is performed for different values of derivative orders results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations most attention is paid to two models a fractional order reaction diffusion system with bonhoeffer van der pol kinetics and to the brusselator model

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

multifunctional variational method for description of evolution and dynamics of dissipative structures in nonequilibrium systems a variational principle based on the introduction of a vector functional each component whereof has its extremum with respect to variation of only one or a group of macroscopic parameter s of a system is presented the suggested method greatly facilitates the description of the characteristic parameters of stationary and time dependent complex inhomogeneous macroscopic states occurring in nonequilibrium systems the fruitfulness and simplicity of the presented method are illustrated by analysis of bifurcation types and by studying the shape stability and evolution of spike strata and autosolitions a nonlinear theory of pulsating spike strata and autosolitons of large amplitude is developed it is shown that the variations of the characteristic parameters of pulsating strata and autosolitons are relaxational spike auto oscillations which may be of periodical as well as of apparently chaotic character a simple method of analysis of quasiharmonic states is developed and for a concrete model it is shown that supercritical solution bifurcations take place only if parameters of the system meet very rigorous requirements i e it is confirmed that instability of the system s homogeneous states leads as a rule to abrupt formation of large amplitude structures the proposed variational principle is shown to be useful for deriving a set of ordinary differential equations which describe in a simple way the interaction between autosolitons and turbulence in active distributed media

successive matrix squaring algorithm for computing the drazin inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 this paper derives a successive matrix squaring sms algorithm to approximate the drazin inverse which can be expressed in the form of successive squaring of a composite matrix t given an n by n matrix a the study shows that the drazin inverse of a can be computed in parallel time ranging from o logn to o log 2n provided that there are enough processors to support matrix multiplication in time o logn the sms algorithm is generalized to higher order schemes where the composite matrix is repeatedly raised to an integer power l 2 this form of expression leads to a simplified notation compared to that of earlier methods we argue that there is no obvious advantage in choosing l other than 2 our derived error bound for the approximation of a d is new

pattern formation in a fractional reaction diffusion system abstract we investigate pattern formation in a fractional reaction diffusion system by the method of computer simulation of the model of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and space fractional derivatives we further compare the patterns obtained by computer simulation with those obtained by simulation of the similar system without fractional derivatives as a result we are able to show that nonlinearity plays the main role in structure formation and fractional derivative terms change the transient dynamics so when the order of time derivative increases and approaches the value of 1 5 the special structure formation switches to homogeneous oscillations in the case of space fractional derivatives the decrease of the order of these derivatives leads to more contrast dissipative structures the variational principle is used to find the approximate solution of such fractional reaction diffusion model in addition we provide a detailed analysis of the characteristic dissipative structures in the system under consideration

perturbation theory for the generalized bott duffin inverse and its applications a perturbation theory for the bott duffin inverse a l of a with respect to a subspace l is developed the perturbation bound for the solution of the constrained system ax y b email protected l email protected l is established where email protected m n c a subspace email protected c n and email protected c n meanwhile this paper shows the generalized bott duffin condition number k g b d a email protected email protected email protected a l to be minimum in the inequality of error analysis

a cramer rule for solution of the general restricted matrix equation a cramer rule for finding the unique w weighted drazin inverse solution of a class of restricted matrix equationswawxwbw d r x r aw k 1 n x n wb k 2 is presented and the results in linear algebra appl 116 1989 27 appl math comput 125 2002 303 can be deduced from this paper where email protected c m x n email protected c n x m ind aw k 1 email protected c p x q email protected c q x p ind wb email protected 2 and email protected c n x p

patterns in reaction diffusion systems generated by global alternation of dynamics abstract we recently proposed a mechanism for inducing a turing instability by alternation of reaction diffusion dynamics each of which is pattern free herein we shed light on the question of which particular switching schemes produce that instability and which do not we also consider a particular reaction diffusion model to illustrate the mechanism

iterative methods for solving restricted linear equations in this paper the iterative methods for computing some solution of the general restricted linear equations ax b x et are established and the iterative methods for finding the solution of several special linear equations considered are constructed

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

discrete mittag leffler kernel type fractional difference initial value problems and gronwall s inequality abstract in this article we studied the caputo and riemann liouville type discrete fractional difference initial value problems with discrete mittag leffler kernels the existence and uniqueness of the solution is proved by using banach contraction principle the linear type equations are used to prove new discrete fractional versions of the gronwall s inequality the nabla discrete laplace transform is used to obtain solution representations the proven gronwall s inequality under a new defined lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions numerical examples are discussed to demonstrate the reliability of the theoretical results

fast tensor product solvers for optimization problems with fractional differential equations as constraints fractional differential equations have recently received much attention within computational mathematics and applied science and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms an optimization problem with constraints given by fractional differential equations is considered which in its discretized form leads to a high dimensional tensor equation to reduce the computation time and storage the solution is sought in the tensor train format we compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned krylov solvers or tailored alternating schemes the competitiveness of these approaches is presented using several examples with constant and variable coefficients

disordered totally asymmetric simple exclusion process exact results we study the effect of quenched spatial disorder on the current carrying steady states of a totally asymmetric simple exclusion process with spatially disordered jump rates expressions for the steady state weights and the current are derived for this model in one dimension these solutions are exploited to study analytically the exact symmetries of the system in particular we confirm the recent numerical observation of tripathy and barma phys rev e 58 1911 1998 that the magnitude of the steady state current is left invariant when the direction of all the allowed particle jumps are reversed

the algorithm for computing the drazin inverses of two variable polynomial matrices we propose a finite algorithm for computing drazin inverse of two variable polynomial matrices based on greville s finite algorithm for computing drazin inverse of a constant matrix also a three dimensional recursive algorithm to do that is deduced examples show that these methods are feasible and the implementation is developed in the symbolic package matlab

transcendental bernstein series for solving nonlinear variable order fractional optimal control problems abstract this paper deals with finding an approximate solution of variable order fractional optimal control problems v focps based on lagrange multiplier optimization technique in combination with the new operational matrix of variable order vo fractional derivatives to carry out the proposed approach we firstly develop the well known bernstein polynomials to the new series of functions namely transcendental bernstein series tbs then we implement these basis functions to approximate the solutions of v focps in fact the series expansion in tbs with unknown free coefficients and control parameters is the novel idea for solving the fractional systems with less terms of approximation the convergence analysis of our proposed method will be guaranteed by proving a new theorem for the tbs to investigate the practical computational efficiency and accuracy of the presented method some numerical examples are provided the experimental results confirm the applicability of the method and a good agreement between the approximate and exact solutions

generalized taylor s formula abstract in this paper a new generalized taylor s formula of the kind f x j 0 n a j x a j r n x where a j r x a 0 1 is established such an expression is precisely the classical taylor s formula in case of 1 in addition detailed expressions for r n x and a j that involving caputo fractional derivatives are well constructed some applications including approximation of functions and solutions of fractional differential equations are also given

characteristics of the asymmetric simple exclusion process in the presence of quenched spatial disorder we investigate the effect of quenched spatial disordered hopping rates on the characteristics of the asymmetric simple exclusion process with open boundaries both numerically and by extensive simulations disorder averages of the bulk density and current are obtained in terms of various input and output rates we study the binary and uniform distributions of disorder it is verified that the effect of spatial inhomogeneity is generically to enlarge the size of the maximal current phase this is in accordance with the mean field results obtained by harris and stinchcombe phys rev e 70 016108 2004 furthermore we obtain the dependence of the current and the bulk density on the characteristics of the disorder distribution function it is shown that the impact of disorder crucially depends on the particle input and out rates in some situations disorder can constructively enhance the current

fast implicit integration factor method for nonlinear space riesz fractional reaction diffusion equations abstract after spatially discretizing the nonlinear space riesz fractional reaction diffusion equations by the fractional centered difference formula the resulting semi discrete equations lead to a nonlinear ordinary differential equation system in order to obtain good stability and robustness the implicit integration factor iif method that treats the reaction term implicitly and the diffusion term exactly is employed to solve the system to deal with the extremely expensive cost in iif we also propose the shift invert lanczos method based on the gohberg semencul formula to compute the matrix exponential by using the particular property that the coefficient matrix is symmetric positive definite toeplitz the proposed fast method requires only o m memory storage and o m log m computation cost in each iterative step compared to the o m 2 storage and o m 3 computational complexity of the direct solution method where m is the size of the spatial grid some numerical experiments are performed to verify the correctness of the theoretical results and demonstrate the efficiency of our fast solution algorithm

distributed order reaction diffusion systems associated with caputo derivatives this paper deals with the investigation of the solution of an unified fractional reaction diffusion equation of distributed order associated with the caputo derivatives as the time derivative and riesz feller fractional derivative as the space derivative the solution is derived by the application of the joint laplace and fourier transforms in compact and closed form in terms of the h function the results derived are of general nature and include the results investigated earlier by other authors notably by mainardi et al the fundamental solution of the space time fractional diffusion equation fractional calculus appl anal 4 153 202 2001 mainardi et al fox h functions in fractional diffusion j comput appl math 178 321 331 2005 for the fundamental solution of the space time fractional equation including haubold et al solutions of reaction diffusion equations in terms of the h function bull astron soc india 35 681 689 2007 and saxena et al fractional reaction diffusio

a characterization and determinantal formula for the generalized inverse at s 2 and its applications in this paper we present a characterization and determinantal formula for the generalized inverse a t s 2 as applications on the one hand we get the corresponding results for many common important generalized inverses such as a a m n a d a and so on on the other hand we apply these results to solve restricted linear equations ax b email protected t

gradient methods for computing the drazin inverse solution the problem of computing the vector a db in the cases when it provides a solution of a singular linear system is reduced to the problem of finding a least squares solution of an appropriate linear system later the presented ideas are extended to arbitrary linear systems as a consequence we present iterative methods for computing the vector a db in general the functionality of the exposed algorithms is based on a specific representation of the drazin inverse solution as well as the properties that we introduce convergence results of the proposed methods are also considered illustrative numerical examples are presented

front propagation in hyperbolic fractional reaction diffusion equations from the continuous time random walk scheme and assuming a l evy waiting time distribution typical of subdiffusive transport processes we study a hyperbolic reaction diffusion equation involving time fractional derivatives the linear speed selection of wave fronts in this equation is analyzed when the reaction diffusion dimensionless number and the fractional index satisfy a certain condition we find fronts exhibiting an unphysical behavior they travel faster in the subdiffusive than in the diffusive regime

grain boundary random walks and diffusion in polycrystals the grain boundary diffusion problem is considered in terms of random walks inside the boundary layer and the grain bulk as we have shown if the time t is not extremely small diffusion in polycrystals can be described by certain random walks on a two dimensional manifold specified by the geometry of the grain boundaries however the time tt of these random walks called grain boundary random walks gbrw does not coincide with the real time t and in general is a random variable when the length l of atom penetration into polycrystal space is substantially larger than the mean grain size r gbrw can be also considered within the framework of random migration in a certain three dimensional continuum in possible different limits specific equations for such random processes are obtained and corresponding properties of atoms are analyzed

condensed cramer rule for solving restricted matrix equations a cramer rule for solving a kind of restricted matrix equationswawxw bw d r x r aw k 1 n x n w b k 2 is presented in g wang jie sun a cramer rule for solution of the general restricted matrix equation appl math comput 154 2004 415 422 this paper gives a more condensed cramer rule for the solution of the matrix equations the results in g wang z l xu solving a kind of restricted matrix equations and cramer rule appl math comput 162 2005 329 338 chao gu g r wang z l xu pcr algorithm for the parallel computation of the solution of a class of singular linear systems appl math comput 176 2006 237 244 are partially the special cases in our paper

condition number related with w weighted drazin inverse and singular linear systems in this paper we introduce a pq norm and discuss the condition number of the w weighted drazin inverse we get some results on the condition number with respect to a d w and singular linear systems wawx b b r wa k 2 x r aw k 1 which are similar to the regular inverse a 1 and nonsingular linear systems ax b

scaling approach to related disordered stochastic and free fermion models motivated by mapping from a stochastic system with spatially random rates we consider disordered non conserving free fermion systems using a scaling procedure for the equations of motion this approach demonstrates disorder induced localization acting in competition with the asymmetric driving we discuss the resulting implications for the original stochastic system

on a time fractional reaction diffusion equation a reaction diffusion equation with a caputo fractional derivative in time and with various boundary conditions is considered under some conditions on the initial data we show that solutions may experience blow up in a finite time however for realistic initial conditions solutions are global in time moreover the asymptotic behavior of bounded solutions will be analyzed

solving a kind of restricted matrix equations and cramer rule the problem of finding determinantal formula for solutions of some restricted linear systems ax b has been discussed in linear multilinear algebra 15 1984 319 linear algebra appl 116 1989 27 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 this paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsax d r x r a k 1 xb d n x n b k 2 axb d r x r a k 1 n x n b k 2 where a c n x n with ind a k 1 b c m x m with ind b k 2 and d c n x m the results in linear multilinear algebra 15 1984 319 linear algebra appl 74 1986 213 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 are partially the special cases in our paper the classic cramer rule is also a special case of our results

an iterative method for solving general restricted linear equations the newton iterative method for computing outer inverses with prescribed range and null space is used in the non stationary richardson iterative method to develop an iterative method for solving general restricted linear equations starting with any suitably chosen initial iterate our method generates a sequence of iterates converging to the solution the necessary and sufficient conditions for the convergence along with the error bounds are established the applications of the iterative method for solving some special linear equations are also discussed a number of numerical examples are worked out they include singular square rectangular randomly generated rank deficient matrices full rank matrices and a set of singular matrices given in matrix computation toolbox mctoolbox with the condition numbers ranging from order 1016 to 1050 the mean cpu time mct and the error bounds are the performance measures used our results when compared with the results obtained by chen 1997 leads to substantial improvement in terms of both computational speed and accuracy

diffusive effects on dispersion in excitable media perturbation and numerical methods are used to study traveling and standing wave solutions for a system of reaction diffusion equations modeled after excitable media and the effect of variation of the diffusion coefficients it is shown that traveling waves and standing waves can coexist and that as a certain diffusion coefficient is varied a branch of traveling waves can bifurcate from a branch of standing waves providing a transition between only traveling solutions in one limit and only standing waves in the other limit key words traveling waves standing waves bifurcation dispersion reaction diffusion equations singular perturbation

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

a condensed cramer s rule for the minimum norm least squares solution of linear equations abstract in this paper we will derive the condensed cramer s rule of werner for minimal norm least squares solution of linear equations ax b from the cramer s rule of ben israel and verghese in addition a new condensed cramer s rule will be obtained in this paper

cramer s rule for a system of quaternion matrix equations with applications abstract in this paper we investigate cramer s rule for the general solution to the system of quaternion matrix equations a 1 x b 1 c 1 a 2 x b 2 c 2 and cramer s rule for the general solution to the generalized sylvester quaternion matrix equation a x b c y d e respectively as applications we derive the determinantal expressions for the hermitian solutions to some quaternion matrix equations the findings of this paper extend some known results in the literature

mathematical modeling of time fractional reaction diffusion systems we study a fractional reaction diffusion system with two types of variables activator and inhibitor the interactions between components are modeled by cubical nonlinearity linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives it is shown that by combining the fractional derivatives index with the ratio of characteristic times it is possible to find the marginal value of the index where the oscillatory instability arises the increase of the value of fractional derivative index leads to the time periodic solutions the domains of existing periodic solutions for different parameters of the problem are obtained a computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented for the fractional reaction diffusion systems it is established that there exists a set of stable spatio temporal structures of the one dimensional system under the neumann and periodic boundary conditions the characteristic features of these solutions consist of the transformation of the steady state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system

a kronecker product splitting preconditioner for two dimensional space fractional diffusion equations abstract we study preconditioned iterative methods for the linear system arising in the numerical discretization of a two dimensional space fractional diffusion equation our approach is based on a formulation of the discrete problem that is shown to be the sum of two kronecker products by making use of an alternating kronecker product splitting iteration technique we establish a class of fixed point iteration methods theoretical analysis shows that the new method converges to the unique solution of the linear system moreover the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real the basic iteration is accelerated by a krylov subspace method like gmres the corresponding preconditioner is in a form of a kronecker product structure and requires at each iteration the solution of a set of discrete one dimensional fractional diffusion equations we use structure preserving approximations to the discrete one dimensional fractional diffusion operators in the action of the preconditioning matrix numerical examples are presented to illustrate the effectiveness of this approach

representation of the drazin inverse for special block matrix abstract for a c n n the matrix x c n n is said to be the drazin inverse of a if it holds that xax x ax xa a r xa a r where r is the smallest nonnegative integer such that rank a r rank a r 1 in this paper we give the representation for the drazin inverse of the block matrix a a b 0 a b c n n a 2 a

a fractional phase field model for two phase flows with tunable sharpness algorithms and simulations abstract we develop a fractional extension of a mass conserving allen cahn phase field model that describes the mixture of two incompressible fluids the fractional order controls the sharpness of the interface which is typically diffusive in integer order phase field models the model is derived based on an energy variational formulation an additional constraint is employed to make the allen cahn formulation mass conserving and comparable to the cahn hilliard formulation but at reduced cost the spatial discretization is based on a petrov galerkin spectral method whereas the temporal discretization is based on a stabilized adi scheme both for the phase field equation and for the navier stokes equation we demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two phase flow in a pipe and of a rising bubble we also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

existence of solutions of abstract fractional integrodifferential equations of sobolev type this paper deals with the study of existence of solutions of nonlinear fractional integrodifferential equations of sobolev type with nonlocal condition in banach spaces the results are obtained by using resolvent operators fractional calculus and fixed point technique an example is provided to illustrate the theory

a cramer rule for finding the solution of a class of singular equations a cramer rule for finding the unique solution x r ak of singular equations ax b ind a k b r ak is given and reduces to the common cramer rule if a is a nonsingular

further solutions of fractional reaction diffusion equations in terms of the h function this paper is in continuation of our earlier paper in which we have derived the solution of a unified fractional reaction diffusion equation associated with the caputo derivative as the time derivative and riesz feller fractional derivative as the space derivative in this paper we consider a unified reaction diffusion equation with the riemann liouville fractional derivative as the time derivative and riesz feller derivative as the space derivative the solution is derived by the application of the laplace and fourier transforms in a compact and closed form in terms of the h function the results derived are of general character and include the results investigated earlier in 7 8 the main result is given in the form of a theorem a number of interesting special cases of the theorem are also given as corollaries

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

on the perturbation and subproper splittings for the generalized inverse at s 2 of rectangular matrix a abstract in this paper the perturbation and subproper splittings for the generalized inverse at s 2 the unique matrix x such that xax x r x t and n x s are considered we present lower and upper bounds for the perturbation of at s 2 convergence of subproper splittings for computing the special solution at s 2 b of restricted rectangular linear system ax b x t are studied for the solution at s 2 b we develop a characterization therefore we give a unified treatment of the related problems considered in literature by ben israel berman hanke neumann plemmons etc

stability of the static spike autosolitons in the gray scott model we performed an asymptotic linear stability analysis of the static spike autosolitons ass self sustained solitary inhomogeneous states in the gray scott model of an autocatalytic chemical reaction we found that in one dimension these ass destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough in higher dimensions the one dimensional static spike ass are always unstable with respect to corrugation and wriggling the higher dimensional radially symmetric static spike ass may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow

approximate solutions of nonlinear fractional equations fractional exponential that are invariant under fractional derivatives elementary and special fractional functions are introduced approximate solutions to fractional burgers equation by using the homotopy perturbation method are obtained furthermore real integral representations for some h functions are found that may be very helpful in numerical computations

a characterization for the w weighted drazin inverse and a cramer rule for the w weighted drazin inverse solution we establish characterization for the w weighted drazin inverse of an arbitrary rectangular matrix which reduces to the well known result if the matrix is nonsingular also a cramer rule for finding the unique w weighted drazin inverse solution x r aw k 1 of special restricted linear equationswawx b b r wa k 2 is presented and reduces to the classical cramer rule if a is invertible

variational representation of the projection dynamics and random motion of highly dissipative systems we propose the method of investigation of highly dissipative systems which are based on the approximation of the attractor by some manifold the projection dynamic equations for the general form of such manifold of the dissipative dynamic system are obtained variational principles for the projection dynamics are considered on the basis of the projection dynamic equations we investigate the influence of the random forces on the behaviour of the system

steady state properties of a totally asymmetric exclusion process with periodic structure we study the steady state behavior of totally asymmetric simple exclusion processes taseps that contain periodically varying movement rates in this model particles move to the right at one of two rates p 2 if the particle occupies one of a periodically arranged set of lattice sites p 1 otherwise approximate mean field approaches are used to study the steady state currents and bulk densities of this model these mean field methods are found to provide results in good agreement with data derived from monte carlo simulations finally the condition for particle hole symmetry in the tasep with periodically varying movement rates is specified and the changes in the locations of the boundary limited to maximal current transition lines due to symmetry violation are investigated

weighted moore penrose drazin and group inverses of the kronecker product a b and some applications abstract the representations for the weighted moore penrose inverse the drazin inverse and the group inverse of the kronecker product a b of the two matrices a and b are given and by using these results the determinant forms for projectors aa mn a mn a a d a and a g a are deduced

explicit expressions of the generalized inverses and condensed cramer rules in this paper we obtain an explicit representation of the 2 inverse a 2 t s of a matrix a c m n with the prescribed range t and null space s as special cases new expressions for the moore penrose inverse a and drazin inverse a d are derived through explicit expressions we re derive the condensed cramer rules of werner for minimal norm least squares solution of linear equations ax b and propose two new condensed cramer rules for the unique solution of a class of singular system ax b x r a k b r a k k ind a finally condensed determinantal expressions for a a d a a a a a ndaa d are also presented

iterative solutions for a coupled system of fractional differential integral equations with two point boundary conditions abstract in this paper we deal with a coupled system of fractional differential integral equations with two point boundary conditions the maximal and minimal solutions are obtained by using the monotone iterative technique combined with the method of upper and lower solutions meantime the error estimate of the solutions are given in the end some examples are given to illustrate the results

fractional diffusion equations coupled by reaction terms we investigate the behavior for a set of fractional reaction diffusion equations that extend the usual ones by the presence of spatial fractional derivatives of distributed order in the diffusive term these equations are coupled via the reaction terms which may represent reversible or irreversible processes for these equations we find exact solutions and show that the spreading of the distributions is asymptotically governed by the same the long tailed distribution furthermore we observe that the coupling introduced by reaction terms creates an interplay between different diffusive regimes leading us to a rich class of behaviors related to anomalous diffusion

disordered asymmetric simple exclusion process mean field treatment we provide two complementary approaches to the treatment of disorder in a fundamental nonequilibrium model the asymmetric simple exclusion process first a mean field steady state mapping is generalized to the disordered case where it provides a mapping of probability distributions and demonstrates how disorder results in a new flat regime in the steady state current density plot for periodic boundary conditions this effect was earlier observed by phys rev e 58 1911 1998 but we provide a treatment for more general distributions of disorder including both numerical results and analytic expressions for the width 2 delta c of the flat section we then apply an argument based on moving shock fronts europhys lett 48 257 1999 to show how this leads to an increase in the high current region of the phase diagram for open boundary conditions second we show how equivalent results can be obtained easily by taking the continuum limit of the problem and then using a disordered version of the well known cole hopf mapping to linearize the equation within this approach we show that adding disorder induces a localization transformation verified by numerical scaling and delta c maps to an inverse localization length helping to give a physical interpretation to the problem

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

flow on regular and disordered networks abstract the paper concerns processes on networks the processes considered are linear and collective flow since they are the prime function of many real networks the study has bearing on and how a network is probed by a process and in particular how it may be categorised through its dynamic and steady state flow characteristics the linear flow processes considered include both symmetric and biassed flow e g of single particles or currents related linearly to density and for these we exploit existing techniques to deal with background geometry and disorder the nonlinear collective flow considered is driven flow of vehicles and other excluding particles as it is captured by a biased hard core hopping model this stochastic nonequilibrium process has been much studied in the one dimensional case where it provides the simplest example of a nonequilibrium phase transition recent work has given some insights into the typically severe effects of disorder in such collective models mainly in one dimension correspondingly striking effects are expected for such processes on structured networks but so far little has been published even for the pure case the paper uses linear flow to provide a basic introduction to the subject then briefly discusses disorder effects before moving on to nonlinear flow on pure structured networks including the cayley tree and finally comments on the combined effects of structure and disorder

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular

stability and convergence of a new explicit finite difference approximation for the variable order nonlinear fractional diffusion equation in this paper we consider the variable order nonlinear fractional diffusion equation u x t t b x t xr a x t u x t f u x t where xr a x t is a generalized riesz fractional derivative of variable order a x t 1 a x t 2 and the nonlinear reaction term f u x t satisfies the lipschitz condition f u 1 x t f u 2 x t

extensions and solutions for nonlinear diffusion equations and random walks we investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by einstein to explain the brownian motion we show here how to properly modify that framework in order to handle different physical scenarios we obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism finally we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations depending on the choice of distribution functions related to the spreading of systems

finite difference spectral approximation for a time space fractional equation on two and three space dimensions abstract we consider the numerical method for solving a time space fractional equation whose solution has a weak initial singularity l 1 scheme on graded mesh is used to capture the initial weak singularity it is shown that the method can recover order 2 convergence by choosing some appropriate grading parameters where 0 1 is the order of temporal caputo fractional derivative a fully discrete scheme is proposed by combining spectral approximation for the discretization of space in 2d and 3d cases stability and convergence of the fully discrete scheme are rigorously established numerical results are presented to show the sharpness of the error estimate

mixing driven equilibrium reactions in multidimensional fractional advection dispersion systems we study instantaneous mixing driven bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation the superdiffusive nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical fickian diffusion model in particular regions where reaction rates would be zero for the fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates we also study a global metric of mixing in the system the scalar dissipation rate and compute its asymptotic scaling rates analytically the scalar dissipation rate scales asymptotically as t d where d is the number of spatial dimensions and is the fractional derivative exponent