galerkin approximations for the stochastic burgers equation existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional galerkin approximations is established and the convergence rate of the galerkin approximations to the solution of the stochastic evolution equation is estimated these abstract results are applied to several examples of stochastic partial differential equations spdes of evolutionary type including a stochastic heat equation a stochastic reaction diffusion equation and a stochastic burgers equation the estimated convergence rates are illustrated by numerical simulations the main novelty in this article is the estimation of the difference of the finite dimensional galerkin approximations and of the solution of the infinite dimensional spde uniformly in space i e in the l infty topology instead of the usual hilbert space estimates in the l 2 topology that were shown before

the exponential integrator scheme for stochastic partial differential equations pathwise error bounds we present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions we discretise in space by a galerkin method and in time by using a stochastic exponential integrator we show that for spatially regular smooth noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces and the noise becomes rougher

fast evaluation of gaunt coefficients recursive approach solution of electromagnetic multisphere scattering problems involves the determination of a large number of gaunt coefficients that are connected with definite integrals of products of three associated legendre functions this paper presents general recursion formulae cycling a single index for the gaunt coefficient with the use of the recursive scheme gaunt coefficients of either low or high degree and order can be evaluated accurately and expeditiously

convergence properties of the sbar d transformation and a fast and accurate numerical evaluation of molecular integrals the present work concerns the study of the convergence properties of the nonlinear s transformation a fast and accurate numerical evaluation of highly oscillatory semi infinite integrals involved in the analytic expressions of molecular integrals over slater type orbitals is obtained using the s transformation these semi infinite integrals involve spherical and reduced bessel functions the convergence properties are analysed and they show that the approximations obtained using the s approach converge to the exact values of the semi infinite integrals without any constraint the numerical tables show that the s method gives unprecedented accuracy with speed up by a factor 2 over the h method and a factor 10 over the transformation this illustrates the superiority of this new approach

a new method for approximating improper integrals in this paper a new transformation referred to as the g n m transform is introduced for the purpose of increasing the rate of convergence of a sequence to its limit in particular if the sequence is a sequence of partial integrals f x then g n m f x is shown to converge super fast to f infty under very general conditions the g n m transform is shown to be closely related to levin and sidi s d transform appl math comp 9 1981 pp 175 215 and the b n transform introduced by gray atchinson and mcwilliams siam j numer anal 8 1971 pp 365 381 several examples are given

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

analytical and numerical development for the two centre overlap like quantum similarity integrals over slater type functions in molecular quantum similarity measure between two molecules using the molecular electron density the major task involves the accurate numerical evaluation of overlap between the two electron densities in the integral measure by expanding the electron densities in terms of atomic orbitals using the linear combination of atomic orbitals approach a large number of overlap like quantum similarity integrals over the basis functions in the two molecules will be required accurately for the calculation of a meaningful quantum similarity measure improvement of the computational methods of these integrals would be indispensable to a further development in computational studies of large molecular systems analytic expressions were obtained for these overlap like quantum similarity integrals over slater type functions in terms of the usual two centre overlap integrals over b functions different approaches were developed for the numerical evaluation of these two centre overlap integrals over b functions which can be used for the numerical evaluation of the two centre overlap like quantum similarity integrals over slater type functions in this work we present our approach which is based on the use of nonlinear transformations for improving convergence of highly oscillatory integrals in the case of more complicated multicentre integrals this approach is shown to be able to produce remarkably good results we also present different representations which were obtained by steinborn group and which can be used for a fast and accurate computation of the two centre overlap like quantum similarity integrals over slater type functions

numerical integration of functions with a sharp peak at or near one boundary using mo bius transformations abstract a class of coordinate transformations depending on a single parameter is studied as quadrature tool working rules for the choice of the parameter are proposed numerical tests for the method are presented they show that these coordinate transformations when combined with gauss legendre quadrature rules are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of the interval of integration a method to combine the transformations with automatic quadrature routines is also proposed it seems to be useful for the numerical evaluation of integrals with the same kind of integrand behavior

the numerical evaluation of very oscillatory infinite integrals by extrapolation recently the author has given two modifications of a nonlinear extrapolation method due to levin and sidi which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity in this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity the new method is applied to a number of complicated integrals among them the solution to a problem in viscoelasticity some convergence results for this method are presented

the hd and hd methods for accelerating the convergence of three center nuclear attraction and four center two electron coulomb integrals over b functions and their convergence properties abstract three center nuclear attraction and four center two electron coulomb integrals over slater type orbitals are required for ab initio and density functional theory dft molecular structure calculations they occur in many millions of terms even for small molecules and require rapid and accurate evaluation the b functions are used as a basis set of atomic orbitals these functions are well adapted to the fourier transform method that allowed analytical expressions for the integrals of interest to be developed rapid and accurate evaluation of these analytical expressions is now made possible by applying the hd and hd methods for accelerating the convergence of the semi infinite oscillatory integrals the convergence properties of the new methods are analysed the numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods

two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series two new classes of nonlinear transformations the d transformation to accelerate the convergence of infinite integrals and the d transformation to accelerate the convergence of infinite series are presented in the course of the development of these transformations two interesting asymptotic expansions one for infinite integrals and the other for infinite series are derived the transformations d and d can easily be applied to infinite integrals 0f t dt whose integrands f t satisfy linear differential equations of the form f t s m k 1p k t f k t and to infinite series s r 1f r w terms f r satisfy a linear difference equation of the form f r s m k 1p k r d kf r such that in both cases the p k have asymptotic expansions in inverse powers of their arguments in order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy mere knowledge of the existence of such a differential or difference equation and its order m is enough this broadens the areas to which these methods can be applied the connection between the d and d transformations with some known transformations in shown the use and the remarkable efficiency of the d and d transformations are demonstrated through several numerical examples the computational aspects of these transformations are described in detail

integrated semigroups and linear partial differential equations with delay abstract we study existence and uniqueness of solutions for linear partial differential equations with delay in l p spaces using an approach of batkai and piazzera and a recent perturbation result for integrated semigroups we apply our result to an equation with delay in the highest order derivatives

an extremely efficient approach for accurate and rapid evaluation of three centre two electron coulomb and hybrid integrals over b functions analytic expressions of three centre two electron coulomb and hybrid integrals over b functions are obtained using the fourier transform method thoroughly explored by steinborn s group these analytic expressions involve semi infinite integrals which are slowly convergent due to the presence of hypergeometric and spherical bessel functions in the integrands we have proven that these hypergeometric functions can be expressed as finite expansions and the integrands involving these series satisfy all the conditions required to apply the h approach which greatly simplifies the application of the nonlinear transformation this work presents a rapid and accurate evaluation of these integrals obtained by using a new approach which we called s this new method is based on the h and methods and some practical properties of spherical bessel reduced bessel and sine functions the s method has greatly simplified the calculations avoiding the long and difficult implementation of the successive zeros of the spherical bessel function and a method for solving linear systems which are required by h and

l2 regularity for parabolic partial integrodifferential equations with delay in the highest order derivatives abstract a partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay by means of abstract regularity results global existence and uniqueness of a strict solution are obtained moreover a characterization of the infinitesimal generator of the solution operator is given

strong convergence rate of finite difference approximations for stochastic cubic schrodinger equations abstract in this paper we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic schrodinger equations driven by a multiplicative q wiener process beyond the uniform boundedness of moments for high order derivatives of the exact solution the key requirement of our approach is the exponential integrability of both the exact and numerical solutions by constructing and analyzing a lyapunov functional and its discrete correspondence we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution which immediately yields the well posedness of both the continuous and discrete problems the latter exponential integrability is obtained through a variant of a criterion given by cox hutzenthaler and jentzen arxiv 1309 5595 as a by product of this exponential integrability we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation type result on the dependence of the noise with first order strong convergence rate

efficient evaluation of coulomb integrals the nonlinear d and transformations the two electron multicentre coulomb integrals constitute the rate limiting step of ab initio and density functional theory dft molecular structure calculations speed up can be achieved by limiting the number of integrals to evaluate analytically but these analytical evaluations remain rate limiting for large molecules here we apply the nonlinear d and transformations to evaluate coulomb integrals over b functions more rapidly than the alternative transformation methods to a given predetermined high accuracy

computation of tail probabilities via extrapolation methods and connection with rational and pade approximants we use the recently developed algorithm for the g n 1 transformation to approximate tail probabilities of the normal distribution the gamma distribution the student s t distribution the inverse gaussian distribution and fisher s f distribution using this algorithm which can be computed recursively when using symbolic programming languages we are able to compute these integrals to high predetermined accuracies previous to this contribution the evaluation of these tail probabilities using the g n 1 transformation required symbolic computation of large determinants with the use of our algorithm the g n 1 transformation can be performed relatively easily to produce explicit approximations after a brief theoretical study a connection between the g n 1 transformation and rational and pade approximants is established

efficient evaluation of vector translation coefficients in multiparticle light scattering theories vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle scattering problems these theorems include a large number of vector addition coefficients there exist three basic types of analytical expressions for vector translation coefficients stein s quart appl math 19 15 1961 cruzan s quart appl math 20 33 1962 and xu s j comput phys 127 285 1996 stein s formulation relates vector translation coefficients with scalar translation coefficients cruzan s formulas use the wigner 3jm symbol xu s expressions are based on the gaunt coefficient since the scalar translation coefficient can also be expressed in terms of the gaunt coefficient the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the wigner 3jm symbol or the gaunt coefficient we present highly efficient recursive approaches to accurately evaluating wigner 3jm symbols and gaunt coefficients armed with these recursive approaches we discuss several schemes for the calculation of the vector translation coefficients using the three general types of formulation respectively our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of stein s type is less accurate in some particular cases these extensive test calculations also show that the scheme using the formulation based on the gaunt coefficient is the most efficient in practical computations

pathwise holder convergence of the implicit linear euler scheme for semi linear spdes with multiplicative noise in this article we prove pathwise holder convergence with optimal rates of the implicit euler scheme for the abstract stochastic cauchy problem begin aligned left begin aligned du t au t dt f t u t dt g t u t dw h t quad t in 0 t u 0 x 0 end aligned right end aligned here a is the generator of an analytic c 0 semigroup on a umd banach space x w h is a cylindrical brownian motion in a hilbert space h and the functions f 0 t times x rightarrow x theta f and g 0 t times x rightarrow fancyscript l h x theta g satisfy appropriate local lipschitz conditions the results are applied to a class of second order parabolic spdes driven by multiplicative space time white noise

symmetries travelling wave and self similar solutions of the burgers hierarchy we examine the general element of the burgers hierarchy ut x xu nu 0 n 0 1 2 for its lie point symmetries we use these symmetries to construct traveling wave and self similar solutions we observe that the general member of the hierarchy can be rendered as a linear 1 1 evolution equation by means of an elementary riccati transformation and examine this equation for its lie point symmetries with the use of these symmetries we can construct the traveling wave and self similar solutions in closed form

postprocessing for stochastic parabolic partial differential equations we investigate the strong approximation of stochastic parabolic partial differential equations with additive noise we introduce postprocessing in the context of a standard galerkin approximation although other spatial discretizations are possible in time we follow g j lord and j rougemont ima j numer anal 24 2004 pp 587 604 and use an exponential integrator we prove strong error estimates and discuss the best number of postprocessing terms to take numerically we evaluate the efficiency of the methods and observe rates of convergence some experiments with the implicit euler maruyama method are described

a new algorithm for accurate and fast numerical evaluation of hybrid and three centre two electron coulomb integrals over slater type functions recently we developed a new algorithm for a fast and accurate numerical evaluation of three centre nuclear attraction integrals over slater type functions the results obtained were very satisfactory now it is shown that this new algorithm can also be applied to hybrid and three centre two electron coulomb integrals over slater type functions these integrals which are numerous are very difficult to evaluate to a high accuracy because of the presence of spherical bessel functions and hypergeometric series in the integrands we have proved that the integrands that occur in the analytic expressions of the integrals under consideration satisfy all the conditions to apply the approach the hypergeometric functions which occur in the semi infinite integrals can be expressed as a finite expansion and the semi infinite integrals involving the spherical bessel functions can be transformed into semi infinite integrals involving the simple sine function the numerical results obtained with linear and non linear systems illustrate clearly a further improvement of accuracy and a substantial reduction in calculation times comparisons with existing codes stop developed by bouferguene et al 1996 int j quantum chem 57 801 and adggstngint developed by rico et al 1997 comp phys commun 105 216 are listed

addition theorems for b functions and other exponentially declining functions in this paper addition theorems are derived for a special class of exponentially declining functions the so called b functions bmn l r e filter and e o steinborn phys rev a 18 1 1978 although these b functions have a relatively complicated analytical structure they nevertheless have some mathematical properties that are particularly advantageous in connection with multicenter problems also all the commonly occurring exponentially declining functions like for instance bound state hydrogen eigenfunctions and slater type functions can be expressed by simple finite sums of b functions consequently addition theorems for these functions can also be written down immediately the various addition theorems for b functions are derived by applying suitable generating differential operators to the well known addition theorem of the special b function bm l l which is that solution of the modified helmholtz equation that is irregular at the origin and regular at infinity e j weniger and e o

a complexity analysis of the gauss bessel quadrature as applied to the evaluation of multi centre integrals over stfs in a previous work bouferguene 2005 j phys a math gen 38 3923 we have shown that in the framework of the gaussian integral transform multi centre integrals over slater type functions can be evaluated to an acceptable accuracy using a tailored gauss quadrature in which the weight function has the form w z z exp z z to be considered a solution worth implementing within a software for routine use in ab initio molecular simulations the method must also prove to be at least as efficient as those methods previously published in the literature two major results are provided in this paper firstly an improvement of the procedure used to generate the roots and weights of the gauss bessel quadrature is proposed secondly a computational cost analysis of the present method and the safouhi 2001 j phys a math gen 34 2801 based approach are compared hence proving the equivalence of the two from a complexity point of view

statistical analysis and simulation of random shocks in stochastic burgers equation we study the statistical properties of random shock waves in stochastic burgers equation subject to random space time perturbations and random initial conditions by using the response excitation probability density function pdf method and the mori zwanzig mz formulation of irreversible statistical mechanics we derive exact reduced order equations for the one point and two point pdfs of the solution field in particular we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive shock capturing discontinuous galerkin method in both physical and probability spaces we consider stochastic flows generated by high dimensional random initial conditions and random additive forcing terms yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state we also address the question of how random shock waves in space and time manifest themselves in probability space the proposed new mathematical framework can be applied to different conservation laws potentially leading to new insights into high dimensional stochastic dynamical systems and more efficient computational algorithms

convergence of the spectral galerkin method for the stochastic reaction diffusion advection equation abstract we study the convergence of the spectral galerkin method in solving the stochastic reaction diffusion advection equation under different lipschitz conditions of the reaction function f when f is globally locally lipschitz continuous we prove that the spectral galerkin approximation strongly weakly converges to the mild solution of the stochastic reaction diffusion advection equation and the rate of convergence in h r norm is 1 2 r for any r 0 1 2 r 1 2 1 2 d 1 2 the convergence analysis in the local lipschitz case is challenging especially in the presence of an advection term we propose a new approach based on the truncation techniques which can be easily applied to study other stochastic partial differential equations numerical simulations are also provided to study the convergence of galerkin approximations

an extremely efficient and rapid algorithm for numerical evaluation of three centre nuclear attraction integrals over slater type functions the present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of the three centre nuclear attraction integrals over slater type functions these integrals are numerous they occur in many millions of terms even for small molecules and they require rapid and accurate evaluation the new algorithm is based on nonlinear transformation methods on numerical quadrature and on properties of the sine and bessel functions the section with numerical results shows the high accuracy and the substantial gain in calculation time realized using the new algorithm the complete expressions of the three centre nuclear attraction integrals over b functions and over slater type functions are evaluated for different values of the quantum numbers to show the efficiency of the new approach numerical results obtained with linear and nonlinear systems and comparisons with numerical results from the literature are listed

markovianity and ergodicity for a surface growth pde the paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach we prove existence of a weak martingale solution satisfying energy inequalities and having the markov property furthermore under nondegeneracy conditions on the noise we establish that any such solution is strong feller and has a unique invariant measure

multicentre two electron coulomb and exchange integrals over slater functions evaluated using a generalized algorithm based on nonlinear transformations when calculating molecular electronic energies the contributions involving the coulomb operator for bielectronic terms are required rapidly and to high chemically significant accuracy the atomic orbital basis functions chosen in the present work are slater type functions stfs these functions can be expressed as finite linear combinations of b functions which are suitable to apply the fourier transform method the difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of two or three dimensional integral representations in this work we present a generalized algorithm for a precise and fast numerical evaluation of molecular integrals over stfs numerical results obtained with c2h2 c2h4 and ch4 molecules show the efficiency of the approach presented in this work comparisons with the existing codes are also listed

hybrid polynomial approximation to higher derivatives of rational curves abstract in this paper we extend the results published in jcam volume 214 pp 163 174 in 2008 based on the bound estimates of higher derivatives of both bernstein basis functions and rational bezier curves we prove that for any given rational bezier curve if the convergence condition of the corresponding hybrid polynomial approximation is satisfied then not only the l th l 1 2 3 derivatives of its hybrid polynomial approximation curve uniformly converge to the corresponding derivatives of the rational bezier curve but also this conclusion is tenable in the case of any order derivative this result can expand the area of applications of hybrid polynomial approximation to rational curves in geometric design and geometric computation

the properties of sine spherical bessel and reduced bessel functions for improving convergence of semi infinite very oscillatory integrals the evaluation of three centre nuclear attraction integrals over b functions this paper focuses on the use of useful properties of sine spherical bessel and reduced bessel functions to simplify the application of the nonlinear d and transformations for accelerating the convergence of semi infinite very oscillatory integrals and to reduce the calculation times keeping a high predetermined accuracy three centre nuclear attraction integrals which are one of the most difficult type involved in density functional theory methods when using a basis set of b functions are evaluated using the new approach the numerical results show the efficiency of the new method compared with other alternatives

semigroup splitting and cubature approximations for the stochastic navier stokes equations approximation of the marginal distribution of the solution of the stochastic navier stokes equations on the two dimensional torus by high order numerical methods is considered the corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on wiener space applied to a spectral galerkin discretization of degree n while the estimates exhibit a strong n dependence convergence is obtained for appropriately chosen time step sizes results of numerical simulations are provided and confirm the applicability of the methods

fast evaluation of the gaunt coefficients addition theorems for vector spherical harmonics require the determination of the gaunt coefficients that appear in a linearization expansion of the product of two associated legendre functions this paper presents an algorithm for the efficient calculation of these coefficients through solving the most appropriate lower triangular linear system and derives all relevant recurrence relations needed in the calculation this algorithm is also applicable to the calculation of the clebsch gordan coefficients that are closely related to the gaunt coefficients and are frequently encountered in the quantum theory of angular momentum the new method described in this paper reduces the computing time to 1 compared to the existing formulation that is widely used this new method can be applied to the calculation of both low and high degree coefficients while the existing formulation works well only for low degrees

lipschitzian semigroups and abstract functional differential equations abstract the paper is devoted to studying an abstract functional differential equation by a nonlinear semigroup approach we first prove in details the equivalence of the well posedness of an abstract functional differential equation and an associated abstract cauchy problem in the sense of strong solutions secondly a sufficient condition is derived for well posedness of the abstract functional differential equation thirdly we present principles of linearized stability for the abstract functional differential equation finally the results obtained are applied to a reaction diffusion equation with delays

numerical approximation of stochastic evolution equations convergence in scale of hilbert spaces the research of hakima bessaih was supported by the nsf united states grant dms 1418838 the research of erika hausenblas was supported by the austrian science fund fwf austria p25968 the research of tsiry randrianasolo was supported by the austrian science fund fwf austria p25968 and the german research council germany as part of the collaborative research center sfb 1283

the fourier transform method and the sd approach for the analytical and numerical treatment of multicenter overlap like quantum similarity integrals molecular overlap like quantum similarity measurements imply the evaluation of overlap integrals of two molecular electronic densities related by dirac delta function when the electronic densities are expanded over atomic orbitals using the usual lcao mo approach linear combination of atomic orbitals overlap like quantum similarity integrals could be expressed in terms of four center overlap integrals it is shown that by introducing the fourier transform of delta dirac function in the integrals and using the fourier transform approach combined with the so called b functions one can obtain analytic expressions of the integrals under consideration these analytic expressions involve highly oscillatory semi infinite spherical bessel functions which are the principal source of severe numerical and computational difficulties in this work we present a highly efficient algorithm for a fast and accurate numerical evaluation of these multicenter overlap like quantum similarity integrals over slater type functions this algorithm is based on the s d approach due to safouhi recurrence formulae are used for a better control of the degree of accuracy and for a better stability of the algorithm the numerical result section shows the efficiency of our algorithm compared with the alternatives using the one center two range expansion method which led to very complicated analytic expressions the epsilon algorithm and the nonlinear d transformation

new formulae for higher order derivatives and applications we present new formulae the slevinsky safouhi formulae i and ii for the analytical development of higher order derivatives these formulae which are analytic and exact represent the kth derivative as a discrete sum of only k 1 terms involved in the expression for the kth derivative are coefficients of the terms in the summation these coefficients can be computed recursively and they are not subject to any computational instability as examples of applications we develop higher order derivatives of legendre functions chebyshev polynomials of the first kind hermite functions and bessel functions we also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations we also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical bessel integral functions

numerical analysis of semilinear stochastic evolution equations in banach spaces the solution of stochastic evolution equations generally relies on numerical computation here usually the main idea is to discretize the spde spatially obtaining a system of sdes that can be solved by e g the euler scheme in this paper we investigate the discretization error of semilinear stochastic evolution equations in lp spaces resp banach spaces the space discretization may be done by galerkin approximation for the time discretization we consider the implicit euler the explicit euler scheme and the crank nicholson scheme in the last section we give some examples i e we consider an spdes driven by nuclear wiener noise approximated by wavelets and delay equation approximated by finite differences

an implicit interface boundary integral method for poisson s equation on arbitrary domains abstract we propose a simple formulation for constructing boundary integral methods to solve poisson s equation on domains with smooth boundaries defined through their signed distance function our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain where the integrations are carried out in the level set framework using an appropriate jacobian by the coarea formula the algorithm operates in the euclidean space and does not require any explicit parameterization of the boundaries we present numerical results in two and three dimensions

semigroups and linear partial differential equations with delay abstract we prove the equivalence of the well posedness of a partial differential equation with delay and an associated abstract cauchy problem this is used to derive sufficient conditions for well posedness exponential stability and norm continuity of the solutions applications to a reaction diffusion equation with delay are given

efficient and rapid numerical evaluation of the two electron four center coulomb integrals using nonlinear transformations and useful properties of sine and bessel functions two electron four center coulomb integrals are undoubtedly the most difficult type involved in ab initio and density functional theory molecular structure calculations millions of such integrals are required for molecules of interest therefore rapidity is the primordial criterion when the precision has been reached this work presents an extremely efficient approach for improving convergence of semi infinite very oscillatory integrals based on the nonlinear d transformation and some useful properties of spherical bessel reduced bessel and sine functions the new method is now shown to be applicable to evaluating the two electron four center coulomb integrals over b functions the section with numerical results illustrates the unprecedented efficiency of the new approach in evaluating the integrals of interest

field theoretic thermodynamic uncertainty relation general formulation exemplified with the kardar parisi zhang equation we propose a field theoretic thermodynamic uncertainty relation as an extension of the one derived so far for a markovian dynamics on a discrete set of states and for overdamped langevin equations we first formulate a framework which describes quantities like current entropy production and diffusivity in the case of a generic field theory we will then apply this general setting to the one dimensional kardar parisi zhang equation a paradigmatic example of a non linear field theoretic langevin equation in particular we will treat the dimensionless kardar parisi zhang equation with an effective coupling parameter measuring the strength of the non linearity it will be shown that a field theoretic thermodynamic uncertainty relation holds up to second order in a perturbation expansion with respect to a small effective coupling constant the calculations show that the field theoretic variant of the thermodynamic uncertainty relation is not saturated for the case of the kardar parisi zhang equation due to an excess term stemming from its non linearity

overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise we consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space time white noise we introduce a new numerical scheme for the time discretization of the finite dimensional galerkin stochastic differential equations which we call the exponential euler scheme and show that it converges in the strong sense faster than the classical numerical schemes such as the linear implicit euler scheme or the crank nicholson scheme for this equation with the general noise in particular we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1 3 which exceeds the barrier order 1 6 for numerical schemes using only basic increments of the noise process reported previously by contrast our scheme takes advantage of the smoothening effect of the laplace operator and of a linear functional of the noise and therefore overcomes this order barrier

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local lipschitz condition abstract in this paper we obtain a general statement concerning pathwise convergence of the full discretization of certain stochastic partial differential equations spdes with non globally lipschitz continuous drift coefficients we focus on non diagonal colored noise instead of the usual space time white noise by applying a spectral galerkin method for spatial discretization and a numerical scheme in time introduced by jentzen kloeden and winkel we obtain the rate of path wise convergence in the uniform topology the main assumptions are either uniform bounds on the spectral galerkin approximation or uniform bounds on the numerical data numerical examples illustrate the theoretically predicted convergence rate