mesoscopic simulation of ostwald ripening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening this phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange ising model this model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations the typical cluster size is observed to grow like t1 3 over range of times with faster growth at later times the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

theory of morphological evolution in ostwald ripening abstract the multiple expansion theory for ostwald ripening in a three dimensional system is presented analogously to the case in two dimensional coarsening t imaeda and k kawasaki physica a 164 1990 335 the migration and the shape deformation of droplets caused by the diffusional interaction are taken into account in our kinetic equations of motion the derived equations can be solved numerically within a fairly resonable computation time the role of the mullins sekerka instability in the coarsening kinetics is examined

the gelfand cetlin system and quantization of the complex flag manifolds the construction of angle action variables for collective completely integrable systems is described and the associated bohr sommerfeld sets are determined the quantization method of sniatycki applied to such systems gives formulas for multiplicities for the gelfand cetlin system on complex flag manifolds we show that these formulas give the correct answers for the multiplicities of the associated representations

spectral method study of domain coarsening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening the late stage of ostwald ripening is studied here through the use of a mesoscopic model in conjunction with a special configuration that allows for the direct measurement of system characteristics such as domain size the mesoscopic model is a stochastic partial integrodifferential equation and is studied through the use of recently developed and benchmarked spectral schemes the size of the growing region is not observed to increase as a power law in this model during the late stages of self organization in contrast to the power law growth observed in simulations of the earlier stages of self organization the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

comparative analysis of multiscale gaussian random field simulation algorithms abstract we analyze and compare the efficiency and accuracy of two simulation methods for homogeneous random fields with multiscale resolution we consider in particular the fourier wavelet method and three variants of the randomization method a without any stratified sampling of wavenumber space b with stratified sampling of wavenumbers with equal energy subdivision c with stratified sampling with a logarithmically uniform subdivision we focus primarily on fractal gaussian random fields with kolmogorov type spectra previous work has shown that variants a and b of the randomization method are only able to generate a self similar structure function over three to four decades with reasonable computational effort by contrast variant c along with the fourier wavelet method is able to reproduce accurate self similar scaling of the structure function over a number of decades increasing linearly with computational effort for our examples we will show that nine decades can be reproduced we provide some conceptual and numerical comparison of the various cost contributions to each random field simulation method we find that when evaluating ensemble averaged quantities like the correlation and structure functions as well as some multi point statistical characteristics the randomization method can provide good accuracy with less cost than the fourier wavelet method the cost of the randomization method relative to the fourier wavelet method however appears to increase with the complexity of the random field statistics which are to be calculated accurately moreover the fourier wavelet method has better ergodic properties and hence becomes more efficient for the computation of spatial rather than ensemble averages which may be important in simulating the solutions to partial differential equations with random field coefficients

numerical simulation of randomly forced turbulent flows several authors have proposed studying randomly forced turbulent flows e g e a novikov soviet physics jetp 20 5 1290 1965 more recently theoretical investigations e g renormalization group have focused on white noise forced navier stokes equations v yakhot and s a orszag j sci comput 1 1 3 1986 the present article aims to provide an appropriate numerical method for the simulation of randomly forced turbulent systems the spatial discretization is based on the classical fourier spectral method the time integration is performed by a second order runge kutta scheme the consistency of an extension of this scheme to treat additive noise stochastic differential equations is proved the random number generator is based on lagged fibonacci series results are presented for two randomly forced problems the burgers and the incompressible navier stokes equations with a white noise in time forcing term characterized by a power law correlation function in spectral space a variety of statistics are computed for both problems including the structure functions the third order structure functions are compared with their exact expressions in the inertial subrange the influence of the dissipation mechanism viscous or hyperviscous on the inertial subrange is discussed in particular probability density functions of velocity increments are computed for the navier stokes simulation finally for both burgers and navier stokes problems our results support the view that random sweeping is the dominant effect of the large scale motion on the small scales

finite element approximation of the cahn hilliard equation with concentration dependent mobility we consider the cahn hilliard equation with a logarithmic free energy and non degenerate concentration dependent mobility in particular we prove that there exists a unique solution for sufficiently smooth initial data further we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions finally some numerical experiments are presented

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

hierarchical monte carlo methods for fractal random fields two hierarchical monte carlo methods for the generation of self similar fractal random fields are compared and contrasted the first technique successive random addition sra is currently popular in the physics community despite the intuitive appeal of sra rigorous mathematical reasoning reveals that sra cannot be consistent with any stationary power law gaussian random field for any hurst exponent furthermore there is an inherent ratio of largest to smallest putative scaling constant necessarily exceeding a factor of 2 for a wide range of hurst exponentsh with 0 30 h 0 85 thus sra is inconsistent with a stationary power law fractal random field and would not be useful for problems that do not utilize additional spatial averaging of the velocity field the second hierarchical method for fractal random fields has recently been introduced by two of the authors and relies on a suitable explicit multiwavelet expansion mwe with high moment cancellation this method is described briefly including a demonstration that unlike sra mwe is consistent with a stationary power law random field over many decades of scaling and has low variance

the arnol d formula for algebraically completely integrable systems soit f v n r m une application algebrique reelle et d r m l ensemble de ses valeurs critiques on etudie la monodromie reelle des applications f definies sur une variete symplectique v 2m et dont les fibres generiques sont lagrangiennes pour la forme symplectique de tels applications sont des applications impulsion de systemes hamiltoniens integrables

coarsening kinetics in finite clusters we address the problem of diffusional interactions in a finite sized cluster of spherical particles for volume fractions v v in the range 0 0 01 we determined the quasistatic monopole diffusion solution for n particles distributed at random in a continuous matrix a global mass conservation condition is employed obviating the need for any external boundary condition the numerical results provide the instantaneous snapshot growth or shrinkage rate of each particle precluding the need for extensive time dependent computations the close connection between these snapshot results and the coarse grained kinetic constants are discussed a square root dependence of the deviations of the rate constants from their zero volume fraction value is found for the higher v v investigated the behavior is consistent with predictions from the diffusion debye h uckel screening theory by contrast a cube root dependence reported in earlier numerical studies is found for the lower v v investigated the roll over region of the volume fraction where the two asymptotics merge depends on the number of particles n alone a theoretical estimate for the roll over point predicts that the corresponding v v varies as n ensuremath 2 in good agreement with the numerical results

non self similar behavior in the lsw theory of ostwald ripening the classical lifshitz slyozov wagner theory of domain coarsening predicts asymptotically self similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field here we consider the long time behavior of measure valued solutions for systems in which particle size is uniformly bounded i e for initial measures of compact support we prove that the long time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support we give a necessary criterion for convergence to other self similar solutions and conditional stability theorems for some such solutions for a dense set of initial data convergence to any self similar solution is impossible

numerical approximation of the lifshitz slyozov wagner equation the lifshitz slyozov wagner theory of coarsening ostwald ripening describes the late stages of the growth by diffusional mass transfer of the grains of a new phase from a supersaturated solution it results in a nonlinear transport equation with a nonlocal nonlinearity for the volume distribution function of the grains a time explicit finite volume numerical scheme is proposed to solve this equation in self similar variables and is shown to converge under a cfl condition numerical simulations are also presented

numerical study of the cahn hilliard equation in one two and three dimensions abstract the cahn hilliard ch equation is related with a number of interesting physical phenomena like the spinodal decomposition phase separation and phase ordering dynamics on the other hand this equation is very stiff and the difficulty to solve it numerically increases with the dimensionality and therefore there are several published numerical studies in one dimension 1d dealing with different approaches and much fewer in two dimensions 2d in three dimensions 3d there are very few publications usually concentrate in some specific result without the details of the used numerical scheme we present here a stable and fast conservative finite difference scheme to solve the ch with two improvements a splitting potential into an implicit and explicit in time part and the use of free boundary conditions we show that gradient stability is achieved in one two and three dimensions with large time marching steps than normal methods

a particle model for spinodal decomposition we study a one dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a small drift toward regions of higher concentration we prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift there is a critical valuea c ofa fora a c the density values are unrestricted while fora a c they should all be to the right or to the left of a given interval a the diffusion coefficient of the limiting equation can be continued analytically to a and in the interior of a it has negative values which should correspond to particle aggregation phenomena we also show that the dynamics can be obtained as a limit of a kawasaki evolution associated to a kac potential the coefficienta plays the role of the inverse temperature the critical value ofa coincides with the critical inverse temperature in the van der waals limit and a with the spinodal region it is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic the system evolves according to an integrodifferential equation the instanton solutions of this equation as studied by dal passo and de mottoni are then related to the phase transition region in the thermodynamic phase diagram analogies with the cahn hilliard equations are also discussed

numerical approximation of some linear stochastic partial differential equations driven by special additive noises this paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises a special representation of the noise is considered and it is compared with general representations of noises in the infinite dimensional setting convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods the effects of the noises on the accuracy of the approximations are illustrated results of the numerical experiments are provided

on the splitting up method and stochastic partial differential equations we consider two stochastic partial differential equations du varepsilon t l ru varepsilon t f r t dv varepsilon t r m k u varepsilon t g k t circ dy t k qquad hspace 5pt varepsilon 0 1 driven by the same multidimensional martingale y yk and by different increasing processes vr0 vr1 r 1 2 d1 where lr and mk are second and first order partial differential operators and stands for the stratonovich differential we estimate the moments of the supremum in t of the sobolev norms of u1 t u0 t in terms of the supremum of the differences break vr0t vr1t hence we obtain moment estimates for the error of a multistage splitting up method for stochastic pdes in particular for the equation of the unnormalized conditional density in nonlinear filtering

a fourier wavelet monte carlo method for fractal random fields a new hierarchical method for the monte carlo simulation of random fields called the fourier wavelet method is developed and applied to isotropic gaussian random fields with power law spectral density functions this technique is based upon the orthogonal decomposition of the fourier stochastic integral representation of the field using wavelets the meyer wavelet is used here because its rapid decay properties allow for a very compact representation of the field the fourier wavelet method is shown to be straightforward to implement given the nature of the necessary precomputations and the run time calculations and yields comparable results with scaling behavior over as many decades as the physical space multiwavelet methods developed recently by two of the authors however the fourier wavelet method developed here is more flexible and in particular applies to anisotropic spectra generated through solutions of differential equations simulation results using this new technique and the well known nonhierarchical simulation technique the randomization method are given and compared for both a simple shear layer model problem as well as a two dimensional isotropic gaussian random field the fourier wavelet method results are more accurate for statistical quantities depending on moments higher than order 2 in addition to showing a quite smooth decay to zero on the scales smaller than the scaling regime when compared with the randomization method results the only situation in which the nonhierarchical randomization method is more computationally efficient occurs when no more than four decades of scaling behavior are needed and the statistical quantities of interest depend only on second moments

on first order corrections to the lsw theory ii finite systems we consider first order corrections to the classical theory by lifshitz slyozov and wagner lsw for systems with a finite number of particles numerical simulations in v e fradkov etal phys rev e 53 3925 3932 1996 show a cross over in the scaling of the correction term from 1 3 to 1 2 is the volume fraction of particles when the system size becomes larger than the screening length we rigorously derive this cross over for the rate of change of the energy starting from the monopole approximation the proof exploits the fact that the rate of change of energy has a variational characterization

the stackel systems and algebraic curves we show how the abel jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves we prove that derivative of the abel jacobi map is just the stackel matrix which determines n orthogonal curvilinear coordinate systems in a flat space the lax pairs r matrix algebras and explicit form of the flat coordinates are constructed an application of the weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space

semi discretization of stochastic partial differential equations on r 1 by a finite difference method the paper concerns finite difference scheme for the approximation of partial differential equations in r 1 with additional stochastic noise by replacing the space derivatives in the original stochastic partial differential equation spde for short with difference quotients we obtain a system of stochastic ordinary differential equations we study the difference between the solution of the original spde and the solution to the corresponding equation obtained by discretizing the space variable the need to approximate the solution in r 1 with functions of compact support requires us to introduce a scale of weighted sobolev spaces employing the weighted lp theory of spde a sup norm error estimate is derived and the rate of convergence is given