discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

finite difference preconditioning cubic spline collocation method of elliptic equations we discuss a finite difference preconditioner for the c 1 interpolatory cubic spline collocation method for a uniformly elliptic operator a defined by au delta u a 0 u in omega the unit square with homogeneous dirichlet boundary conditions using the generalized field of values arguments we discuss the eigenvalues of the preconditioned matrix hat l n 2 1 hat a n 2 where hat a n 2 is the matrix of the collocation discretization operator a n 2 corresponding to a and hat l n 2 is the matrix of the finite difference operator l n 2 corresponding to the uniformly elliptic operator l given by l v delta v v in omega with homogeneous dirichlet boundary conditions finally we mention a bound of h 1 singular values of hat l 1 n 2 hat a n 2 for a general elliptic operator au delta u a 1 u x a 2 u y a 0 u in omega

tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

preconditioning the poincare steklov operator by using green s function this paper is concerned with the poincare steklov operator that is widely used in domain decomposition methods it is proved that the inverse of the poincare steklov operator can be expressed explicitly by an integral operator with a kernel being the green s function restricted to the interface as an application for the discrete poincare steklov operator with respect to either a line edge or a star shaped web associated with a single vertex point a preconditioner can be constructed by first imbedding the line as the diameter of a disk or the web as a union of radii of a disk and then using the green s function on the disk the proposed technique can be effectively used in conjunction with various existing domain decomposition techniques especially with the methods based on vertex spaces from multi subdomain decomposition some numerical results are reported

a nonoverlapping domain decomposition method for orthogonal spline collocation problems a nonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the dirichlet boundary value problem for poisson s equation on an l shaped region we prove existence and uniqueness of the collocation solution and derive optimal order hs norm error bounds for s 0 1 2 the collocation solution on two interfaces is computed using the preconditioned conjugate gradient method and the collocation solution on three squares is computed by a matrix decomposition method that uses fast fourier transforms the total cost of the algorithm is o n2 log n where the number of unknowns in the collocation solution is o n2

a fast domain decomposition poisson solver on a rectangle for hermite bicubic orthogonal spline collocation a fast domain decomposition solver is presented for the piecewise hermite bicubic orthogonal spline collocation solution of poisson s equation on a rectangle the rectangle is divided into parallel strips and the collocation solution is first obtained on the interfaces by solving a collection of independent tridiagonal linear systems a recently developed fast fourier transform solver for piecewise hermite bicubic orthogonal spline collocation is then used to compute the collocation solution on each strip on an n times n uniform partition the proposed domain decomposition solver requires o n 2 log log n arithmetic operations assuming that the strips have the same width and that their number is proportional to n log n the solver is also highly parallel in nature

scaling behaviour for excitation trapping on fractals the authors study the decay behaviour of a nearest neighbour random walker which gets trapped at the first encounter of a sink as underlying structures they consider fractals sierpinski gaskets and regular lattices and establish the scaling behaviour of the decay law with respect to x nd2 ln 1 p where n is the number of steps p the sink concentration and d the spectral dimension the decay scales well for x large for small x scaling is reasonable for the gaskets and the linear chain and rather poor for the square lattice

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

matrix decomposition algorithms in orthogonal spline collocation for separable elliptic boundary value problems fast direct methods are presented for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to separable second order linear elliptic partial

analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables optimal order error estimates are derived for the continuous time collocation method and a discrete time collocation method the crank nicolson collocation method for approximating the solution of a semilinear parabolic initial boundary value problem in r times 0 t where r is the unit square in each case at each time level the approximation is a c 1 piecewise polynomial of degree r geqq 3 defined by collocation at gauss points in r optimal order error estimates are also derived for a family of discrete time collocation methods for a semilinear second order hyperbolic initial boundary value problem in r times 0 t

spline collocation differentiation matrices spline collocation differentiation matrices arising when solving pdes with periodic boundary conditions have block circulant structure in this paper we use this fact to present an eigenvalue analysis for a large class of such spline collocation differentiation matrices we use this to discuss the stability of collocation algorithms for parabolic and hyperbolic problems and fast collocation algorithms for elliptic pdes

a schwarz alternating procedure using spline collocation methods a collocation method is described which obtains an approximate solution to poisson s equation u f on an l shaped region the l shaped region is viewed as the union of two overlapping rectangles the schwarz alternating procedure may be employed to reduce the problem to one in which collocation techniques are used on each rectangle the convergence of the alternating scheme depends upon a discrete maximum principle for collocation methods several sample problems are presented to illustrate the order of convergence of this method and to compare it with several existing numerical methods for the l shaped region

a large deviation result for the range of random walk and for the wiener sausage let s n be a random walk on d and let r n be the number of different points among 0 s1 s n 1 we prove here that if d 2 then x lim n 1 n logp r n nx exists for x 0 and establish some convexity and monotonicity properties of x the one dimensional case will be treated in a separate paper

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

exact enumeration of random walks with traps a new useful method of exact enumeration of random walks with traps is presented the method is applied to the one dimensional case and is shown to yield excellent agreement with an exact solution the advantage of the method is that it can be applied in higher dimensions numerical results confirm the known asymptotic solution and can be used to estimate the time and concentration needed to approach this limit

a generalised self avoiding walk the authors study a generalised self avoiding walk on a lattice in which each vertex may be visited less than k time turban 1983 has argued that for this model the critical exponents should change continuously with k from the standard self avoiding walk exponents k 2 to mean field exponents by generating series expansions for both two and three dimensional lattices the authors find strong support for the conventional and opposing view that the critical exponents remain unchanged for finite k only taking on their mean field values in the limit k to infinity

non universal behaviour of self attracting walks a recently proposed model for self attracting walks is studied using exact enumeration techniques the probability of a step is proportional to exp nu where n 1 for sites already visited by the walker and n 0 for the others with u 0 series for the mean square displacement rn2 approximately n2 nu and the mean number of visited sites sn approximately ns of n step walks are calculated in one to four dimensions in all dimensions anomalous diffusion is observed and exponents nu and s vary continuously with the strength parameter u the results are compared with simulations and with previous results for static and dynamic models of generalized random walks in one dimension the behaviour in two and three dimensions may describe anomalous diffusion in real systems

fast numerical solution of the biharmonic dirichlet problem on rectangles a new method for the numerical solution of the first biharmonic dirichlet problem in a rectangular domain is presented for an n times n mesh the complexity of this algorithm is on the order of

random walks on lattices with randomly distributed traps i the average number of steps until trapping for a random walk on a lattice with a random distribution of traps we derive an asymptotic expansion valid for smallq for the average number of steps until trapping whereq is the probability that a lattice point is a trap we study the case of perfect traps where the walk comes to an end and the extension obtained by letting the traps be imperfect i e by giving the walker a finite probability to remain free when stepping on a trap several classes of random walks of varying dimensionality are considered and special care is taken to show that the expansion derived is exact up to and including the last term calculated the numerical accuracy of the expansion is discussed

a parallel robin robin domain decomposition method for the stokes darcy system we propose a new parallel robin robin domain decomposition method for the coupled stokes darcy system with beavers joseph saffman jones interface boundary condition in particular we prove that with an appropriate choice of parameters the scheme converges geometrically independent of the mesh size

a robin type domain decomposition method with red black partition in this paper we present a new robin type nonoverlapping domain decomposition dd preconditioner the unknown variables to be solved in this preconditioned algebraic system are the robin transmission data on the interface which are different from the well known dd methods like substructuring nonoverlapping dd method and feti method through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate for the second order elliptic problem we prove that the condition number of the preconditioned system is c 1 log frac h h 2 where h is the coarse mesh size and h is the fine mesh size moreover in this paper we always assume that there is a red black partition for the whole domain omega numerical results are given to illustrate the efficiency of our dd preconditioner

moderate deviations for the volume of the wiener sausage for a 0 let wa t be the a neighbourhood of standard brownian motion in rd starting at 0 and observed until time t it is well known that elwa t l ikat t oc for d 3 with kea the newtonian capacity of the ball with radius a we prove that lm t d 2 d log p wa t l oc this is markedly different from the optimal strategy for large deviations iwa t i 5 is nonanalytic at some critical point in 0 kia above which it follows a pure power law this crossover is associated with a collapse transition in the optimal strategy we also derive the analogous moderate deviation result for d 2 in this case eiwa t 2wrt logt t oc and we prove that

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

on a parallel robin type nonoverlapping domain decomposition method in recent years a nonoverlapping robin type domain decomposition method ddm for the finite element discretization systems of the second order elliptic equations which is based on using robin type boundary conditions as information transmission conditions on the subdomain interfaces has been developed and analyzed since it was first proposed by p l lions in on the schwarz alternating method iii a variant for nonoverlapping subdomains in proceedings of the 3rd international symposium on domain decomposition methods for partial differential equations siam philadelphia pa 1990 pp 202 223 however the convergence rate of this ddm with many subdomains remains open when the lower term of equations vanishes this open problem will be considered in this paper the convergence rate is almost 1 o h 1 2 h 1 2 in certain cases for example the case of a small number of subdomains where h is the mesh size and h is the size of subdomain in order to get the desirous convergence results two ma

iterative algorithms for orthogonal spline collocation linear systems in this paper we present several block iterative algorithms for solving the linear systems which arise from the discretization of second order separable elliptic partial differential equations with hermite cubic collocation the convergence rates of these algorithms are estimated

solution of a model of self avoiding walks with multiple monomers per site on the husimi lattice we solve a model of self avoiding walks which allows for a site to be visited up to two times by the walk on the husimi lattice this model is inspired in the domb joyce model and was proposed to describe the collapse transition of polymers with one site interactions only we consider the version in which immediate self reversals of the walk are forbidden the phase diagram we obtain for the grand canonical version of the model is similar to the one found in the solution of the bethe lattice with two distinct polymerized phases a tricritical point and a critical endpoint

monte carlo renormalisation group for the true self avoiding walk following the general ideas by swendsen 1982 a monte carlo renormalisation group mcrg estimation of the end to end exponent nu for the one dimensional true self avoiding walk tsaw has been carried out because the tsaw is known to reach its asymptotic behaviour only very slowly an approximate expression for the fixed point has been determined by monte carlo techniques this approximate fixed point has been used as a starting point for the mcrg the final result for the exponent nu is 0 665 or 0 004

asymptotics of the generating function for the volume of the wiener sausage we consider the generating function mathbb e exp lambda c varepsilon t of the voltime of the wiener sausagec t which is the e neighbourhood of the wiener path in the time interval 0 t for 0 it had been investigated by van den berg and toth but in contrast to the case 0 there is no simple expression for the exponential rate known we determine the asymptotic behaviour of this rate for small and large

an alternating direction implicit method for orthogonal spline collocation linear systems an alternating direction implicit method is analyzed for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to some separable second order linear elliptic partial differential equations in rectangles on annxn partition with jordan s selection of the acceleration parameters the method requireso n 2 ln 2 n arithmetic operations to produce an approximation whose accuracy in theh 1 norm is that of the collocation solution

basis spline collocation method for solving the schrodinger equation in axillary symmetric systems we use basis spline collocation to solve the schrodinger equation on a lattice for axially symmetric systems with and without spin the focus of the present work is on systems which have continuum states weakly bound states or strong spin orbit coupling since these are the most difficult to solve on the lattice a brief overview of the basis spline collocation method is included which concentrates primarily on those aspects of the theory which are relevant to its application in cylindrical coordinates to demonstrate the method we solve several model problems selected from the fields of atomic and nuclear physics

a unifying model of generalised random walks the authors propose a unifying model for generalised static random walks in which each walk has a weight exp g sigma ni alpha where ni is the occupation number of a site and alpha and g are variable parameters special cases of this general walk include the ordinary brownian random walk the self avoiding walk the lattice domb joyce model 1972 and the interacting random walk model recently introduced by stanley et al 1983 the asymptotic properties of the walk are studied in one dimension by effective medium arguments and exact enumeration methods for repulsive correlations the authors find saw behaviour while for attractive correlations the model is self trapping for 1 alpha or 2 and exhibits anomalous diffusion with continuously varying exponents for 0 or alpha 1 in higher dimensions comments are made about effective medium predictions and their relationship to other generalised random walk models

spectral analysis of dirichlet neumann operators and optimized schwarz methods with robin transmission conditions in this paper the tight relationship between dirichlet neumann d n operators and optimized schwarz methods with robin transmission conditions is disclosed we describe the spectral distribution of continuous d n operators and give a rigorous spectral analysis of discrete d n operators by these results we prove that the optimized schwarz methods with robin transmission conditions cannot converge geometrically in the case of continuous problems furthermore we get the accurate convergence rate of the two subdomain case in addition an estimation of convergence rate of the optimized schwarz methods is presented in the general case most of our results are asymptotically sharp

optimized schwarz methods for maxwell s equations over the last two decades classical schwarz methods have been extended to systems of hyperbolic partial differential equations using characteristic transmission conditions and it has been observed that the classical schwarz method can be convergent even without overlap in certain cases this is in strong contrast to the behavior of classical schwarz methods applied to elliptic problems for which overlap is essential for convergence more recently optimized schwarz methods have been developed for elliptic partial differential equations these methods use more effective transmission conditions between subdomains than the classical dirichlet conditions and optimized schwarz methods can be used both with and without overlap for elliptic problems we show here why the classical schwarz method applied to both the time harmonic and time discretized maxwell s equations converges without overlap the method has the same convergence factor as a simple optimized schwarz method for a scalar elliptic equation based on this insight we develop an entire new hierarchy of optimized overlapping and nonoverlapping schwarz methods for maxwell s equations with greatly enhanced performance compared to the classical schwarz method we also derive for each algorithm asymptotic formulas for the optimized transmission conditions which can easily be used in implementations of the algorithms for problems with variable coefficients we illustrate our findings with numerical experiments

self attracting walk on lattices we have studied a model of self attracting walk proposed by sapozhnikov using the monte carlo method the mean square displacement and the mean number of visited sites are calculated for one two and three dimensional lattices in one dimension the walk shows diffusive behaviour with however in two and three dimensions we observed a non universal behaviour i e the exponent varies continuously with the strength of the attracting interaction

wiener sausage volume moments the statistical characteristics of a spatial region visited by a spherical brownian particle during timet wiener sausage are investigated the expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension in the one dimensional case the distribution of probability density and the moments of any order are determined for this quantity

block iterative algorithms for solving hermite bicubic collocation equations in this paper we present a so called local elimination technique by which a nonsymmetric system arising in the discretization of the poisson equation with a hermite bicubic collocation approximation is reduced to a block tridiagonal and block symmetric system a class of block iterative algorithms is developed for solving the resulting system the convergence rates of algorithms are discussed

multilevel additive and multiplicative methods for orthogonal spline collocation problems multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise hermite bicubics is applied to the dirichlet boundary value problem for a self adjoint elliptic partial differential equation on a rectangle additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space a general theory of additive and multiplicative schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the laplacian with the spectral constants independent of the fine partition stepsize and the number of levels the preconditioned conjugate gradient and preconditioned orthomin methods are considered for the solution of collocation problems an implementation of the methods is discussed and the results of numerical experiments are presented

second order splitting combined with orthogonal cubic spline collocation method for the kuramoto sivashinsky equation in this paper a second order splitting method is applied to the kuramoto sivashinsky equation and then an orthogonal cubic spline collocation procedure is employed to the approximate resulting system this semidiscrete method yields a system of differential algebraic equations daes of index 1 error estimates in l 2 and l c norms are obtained for the semidiscrete approximation for the time discretization the time integrator radau5 is used the results of numerical experiments are presented to validate the theoretical findings

discrete time orthogonal spline collocation methods for vibration problems discrete time orthogonal spline collocation schemes are formulated and analyzed for vibration problems involving various boundary conditions each problem is written as a schrodinger type system which is then approximated by crank nicolson and or alternating direction implicit orthogonal spline collocation schemes these schemes are shown to be second order accurate in time and of optimal order accuracy in space in the hm norm m 1 2

generalizations and accelerations of lions nonoverlapping domain decomposition method for linear elliptic pde in this paper a one parameter generalization of lions nonoverlapping domain decomposition method for linear elliptic pdes is proposed and studied the generalized methods are shown to be descent direction methods for minimizing an interface bias functional iteration convergence of both the continuous and finite element versions of the proposed methods is established it is theoretically and numerically demonstrated that for generic choices of the parameter the generalized methods converge faster than lions original method algorithms are given and numerical results are presented

generalised self avoiding walk a generalisation of the self avoiding walk is introduced in which k or higher multiple points are forbidden k 2 corresponds to the standard self avoiding walk the flory theory gives the radius of gyration exponent nu k k 1 k 1 e 2 when e or ec k 2k k 1 e is the euclidean dimension of the problem and ec k the upper critical dimension which is also obtained using the fractal set theory

on a random process interpolating between markovian and non markovian random walks the authors give the exact solution in one dimension of the continuum analogue of a model of stanley et al 1983 which interpolates between markovian polya and self avoiding random walks on lattices and make some heuristic comments on the nature of phase transitions in the model in higher dimensions

alternating direction collocation for rectangular regions abstract an alternating direction procedure for finite element collocation equations can be formulated on rectangular regions finite element collocation offers a great savings over finite element techniques in the matrix formation stage of the numerical process moreover with an alternating direction procedure multidimensional problems can be solved as a series of one dimensional problems this greatly reduces the amount of computer time and storage required to solve large problems

on the volume of the intersection of two wiener sausages for a 0 let w a 1 t and w a 2 t be the a neighbourhoods of two independent standard brownian motions in r d starting at 0 and observed until time t we prove that for d 3 and c 0 lim t 1 1 t d 2 d logp jw a 1 ct w a 2 ct j t i a d c and derive a variational representation for the rate constant i a d c here a is the newtonian capacity of the ball with radius a we show that the optimal strategy to realise the above large deviation is for w a 1 ct and w a 2 ct to form a swiss cheese the two wiener sausages cover part of the space leaving random holes whose sizes are of order 1 and whose density varies on scale t 1 d according to a certain optimal prole we study in detail the function c 7 i a d c it turns out that i a d c

polymers in a disordered environment a polymer chain in equilibrium in a disordered environment is studied using a flory theory and by mapping the problem onto random walks in an environment with traps the asymptotic behaviour of the size of the polymer r as a function of the number of monomers n is obtained if the disorder is weak in comparison with the self repulsion of the chain the self avoiding random walk result is found the random environment leads to effective attractive forces which for sufficiently strong disorder lead to the collapse of the chain the properties of the collapsed chain depend upon the type of disorder and on the self repulsion of the chain if the self repulsion increases sufficiently fast as the density increases then the collapsed chain has a finite density n rd to constant as n to infinity otherwise several other interesting scaling forms are possible

sur une conjecture de m kac we consider the following heat conduction problem let k be a compact set in euclidean space 3 suppose that k is held at the temperature 1 while the surrounding medium is at the temperature 0 at time 0 following spitzer we investigate the asymptotic behaviour of the integral e k t which represents the total energy flow in time t from the set k to the surrounding medium 3 k an asymptotic expansion is given for e k t which refines a theorem due to spitzer this expansion also verifies and improves a formal calculation of kac similar results are proved in higher dimensions up to the constant m k the quantity e k t can be interpreted as the expected value of the volume of the wiener sausage associated with k and a d dimensional brownian motion this point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

exponential estimates for the wiener sausage letc e t be the wiener sausage of radius e inr d up to timet we obtain bounds on the asymptotics ofe exp c e t ast for all 0

convergence analysis of orthogonal spline collocation for elliptic boundary value problems existence uniqueness and optimal order h2 h1 and l2 error bounds are established for the orthogonal spline collocation solution of a dirichlet boundary value problem on the unit square the linear elliptic nonself adjoint partial differential equation is given in nondivergence form the approximate solution which is a tensor product of continuously differentiable piecewise polynomials of degree r geq 3 is determined by satisfying the partial differential equation at the nodes of a composite gauss quadrature

a comparative study of interacting random walk models the authors analyse four interacting random walk models from a comparative point of view concentrating on one dimensional versions they consider an interacting random walk model recently introduced by stanley et al 1983 the domb joyce model 1972 and the true self avoiding walk of amit et al 1983 in addition they introduce a model based on the weighting of turning points whose properties are related to those of the ising spin chain fixed points and their stability are identified for both attractive and repulsive correlations in all four cases they compare the mechanisms involved in each of these models and comment on the role played in each by its particular form of correlation between steps the different universal behaviours are shown to arise from competition between short range and long range and particularly cumulative memory effects and between local and global normalisation conditions

efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles piecewise hermite bicubic orthogonal spline collocation laplace modified and alternating direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed the schemes are shown to be unconditionally stable and of optimal order accuracy in the h 1 and discrete maximum norms for space and time respectively implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

trapping of random walks on the line several features of the trapping of random walks on a one dimensional lattice are analyzed the results of this investigation are as follows 1 the correction term to the known asymptotic form for the survival probability ton steps is o 2n 1 3 where ln 1 c andc is the trap concentration 2 the short time form for the survival probability is found to be exp a c n1 2 wherea c is given in eq 21 3 the mean square displacement of a surviving random walker is found to go liken2 3for largen 4 when the distribution of trap free regions is changed so that very large regions are much rarer than for ideally random trap placement the asymptotic survival probability changes its dependence onn one such model is studied

multilevel preconditioners for non self adjoint or indefinite orthogonal spline collocation problems efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation osc discretization of a dirichlet boundary value problem with a non self adjoint or an indefinite operator the osc solution is sought in the hermite space of piecewise bicubic polynomials it is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal osc equation the preconditioners are used with the preconditioned conjugate gradient method and numerical results are presented that demonstrate their efficiency

lifschitz tail and wiener sausage ii the present paper is a continuation of 23 we present here two examples of situations where the method developed in 23 applies the first example is that of a periodic diffusion z in r n 1 with generator l isia dj where a is a c2 z periodic elliptic matrix we consider the wiener sausage w u0 c u i z c where c is a nonpolar compact subset of r for instance a closed ball or disc in r if a denotes the constant matrix of homogenized coefficients of the periodic diffusion see 1 i we show that

spectral analysis of hermite cubic spline collocation systems in this paper we present an eigenvalue analysis of the first order and second order hermite cubic spline collocation differentiation matrices with arbitrary collocation points some important features are explored and compared with some other discrete methods such as finite difference methods

combinatorial formulae for one dimensional generalised random walks the calculation of ensemble averages in one dimensional walks with weights given by exp g sigma ni alpha where ni is the number of times a site is visited is simplified by reducing the problem to one of calculating binomial factors for partitions and their permutations the combinatorial factors are generalised to the case of cayley trees and all lattices without closed loops in one dimension the combinatorial formulae are shown to reduce the known cases when alpha 0 and alpha 1 the authors also use these formulae to extend the exact enumeration results for the one dimensional generalised random walk from 21 to 29 terms and the series are analysed using pade approximant techniques

modified successive overrelaxation msor and equivalent 2 step iterative methods for collocation matrices abstract we consider a class of consistently ordered matrices which arise from the discretization of boundary value problems bvps when the finite element collocation method with hermite elements is used through a recently derived equivalence relationship for the asymptotic rates of convergence of the modified successive overrelaxation msor and a certain 2 step iterative method we determine the optimum values for the parameters of the msor method as it pertains to collocation matrices a geometrical algorithm which utilizes capturing ellipse arguments has been successfully used the fast convergence properties of the optimum msor method are revealed after its comparison to several well known iterative schemes numerical examples which include the solution of poisson s equation are used to verify our results

diffusion with random traps transient one dimensional kinetics in a linear potential the problem of one dimensional diffusion with random traps is solved without and with a constant field of force using an eigenvalue expansion for long times and the method of images for short times we give an exact straightforward solution for the time dependence of the mean survival probability and the mean probability density for returning to the origin using the backward equation approach we determine the mean survival time and the mean residence time density at the origin we comment on the relation between these solutions and those for one dimensional diffusion with random reflectors

transition to localization of biased walkers in a randomly absorbing environment abstract we study biased random walkers on lattices with randomly dispersed static traps in one two and three dimensions as the external bias is increased from zero the system undergoes a phase transition most clearly manifested in the asymptotic drift velocity of survivors which is zero below a critical bias this transition is second order in one dimension but of first order in higher dimensions the model can be mapped to a stretched polymer with attractive interaction between monomers and this phase transition would then describe sudden unfolding of the polymer when the stretching force exceeds a critical value we also present precise simulations of the zero bias case where we show unambiguously that the transition between the rosenstock rs and donsker varadhan dv regimes is first order in dimension 2