on the solution of strong nonlinear oscillators by applying a rational elliptic balance method a rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using jacobi elliptic functions to illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators we first investigate the exact solution of the non homogenous undamped duffing equation then we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators at the end of the paper we compare the numerical integration values of the angular frequencies with approximate solution results based on the proposed rational elliptic balance method

adaptive dimensional reduction numerical solution of monotone quasilinear boundary value problems a feedback extension procedure is developed for the numerical solution of a class of nonlinear boundary value problems associated with antiplane shear hencky s theory of plasticity or maxwell s equations for the magnetic field potential in magnetic shielding a posteriors error estimators are constructed and analyzed under additional smoothness assumptions on the exact and dimensionally reduced solutions the feedback method is proved adaptive i e optimal with respect to convergence rate this extends previous results in s jensen and i babuska siam j numer anal 25 1988 pp 644 669 using dimensional reduction in energy asymptotic format some computational results are given

nonlinear oscillator with discontinuity by generalized harmonic balance method a generalized harmonic balance method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn x this method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form this approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation we find that this method works very well for the whole range of amplitude of oscillation in the case of the antisymmetric piecewise constant force oscillator and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed for the second order approximation we have shown that the relative error in the analytical approximate frequency is 0 24 we also compared the fourier series expansions of the analytical approximate solution and the exact one comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

finite element approximation of a nonlinear heat conduction problem in anisotropic media abstract this paper is a survey of results which we have obtained in solving a stationary nonlinear heat conduction problem by the finite element method in particular we present uniqueness theorems for the classical and weak solutions a comparison theorem existence theorems for the weak and finite element solutions approximation of a curved boundary and numerical integration a discrete maximum principle convergence without any regularity assumptions a priori error estimates nonlinear boundary conditions and numerical algorithms

on an external finite element method for a second order eigenvalue problem on a concave 2d domain with dirichlet boundary conditions abstract this paper deals with a finite element method for a second order elliptic eigenvalue problem on a concave two dimensional domain with dirichlet boundary conditions in the method a variational crime is committed in that the approximate eigenvalue problem is formulated on a domain h which is not included in the original domain hence the finite element approximation space v h used is not a subspace of the sobolev space entering the continuous variational problem we obtain an optimal error estimate from above for the approximate eigenvalues under weaker conditions for the data than in other papers moreover a basic result for the error of the elliptic projection is proved in a different manner

analytical approximations of the period of a generalized nonlinear van der pol oscillator in this paper analytical approximations for the period of a generalized nonlinear van der pol equation will be obtained by using various asymptotic methods

convex hull of n planar brownian motions exact results and an application to ecology we compute exactly the mean perimeter and area of the convex hull of n independent planar brownian paths each of duration t both for open and closed paths we show that the mean perimeter alpha n sqrt t and the mean area beta n t for all t the prefactors alpha n and beta n computed exactly for all n increase very slowly logarithmically with increasing n this slow growth is a consequence of extreme value statistics and has interesting implications in an ecological context in estimating the home range of a herd of animals with a population size n

spherical interface dynamos mathematical theory finite element approximation and application stellar magnetic activities such as the 11 year sunspot cycle are the manifestation of magnetohydrodynamic dynamo processes taking place in the deep interiors of stars this paper is concerned with the mathematical theory and finite element approximation of mean field spherical dynamos and their astrophysical application we first investigate the existence uniqueness and stability of the dynamo system governed by a set of nonlinear pdes with discontinuous physical coefficients in spherical geometry and characterize the system by a saddle point type variational form then we propose a fully discrete finite element approximation to the dynamo system and study its convergence and stability for the astrophysical application we perform some fully three dimensional numerical simulations of a solar interface dynamo using the proposed algorithm which successfully generates the equatorially propagating dynamo wave with a period of about 11 years similar to that of the sun

analysis of a fully discrete finite element method for a nonlinear magnetic field problem in this paper nonlinear stationary magnetic field problems and their approximation by finite elements are studied as a new aspect estimates for the error caused by the approximation of the material function are given a technique for the interpolation of a pointwise given material function that preserves monotonicity is described and error estimates are presented

a two level stochastic collocation method for semilinear elliptic equations with random coefficients in this work we propose a novel two level discretization for solving semilinear elliptic equations with random coefficients motivated by the two grid method for deterministic partial differential equations pdes introduced by xu 1994 our two level stochastic collocation method utilizes a two grid finite element discretization in the physical space and a two level collocation method in the random domain in particular we solve semilinear equations on a coarse mesh t h with a low level stochastic collocation corresponding to the polynomial space p p and solve linearized equations on a fine mesh t h using high level stochastic collocation corresponding to the polynomial space p p we prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with t h and p p the two level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients numerical experiments are provided to verify the theoretical results

finite element analysis of varitional crimes for a quasilinear elliptic problem in 3d we examine a finite element approximation of a quasilinear boundary value elliptic problem in a three dimensional bounded convex domain with a smooth boundary the domain is approximated by a polyhedron and a numerical integration is taken into account we apply linear tetrahedral finite elements and prove the convergence of approximate solutions on polyhedral domains in the w 1 2 norm to the true solution without any additional regularity assumptions

the maximum angle condition in the finite element method for monotone problems with applications in magnetostatics the finite element method for an elliptic equation with discontinuous coefficients obtained for the magnetic potential from maxwell s equations is analyzed in the union of closed domains the boundaries of which form a system of three circles with the same centre as the middle domain is very narrow the triangulations obeying the maximum angle condition are considered in the case of piecewise linear trial functions the maximum rate of convergence o h in the norm of the space h 1 omega h is proved under the following conditions 1 the exact solution u in h 1 omega is piecewise of class h 2 2 the family of subtriangulations cal t h rm a of the narrow subdomain omega rm a satisfies the maximum angle condition expressed by relation 38 the paper extends the results of 24

finite element solution of nonlinear elliptic problems the study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed dirichlet neumann boundary conditions is presented in the discretization variational crimes are commited approximation of the given domain by a polygonal one numerical integration with the assumption that the corresponding operator is strongly monotone and lipschitz continuous and that the exact solutionu h1 the convergence of the method is proved under the additional assumptionu h2 the rate of convergenceo h is derived without the use of green s theorem

on the coupled bem and fem for a nonlinear exterior dirichlet problem in r2 in this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior dirichlet problem in the plane specifically the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region and the laplace equation in the corresponding unbounded exterior region in addition to appropriate boundary and transmission conditions the main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation we provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence uniqueness and approximation results are established then we consider the case where the corresponding operator is strongly monotone and lipschitz continuous and derive asymptotic error estimates for a boundary finite element solution we prove the unique solvability of the discrete operator equations and based on a strang type abstract error estimate we show the strong convergence of the approximated solutions moreover under additional regularity assumptions on the solution of the continous operator equation the asymptotic rate of convergenceo h is obtained

a priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems the effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied under similar assumption on the quadrature formula as for linear problems optimal error estimates in the l 2 and the h 1 norms are proved the numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh the analysis is valid for both simplicial and rectangular finite elements of arbitrary order numerical experiments corroborate the theoretical convergence rates

extremal properties of random systems we find that the probability distribution for the largest intervals p l exhibits universal properties for different systems including random walk and random cutting models in particular p l has an infinite set of singularities at l 1 k with k 2 3 which become weaker and weaker as k ensuremath rightarrow ensuremath infty additionally p l has an essential singularity at l 0 these properties are also found in many dimensional situations

from random walks to spin glasses abstract the talk was a short review on systems which exhibit non self averaging effects sums of random variables when the distribution has a long tail mean field spin glasses random map models and returns of a random walk to the origin non self averaging effects are identical in the case of sums of random variables and in the spin glass problem as predicted by the replica approach also we will see that for the random map models or for the problem of the returns of a random walk to the origin the non self averaging effects coincide with the results of the replica approach when the number n of replica n 1 2 or n 1

analysis of a fem bem coupling method for transonic flow computations a sensitive issue in numerical calculations for exterior flow problems e g around airfoils is the treatment of the far field boundary conditions on a computational domain which is bounded in this paper we investigate this problem for two dimensional transonic potential flows with subsonic far field flow around airfoil profiles we take the artificial far field boundary in the subsonic flow region in the far field we approximate the subsonic potential flow by the prandtl glauert linearization the latter leads via the green representation theorem to a boundary integral equation on the far field boundary this defines a nonlocal boundary condition for the interior ring domain our approach leads naturally to a coupled finite element boundary element method for numerical calculations it is compared with local boundary conditions the error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists

laplace eigenvalues on regular polygons a series in 1 n abstract for regular polygons p n inscribed in a circle the eigenvalues of the laplacian converge as n to the known eigenvalues on a circle we compute the leading terms of n in a series in powers of 1 n by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon the o 1 n 2 term comes from hadamard s formula and reflects the change in area this term disappears if we transcribe the polygon scaling it to have the same area as the circle

joint distributions of partial and global maxima of a brownian bridge we analyze the joint distributions and temporal correlations between the partial maximum m and the global maximum m achieved by a brownian bridge on the subinterval and on the entire interval respectively we determine three probability distribution functions the joint distribution of both maxima the distribution p m of the partial maximum and the distribution of the gap between the maxima we present exact results for the moments of these distributions and quantify the temporal correlations between m and m by calculating the pearson correlation coefficient

first order decoupled finite element method of the three dimensional primitive equations of the ocean this paper is concerned with a first order fully discrete decoupled method for solving the three dimensional 3d primitive equations of the ocean with the dirichlet boundary conditions on the side where a decoupled semi implicit scheme is used for the time discretization and the p 1 p 1 p 1 p 1 p 1 finite element for velocity pressure and density is used for the spatial discretization of these equations the h 1 l 2 h 1 optimal error estimates for the numerical solution u h n p h n theta h n and the l 2 optimal error estimate for u n h theta h n are established under the restriction of 0 h le beta 1 and 0 tau le beta 2 for some positive constants beta 1 and beta 2 moreover numerical investigations are provided to show that the first order decoupled method is of almost unconditional convergence with accuracy mathcal o h tau in the h 1 norm and mathcal o h 2 tau in the l 2 norm for solving the 3d primitive equations of the ocean numerical results are gi

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

on the finite element approximation of a cascade flow problem the finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented the nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements we deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains including the effect of numerical integration

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed

streamline diffusion finite element method for quasilinear elliptic problems we extend the analysis of the streamline diffusion finite element method to quasilinear elliptic problems of second order an existence theorem and error estimates are given in the case of branches of nonsingular solutions following a recent abstract approach in 12 13 26

compactness method in the finite element theory of nonlinear elliptic problems finite element approximation of a nonlinear elliptic pseudomonotone second order boundary value problem in a bounded nonpolygonal domain with mixed dirichlet neumann boundary conditions is studied in the discretization we approximate the domain by a polygonal one use linear conforming triangular elements and evaluate integrals by numerical quadratures we prove the solvability of the discrete problem and on the basis of compactness properties of the corresponding operator which is not monotone in general we prove the convergence of approximate solutions to an exact weak solutionu h1 no additional assumption on the regularity of the exact solution is needed

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

finite element variational crimes in parabolic elliptic problems the study of a completely discretized variational problem corresponding to a nonlinear second order parabolic elliptic initial boundary value problem with an initial value only inl 2 is presented the discretization in time is done by the euler backward method the discretization in space by the finite element method with linear functions on triangular elements the changes in domain are taken into account the convergence and the unconditional stability of the method is proved under the assumption that the boundary is piecewise of classc 3 and the forma v w has a potential and is strongly monotone and lipschitz continuous as a by product the existence and uniqueness theorem is obtained