large deviations and nontrivial exponents in coarsening systems we investigate the statistics of the mean magnetization of its large deviations and persistent large deviations in simple coarsening systems in particular we consider more specifically the case of the diffusion equation of the ising chain at zero temperature and of the two dimensional voter model for the diffusion equation at large times the mean magnetization has a limit law which is studied analytically using the independent interval approximation the probability of persistent large deviations defined as the probability that the mean magnetization was for all previous times greater than some level x decays algebraically at large times with an exponent continuously varying with x when x 1 is the usual persistence exponent similar behaviour is found for the glauber ising chain at zero temperature for the two dimensional voter model large deviations of the mean magnetization are algebraic while the probability of persistent large deviations seem to behave as the usual persistence probability

persistence exponent of the diffusion equation in e dimensions we consider the d dimensional diffusion equation t x t x t with random initial condition and observe that when appropriately scaled 0 t is gaussian and markovian in the limit d 0 this leads via the majumdar sire perturbation theory to a small d expansion for the persistence exponent d we find d 14d 0 12065 d3 2

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

a fast chebyshev s method for quadratic equations a multiresolution transform corresponding to interpolatory technique is used for fast application of chebyshev s method in designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method a convergence theorem is performed finally some numerical results are presented

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

discontinuous galerkin method for an evolution equation with a memory term of positive type we consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by c t s 1 where 0 1 for the time discretization we apply the discontinuous galerkin method using piecewise polynomials of degree at most q 1 for q 1 or 2 for the space discretization we use continuous piecewise linear finite elements the discrete solution satisfies an error bound of order k 9 h 2 l k where k and h are the mesh sizes in time and space respectively and l k max 1 log k 1 in the case q 2 we prove a higher convergence rate of order k 3 h 2 l k at the nodes of the time mesh typically the partial derivatives of the exact solution are singular at t 0 necessitating the use of non uniform time steps we compare our theoretical error bounds with the results of numerical computations

secant like methods for solving nonlinear integral equations of the hammerstein type we consider a one parametric family of secant type iterations for solving nonlinear equations in banach spaces we establish a semilocal convergence result for these iterations by means of a technique based on a new system of recurrence relations this result is then applied to obtain existence and uniqueness results for nonlinear integral equations of the hammerstein type we also present a numerical example where the solution of a particular hammerstein integral equation is approximated by different secant type methods

a modified chebyshev s iterative method with at least sixth order of convergence this paper is devoted to the construction and analysis of a high order variant of the classical chebyshev method the method has order of convergence at least six for simple roots the extension to system of equations and its semilocal convergence for nonlinear equations are presented finally an application to well known algebraic riccati equation is considered

a family of halley chebyshev iterative schemes for non frechet differentiable operators a modification of some classical third order methods is studied the main advantage of these methods is that they do not need evaluate any frechet derivative a convergence theorem in banach spaces just assuming that the second divided difference is bounded by a nondecreasing function and a punctual condition is presented frechet differentiability is not required finally a particular example is analyzed where our conditions are fulfilled and the classical ones fail

the application of an inverse free jarratt type approximation to nonlinear integral equations of hammerstein type abstract we consider an inverse free jarratt type approximation whose order of convergence is four for solving nonlinear equations the convergence of this method is analysed under two different types of conditions we use a new technique based on constructing a system of real sequences finally this method is applied to the study of hammerstein s integral equations

stationary definition of persistence for finite temperature phase ordering for the two dimensional kinetic ising model at finite temperature the local mean magnetization simply related to the fraction of time spent by a given spin in the positive direction has a limiting distribution singular at the onsager spontaneous magnetization the exponent of this singularity defines the persistence exponent we also study first passage exponents associated to persistent large deviations of and their temperature dependence

some variants of cauchy s method with accelerated fourth order convergence in this paper we present some variants of cauchy s method for solving non linear equations analysis of convergence shows that the methods have fourth order convergence per iteration the new methods cost almost the same as cauchy s method numerical results show that the methods can compete with cauchy s method

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

an adaptive version of a fourth order iterative method for quadratic equations a fourth order iterative method for quadratic equations is presented a semilocal convergence theorem is performed a multiresolution transform corresponding to interpolatory technique is used for fast application of the method in designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method finally some numerical results are studied

newton like methods for the computation of fixed points abstract the celebrated banach fixed point theorem provides conditions which assure that the method of successive substitution is convergent the convergence however may take place very slowly so that it may be desirable to use a newton like method for the computation of the fixed point if newton s method itself is applied one ignores the additional information that the problem arises from a frixed point problem with a contraction mapping in the present note some variants of newton s method are discussed which make use of this contraction information it turns out that the convergence of newton s method can be accelerated without any relevant additional computational labour

a two step steffensen s method under modified convergence conditions a simplification of a third order iterative method is proposed the main advantage of this method is that it does not need to evaluate neither any frechet derivative nor any bilinear operator a semilocal convergence theorem in banach spaces under modified kantorovich conditions is analyzed a local convergence analysis is also performed finally some numerical results are presented

perturbation theory for fractional brownian motion in presence of absorbing boundaries fractional brownian motion is a gaussian process x t with zero mean and two time correlations x t 1 x t 2 d t 1 2h t 2 2h t 1 t 2 2h where h with 0 h 1 is called the hurst exponent for h 1 2 x t is a brownian motion while for h 1 2 x t is a non markovian process here we study x t in presence of an absorbing boundary at the origin and focus on the probability density p x t for the process to arrive at x at time t starting near the origin at time 0 given that it has never crossed the origin it has a scaling form p x t t h r x t h our objective is to compute the scaling function r y which up to now was only known for the markov case h 1 2 we develop a systematic perturbation theory around this limit setting h 1 2 e to calculate the scaling function r y to first order in e we find that r y behaves as r y y as y 0 near the absorbing boundary while r y y exp y 2 2 as y with 1 4e o e 2 and 1 2e o e 2 our e expansion result confirms the scaling relation 1 h h proposed in zoia rosso and majumdar phys rev lett 102 120602 2009 we verify our findings via numerical simulations for h 2 3 the tools developed here are versatile powerful and adaptable to different situations

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

some supplementary results on the 1 sqrt 2 order method for the solution of nonlinear equations recently an iterative method for the solution of systems of nonlinear equations having at leastr order 1 sqrt 2 for simple roots has been investigated by the author 7 this method uses as many function evaluations per step as the classical newton method in the present note we deal with several properties of the method such as monotone convergence asymptotic inclusion of the solution and convergence in the case of multiple roots

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

a third order method for fixed points in banach spaces this paper deals with a third order stirling like method used for finding fixed points of nonlinear operator equations in banach spaces the semilocal convergence of the method is established by using recurrence relations under the assumption that the first frechet derivative of the involved operator satisfies the holder continuity condition a theorem is given to establish the error bounds and the existence and uniqueness regions for fixed points the r order of the method is also shown to be equal to at least 2p 1 for p 0 1 the efficacy of our approach is shown by solving three nonlinear elementary scalar functions and two nonlinear integral equations by using both stirling like method and newton like method it is observed that our convergence analysis is more effective and give better results

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

convergence and numerical analysis of a family of two step steffensen s methods we provide sufficient conditions for the semilocal convergence of a family of two step steffensen s iterative methods on a banach space the main advantage of this family is that it does not need to evaluate neither any frechet derivative nor any bilinear operator but having a high speed of convergence some numerical experiments are also presented

third order iterative methods under kantorovich conditions we study a class of third order iterative methods for nonlinear equations on banach spaces a characterization of the convergence under kantorovich type conditions and optimal estimates of the error are found though in general these methods are not very extended due to their computational costs we will show some examples in which they are competitive and even cheaper than other simpler methods we center our analysis in both analytic and computational aspects

compact alternating direction implicit method for two dimensional time fractional diffusion equation high order compact finite difference scheme with operator splitting technique for solving two dimensional time fractional diffusion equation is considered in this paper a grunwald letnikov approximation is used for the riemann liouville time derivative and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme alternating direction implicit adi method is used to split the original problem into two separate one dimensional problems the local truncation error is analyzed and the stability is discussed by the fourier method the proposed scheme is suitable when the order of the time fractional derivative c lies in the interval 12 1 a correction term is added to maintain high accuracy when c 0 12 numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

third order iterative methods without using any frechet derivative a modification of classical third order methods is proposed the main advantage of these methods is they do not need to evaluate any frechet derivative a convergence theorem in banach spaces just assuming the second divided difference is bounded and a punctual condition is analyzed finally some numerical results are presented

a stirling like method with holder continuous first derivative in banach spaces in this paper the convergence of a stirling like method used for finding a solution for a nonlinear operator in a banach space is examined under the relaxed assumption that the first frechet derivative of the involved operator satisfies the holder continuity condition many results exist already in the literature to cover the stronger case when the second frechet derivative of the involved operator satisfies the lipschitz holder continuity condition our convergence analysis is done by using recurrence relations the error bounds and the existence and uniqueness regions for the solution are obtained finally two numerical examples are worked out to show that our convergence analysis leads to better error bounds and existence and uniqueness regions for the fixed points