solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

the justification of approximate bounded solutions of mixed sturm liouville problems the approximate solution of a homogeneous singular integral equation or equivalently a homogeneous infinite algebraic system is fully justified and the error resulting from the truncation is estimated this equation algebraic system is the standard form to which many even mixed sturm liouville problems are reducible

zero temperature dynamics of ising models on the triangular lattice we consider the nature of spin flips of zero temperature dynamics for ferromagnetic ising models on the triangular lattice with nearest neighbor interactions and an initial configuration chosen from a symmetric bernoulli distribution we prove that all spins flip infinitely many times for almost every realization of the dynamics and initial configuration

a numerical comparison of chebyshev methods for solving fourth order semilinear initial boundary value problems in solving semilinear initial boundary value problems with prescribed non periodic boundary conditions using implicit explicit and implicit time stepping schemes both the function and derivatives of the function may need to be computed accurately at each time step to determine the best chebyshev collocation method to do this the accuracy of the real space chebyshev differentiation spectral space preconditioned chebyshev tau real space chebyshev integration and spectral space chebyshev integration methods are compared in the l 2 and w 2 2 norms when solving linear fourth order boundary value problems and in the l 0 t l 2 and l 0 t w 2 2 norms when solving initial boundary value problems we find that the best chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to fourier spectral discretization

the pseudospectral method for solving differential eigenvalue problems abstract a simple pseudospectral method is presented for the numerical solution of linear differential eigenvalue problems the method does not produce the spurious eigenvalues which generally occur when such problems are solved by the spectral tau method the pseudospectral method is presented using two model problems and the presentation contains a useful algorithm for the computation of the spectral differentiation matrices at general collocation points numerical results are offered for a set of benchmark problems including the orr sommerfeld equation for stability of plane poiseuille flow between parallel plates the results indicate that the new pseudospectral method is comparable in accuracy to the tau method

analysis and computation of the pressure field of preatherosclerotic transport in the artery wall prerequisite to atherogenesis is low density lipoprotein ldl cholesterol transport from the blood into the artery wall across its endothelial cell ec lining whereas most ec junctions are tight enough to preclude ldl passage some approx 1 2000 6000 rare ecs have temporarily widened junctions typically approx 10 5 of the problem s radial scale that allow passage of large molecular aggregates like ldl in a purely diffusive transport model laplace s equation describes the steady tracer profile throughout the two major subendothelial wall layers connected by matching conditions in a diffusive and transmural pressure driven advective transport model laplace s equation describes the steady pressure distribution in the wall whose gradient is proportional to the advecting velocity both problems yield the same boundary value problem that has mixed dirichlet robin boundary conditions we construct a linear system of integral equations that approximates the series expansion coefficients o

integrated local petrov galerkin sinc method for structural mechanics problems an integrated local petrov galerkin sinc method is introduced and applied to static structural mechanics problems the method approximates the highest derivative in the local weak form of the governing equation on a rectangular grid and the lower derivatives and unknown function are found by numerical indenite integration we suggest that the essential boundary conditions may be applied by the traditional penalty method or by a method in which the stiness matrix is reduced to eliminate dependent degrees of freedom we propose and compare the performance of three basis functions in terms of their accuracy and convergence properties for two problems a one dimensional tapered bar and a two dimensional plane stress elasticity problem our results indicate that the present method can provide greater accuracy than the sinc method based on interpolation of highest derivative an integration based collocation method suggested by li and wu li c and wu x numerical solution of dierential equations using sinc method based on

a dynamic rotating blade model at an arbitrary stagger angle based on classical plate theory and the hamilton s principle a dynamic model based on classical plate theory is presented to investigate the vibration behavior of a rotating blade at an arbitrary stagger angle and rotation speed the hamilton s principle is

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

the method of integral equation formulation and the unbounded solutions of elastic contact problems abstract it is shown that a modification of the integral equation formulation 1 can be used to find an expression of the unbounded contact stress of problems in the theory of elasticity the modifications consist in reducing the problem to a hilbert type singular integral equation rather than that of the cauchy s kernel one the reduction is carried out in analogous procedures to that followed in 1 but here the unknown function is the contact stress in contrast to the previous formulation in which the unknown function was a necessarily continuous displacement the hilbert equation is inverted to define the contact stress and further reduced to an infinite algebraic system its solution completes the definition with the aid of the physical conditions the truncation of the algebraic system is justified and the error is estimated

the integral equation formulation of mixed finite time dependent elastic problems abstract this paper establishes through a concrete example that the integral equation formulation of time dependent mixed boundary value problems can be extended for problems in the theory of elasticity to this end the method applied to the resulting integral equation is the one begun by cherski 1 and evolved by eckhardt and el sheikh 2 still further to solve intial mixed boundary value problems new relationships between the fundamental coefficients characterizing the technique are obtained these simplify the procedures and reveal more of the nature of the method

solving high order partial differential equations with indirect radial basis function networks this paper reports a new numerical method based on radial basis function networks rbfns for solving high order partial differential equations pdes the variables and their derivatives in the governing equations are represented by integrated rbfns the use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high order derivatives the proposed rbfn method is verified successfully through the solution of thin plate bending and viscous flow problems which are governed by biharmonic equations for thermally driven cavity flows the solutions are obtained up to a high rayleigh number

ising models on hyperbolic graphs we consider ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph v f where v is the number of neighbors of each vertex and f is the number of sides of each face let tc be the critical temperature and t c sup t tc f 2 where f is the free boundary condition b c gibbs state is the plus b c gibbs state and is the minus b c gibbs state we prove that if the hyperbolic graph is self dual i e v f or if v is sufficiently large how large depends on f e g v 35 suffices for any f 3 and v 17 suffices for any f 17 then 0 t c tc in contrast with that t c tc for ising models on the hypercubic lattice zd with d 2 a result due to lebowitz 22 while whenever t t c f 2 the last result is an improvement in comparison with the analogous statement in refs 28 and 33 in which it was only proved that f 2 when t t c and it remains to show in both papers that f 2 whenever t t c therefore t c and tc divide 0 into three intervals 0 t c t c tc and tc in which but f 2 and f 2 and respectively

satisfaction of boundary conditions for chebyshev collocation methods in cuboidal domains abstract a collocation strategy for the satisfaction of boundary conditions in the application of chebyshev spectral methods to poisson and biharmonic type problems in cuboidal domains is presented this strategy leads in both cases to nonsingular linear systems for the determination of the unknown coefficients

phase ordering after a deep quench the stochastic ising and hard core gas models on a tree consider a low temperature stochastic ising model in the phase coexistence regime with markov semigroup p t a fundamental and still largely open problem is the understanding of the long time behavior of p t when the initial configuration is sampled from a highly disordered state e g a product bernoulli measure or a high temperature gibbs measure exploiting recent progresses in the analysis of the mixing time of monte carlo markov chains for discrete spin models on a regular b ary tree open image in new window we study the above problem for the ising and hard core gas independent sets models on open image in new window if is a biased product bernoulli law then under various assumptions on the bias and on the thermodynamic parameters we prove almost sure weak convergence of p t to an extremal gibbs measure pure phase and show that the limit is approached at least as fast as a stretched exponential of the time t in the context of randomized algorithms and if one considers the glauber dynamics on a large finite tree our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

a chebyshev spectral collocation method for the computation of buoyancy induced flows abstract a spectral powell hybrid algorithm is presented for the solution of the boussinesq equations the method is applied to a streamfunction formulation of the governing equations and used to solve the double glazing test problem excellent qualitative agreement with benchmark solutions is obtained given the relatively few degrees of freedom used

some spectral approximations of two dimensional fourth order problems this paper deals with the approximation of the biharmonic equation in a square domain with dirichlet boundary conditions two types of discrete problems are presented the numerical analysis is performed and estimates for the error between the exact and approximate solutions are given

solving high order ordinary differential equations with radial basis function networks this paper is concerned with the application of radial basis function networks rbfns for numerical solution of high order ordinary differential equations odes two unsymmetric rbf collocation schemes namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process are developed to solve high order odes directly and the latter is found to be considerably superior to the former good accuracy and high rate of convergence are obtained with the proposed indirect method

spin models on non euclidean hyperlattices griffiths phases without extrinsic disorder we study short range ferromagnetic models residing on planar manifolds with global negative curvature we show that the local metric properties of the embedding surface induce droplet formation from the boundary resulting in the stability of a griffiths phase at a temperature lower than that of the bulk transition we propose that this behaviour is independent of order parameter and hyperlattice specifics and thus is universal for such non euclidean spin models their temperature curvature phase diagrams are characterized by two distinct bulk and boundary transitions each has mean field critical behaviour and a finite correlation length related to the curvature of the embedding surface the implications for experiments on superconducting hyperlattice networks are also discussed

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

self avoiding walks and polygons on non euclidean lattices self avoiding walks and polygons on four non euclidean lattices are studied by the method of series expansions it appears that the polygons and walks have different critical points from each other and that both are in a different universality class to their cartesian lattice counterparts an exact solution is given for one lattice

a spectral multigrid method for the stokes problem in streamfunction formulation abstract suitable spectral multigrid components for the stokes problem in streamfunction formulation are presented we split the stokes problem into a system of two equations with the laplace operator we discretize by means of a collocation or pseudo spectral method preconditioning with finite differences is employed numerical results for the spectral multigrid method are presented

neumann problem for generalized n poisson equation using a hierarchy of integral operators having higher order neumann functions and their derivatives as kernels the neumann problem for a 2nth order linear partial complex differential equation is discussed the solvability of the problem is obtained

numerical solution of the 3 d buoyancy driven cavity problem by a pseudospectral method the pseudospectral continuation technique introduced in karageorghis and sivaloganathan for 2 d recirculating flows is extended to three dimensions the pseudospectral continuation strategy is applied to the vorticity vector potential formulation of the boussinesq equations and used to successfully solve the three dimensional air filled buoyancy driven cavity problem at various rayleigh numbers

glauber dynamics on trees boundary conditions and mixing time we give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the glauber dynamics in the so called bethe approximation specifically we show that the spectral gap and the log sobolev constant of the glauber dynamics for the ising model on an n vertex regular tree with boundary are bounded below by a constant independent of n at all temperatures and all external fields this implies that the mixing time is o logn in contrast to the free boundary case where it is not bounded by any fixed polynomial at low temperatures in addition our methods yield simpler proofs and stronger results for the spectral gap and log sobolev constant in the regime where the mixing time is insensitive to the boundary condition our techniques also apply to a much wider class of models including those with hard core constraints like the antiferromagnetic potts model at zero temperature proper colorings and the hard core lattice gas independent sets

ising models on the lobachevsky plane we consider the ising model on a lattice which is the orbit of a discrete cocompact group acting on the hyperbolic plane for large values of the inverse temperature we construct an uncountable number of mutually singular gibbs states

metastable behavior for bootstrap percolation on regular trees we examine bootstrap percolation on a regular b 1 ary tree with initial law given by bernoulli p the sites are updated according to the usual rule a vacant site becomes occupied if it has at least occupied neighbors occupied sites remain occupied forever it is known that when b 2 the limiting density q q p of occupied sites exhibits a jump at some p t p t b 0 1 from q t q p t p t we investigate the metastable behavior associated with this transition explicitly we pick p p t h with h 0 and show that as h 0 the system lingers around the critical state for time order h 1 2 and then passes to fully occupied state in time o 1 the law of the entire configuration observed when the occupation density is q q t 1 converges as h 0 to a well defined measure

resolution of fourth order problems by the mortar element method we present a resolution of fourth order problems by the mortar element method in the case of spectral discretizations we prove that this approach leads to a well posed discrete problem and the numerical analysis gives an optimal error estimate

upper transition point for percolation on the enhanced binary tree a sharpened lower bound hyperbolic structures are obtained by tiling a hyperbolic surface with negative gaussian curvature these structures generally exhibit two percolation transitions a system wide connection can be established at a certain occupation probability p p c1 and there emerges a unique giant cluster at p c2 g p c1 there have been debates about locating the upper transition point of a prototypical hyperbolic structure called the enhanced binary tree ebt which is constructed by adding loops to a binary tree this work presents its lower bound as p c2 ensuremath gtrsim 0 55 by using phenomenological renormalization group methods and discusses some solvable models related to the ebt

the finite mixed sturm liouville problems of the dirichlet newton type abstract the mixed sturm liouville problem for the laplace operator in the unit circle is considered the boundary conditions consist in the dirichlet condition prescribed partially at the circumference while the complementary part is subjected to the newton condition the problem is reduced to a homogeneous wing singular integro differential equation and this in turn to an infinite system of algebraic equations whose solution can conveniently be obtained through truncation the procedures of the reduction suggest that almost any problem of the type considered in this paper can be reduced to wing equations of the same class the calculations are carried out right to the numerical steps the first few eigenvalues and their corresponding eigenfunctions are obtained

harmonic solutions of a mixed boundary problem arising in the modeling of macromolecular transport into vessel walls the earliest events leading to atherosclerosis involve the transport of lpw density lipooprotein ldl cholesterol from the blood across endothelial cells that line the artery wall laplace s equation describes the steady state diffusion profile of a tracer through the vessel wall this gives rise to a boundary value problem with mixed dirichlet and robin conditions we construct a linear system of integral equations that approximate the coefficients of the series expansion of the solution we prove the existence of the solution to this problem analytically by using gershgorin s theorem on the location of the eigenvalues of the corresponding matrix we give a uniqueness proof using miranda s theorem c miranda p d e of elliptic type springer verlag berlin 1970 the analytical construction method forms the basis for a numerical calculation algorithm we apply our results to the transport problem above and use them to interpret experimental observations of the growth of localized tracer leakage spots with tracer circulation time

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

conforming spectral approximations for non conforming domain decompositions abstract a spectral domain decomposition scheme is introduced for the numerical solution of second and fourth order elliptic problems the technique is applicable to certain domain decompositions of rectangular or rectangularly decomposable domains it is shown that it yields approximations which are pointwise c 0 continuous across the subdomain interfaces for second order problems and pointwise c 1 continuous across the subdomain interfaces for fourth order problems

the ising model on hyperlattices the authors describe the construction of regular lattices in two dimensional hyperbolic space by means of the action of a discrete subgroups of su 1 1 they consider an ising model on such lattices and show how the thermodynamic limit can be handled they give high and low temperature expansions of the free energy magnetic susceptibility and magnetization and find that these quantities diverge at a critical temperature with mean field exponents beta 1 2 gamma 1 they also conjecture the long distance behaviour of correlation functions at the critical temperature

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

on the mixed initial dirichlet neumann problem for the wave equation in the circle the method of integral equation formulation proposed by eckhardt and el sheikh is extended to solve an even initial mixed problem for the first time the truncation of the integral equation or equivalently the infinite homogeneous algebraic system to which the problem is converted is justified in addition having carried out the progression of the procedures right to the numerical results at several cases a clarified and comprehensive idea about the usefulness of the method is revealed

solving pdes with the aid of two dimensional haar wavelets two dimensional haar wavelets are applied for solution of the partial differential equations pdes the proposed method is mathematically simple and fast to demonstrate the efficiency of the method two test problems solution of the diffusion and poisson equations are discussed computer simulation showed that the method guarantees the necessary exactness already for a small number of grid points

automatic spectral collocation for integral integro differential and integrally reformulated differential equations automatic chebyshev spectral collocation methods for fredholm and volterra integral and integro differential equations have been implemented as part of the chebfun software system this system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization the same objects can be used in matrix free iterative methods in linear algebra in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients as a further application of the ability to implement operator equations a method of greengard 1 for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems in the integral form large condition numbers associated with differentiation matrices in high order problems are avoided the ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun the integral method also can be extended to first order systems although chebfun syntax does not currently allow easy implementation in this case

periodic boundary conditions on the pseudosphere we provide a framework for building periodic boundary conditions on the pseudosphere or hyperbolic plane the infinite two dimensional riemannian space of constant negative curvature starting from the common case of periodic boundary conditions in the euclidean plane we introduce all the required mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane we stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and illustrate how to implement such periodic boundary conditions in two examples the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices

a fully conforming spectral collocation scheme for second and fourth order problems abstract a spectral element scheme is presented for the solution of second and fourth order problems in two and three space dimensions this scheme is based on a collocation formulation and for a certain class of problems results in spectral approximations which are c1 pointwise continuous across the element interfaces for second order problems and c3 pointwise continuous across the element interfaces for fourth order problems

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

geometric effects on critical behaviours of the ising model we investigate the critical behaviour of the two dimensional ising model defined on a curved surface with a constant negative curvature finite size scaling analysis reveals that the critical exponents for the zero field magnetic susceptibility and the correlation length deviate from those for the ising lattice model on a flat plane furthermore when reducing the effects of boundary spins the values of the critical exponents tend to those derived from the mean field theory these findings evidence that the underlying geometric character is responsible for the critical properties of the ising model when the lattice is embedded on negatively curved surfaces

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

the justification of the truncation applied to homogeneous integral equations with cauchy s kernel abstract this paper is devoted to studying the influence of the truncation applied to a homogeneous cauchy type singular integral equation on its eigenvalues and eigenfunctions this equation represents the class to which mixed sturm liouville problems of the dirichlet neuman type are reducible the study is illustrated through considering a concrete problem it has been shown that an eigenvalue of the truncated equation tends to the exact corresponding one on increasing the order of the truncation set on that equation as for a truncated eigenfunction it leads to the designation of the exact one in the limit where the corresponding eigenvalue is sufficiently precise an illustrative example is given

viscous fluid fingering on a negatively curved surface viscous fingering formation in flat hele shaw cells is a classical and widely studied fluid mechanical problem we examine the development of viscous fluid fingering on a two dimensional surface of constant negative gaussian curvature the hyperbolic plane h 2 a perturbative mode coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system fingertip splitting and finger competition we also report on a time dependent control strategy placed on the injection rate which is able to minimize viscous fingering growth on h 2

a fourier method for initial value problems with mixed boundary conditions abstract works concerning solutions of initial problems with mixed boundary conditions are very rare in the literature and essentially depend on the geometry of some individual problems k b ranger for example considered the cooling of a finite needle and strip with prescribed cooling temperature in a medium bounded by two parallel planes perpendicular to the axis and maintained at zero temperature 1 because of the axial symmetries of these problems ranger could employ integral operators which map 2 d harmonics into the solution of the reduced wave equation and into that of the heat equation thereby in this work a systematic method for solving initial plane problems with mixed boundary conditions is proposed it can be considered as a natural generalization of the fourier method that deals with the uniform initial problems on the one hand and of that started by ju i cersi 2 dealing with steady mixed boundary problems on the other by gradual modifications the latter method has become of wide application to solutions of problems in several branches of mathematical physics this has recently been expounded 3 we shall start with presenting the method by means of a concrete example that geometrically as well as physically is rather simplified the heat conductivity in the unit circle with dirichlet neumann boundary conditions the method is applicable to problems with different configurations further it can be extended to deal with more complicated physical situations of mixed character such as dynamical problems of the theory of elasticity with the typical stress strain boundary conditions

finite size scaling in three dimensional bootstrap percolation we consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior bootstrap percolation is an example of cellular automata defined on the d dimensional lattice 1 2 l d in which each site can be empty or occupied by a single particle in the starting configuration each site is occupied with probability p occupied sites remain occupied forever while empty sites are occupied by a particle if at least among their 2d nearest neighbor sites are occupied when d is fixed the most interesting case is the one d this is a sort of threshold in the sense that the critical probability p c for the dynamics on the infinite lattice z d switches from zero to one when this limit is crossed finite size effects in the three dimensional case are already known in the cases 2 in this paper we discuss the case 3 and we show that the finite size scaling function for this problem is of the form f l const in in l we prove a conjecture proposed by a c d van enter

zero temperature glauber dynamics on mathbb z d we study zero temperature glauber dynamics on mathbb z d which is a dynamic version of the ising model of ferromagnetism spins are initially chosen according to a bernoulli distribution with density p and then the states are continuously and randomly updated according to the majority rule this corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field to one at zero temperature with no external field define p c mathbb z d to be the infimum over p such that the system fixates at with probability 1 it is a folklore conjecture that p c mathbb z d 1 2 for every 2 le d in mathbb n we prove that p c mathbb z d to 1 2 as d

generation of pseudospectral differentiation matrices i we present a simple method for computing n times n pseudospectral differentiation matrices of order p in cal o pn 2 operations for the case of quasi polynomial approximation the algorithm is based on fornberg s finite difference algorithm and is numerically stable a fortran implementation is included a necessary and sufficient condition for d p d p 1 is also given

the percolation transition for the zero temperature stochastic ising model on the hexagonal lattice on the planar hexagonal lattice mathbb h we analyze the markov process whose state t in 1 1 mathbb h updates each site v asynchronously in continuous time t 0 so that v t agrees with a majority of its three neighbors the initial v 0 s are i i d with p v 0 1 p 0 1 we study both rigorously and by monte carlo simulation the existence and nature of the percolation transition as t and p 1 2 denoting by t p the expected size of the plus cluster containing the origin we 1 prove that 1 2 and 2 study numerically critical exponents associated with the divergence of p as p 1 2 a detailed finite size scaling analysis suggests that the exponents and of this t dependent percolation model have the same values 4 3 and 43 18 as standard two dimensional independent percolation we also present numerical evidence that the rate at which t as t is exponential

bootstrap percolation on the hypercube in the bootstrap percolation on the n dimensional hypercube in the initial position each of the 2n sites is occupied with probability p and empty with probability 1 p independently of the state of the other sites every occupied site remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial position spans the hypercube we shall show that there are constants c1 c2 0 such that for open image in new window the probability of spanning tends to 1 as n while for open image in new window the probability tends to 0 furthermore we shall show that for each n the transition has a sharp threshold function

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

generation of higher order pseudospectral integration matrices a new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of chebyshev polynomials applications to initial value problems boundary value problems linear integral and integro differential equations are presented the present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods

the estimation of the error resulting due to the truncation applied to homogeneous integral equations with cauchy s kernel abstract many mixed sturm liouville problems can be formulated in a standard form of a discrete riemann problem when it is concerned with the holder continuous entity at the points where the boundary condition changes the discrete problem can be transformed to a homogeneous integral equation with cauchy s kernel this equation can in general be approximately solved and namely through truncation this gives rise to several questions about the justification of the truncation applied to a homogeneous operator as well as the influence of this truncation on the eigenfunctions and eigenvalues in this paper it has been shown that provided an eigenvalue of this integral equation is precisely given the corresponding solution of the truncated integral equations tends to the unique solution of the exact equation on increasing the truncations order indefinitely and the relative error is estimated as for the influence of the truncation on the eigenvalues of that equation it may constitute the subject of another study

spectral differencing with a twist spectral collocation methods have become very useful in providing highly accurate solutions to differential equations a straightforward implementation of these methods involves the use of spectral differentiation matrices to obtain optimal accuracy these matrices must be computed carefully we demonstrate that naive algorithms for computing these matrices suffer from severe loss of accuracy due to roundoff errors several improvements are analyzed and compared a number of numerical examples are provided demonstrating significant differences between the sensitivity of the forward problem and the inverse problem

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

an efficient computational method for solving fractional biharmonic equation in this paper we first introduce the fractional biharmonic equation and an orthogonal system of basis functions for the space of continuous functions on the interval 0 l generated by the shifted chebyshev polynomials moreover we propose a computational method based on the operational matrix of fractional derivatives of these basis functions for solving the fractional biharmonic equation the main characteristic behind this approach is that it reduces the problem under consideration to solving a system of algebraic equations which greatly simplifies the problem convergence of the shifted chebyshev polynomials expansion in two dimensions is investigated also the power of this manageable method is illustrated

multiquadric collocation method with integralformulation for boundary layer problems singularly perturbed boundary value problems often have solutions with very thinlayers in which the solution changes rapidly this paper concentrates on the case where theses layers occur near the boundary although our method can be applied to problems with interior layers one technique to deal with the increased resolution requirements in these layers is the use of domain transformations a coordinate stretching based transform allows to move collocation points into the layer a requirement to resolve the layer accurately previously such transformations have been studied in the context of finite difference and spectral collocation methods in this paper we use radial basis functions rbfs to solve the boundary value problem specifically we present a collocation method based on multiquadric mq functions with an integral formulation combined with a coordinate transformation we find that our scheme is ultimately more accurate than a recently proposed adaptive mq scheme the rbf scheme is also amenable to adaptivity

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

on the coefficients of integrated expansions of ultraspherical polynomials the problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered a formula is proved for the ultraspherical polynomials of which the chebyshev and legendre polynomials are important special cases