a cubic spline method for the solution of a linear fourth order two point boundary value problem abstract a cubic spline method is described for the numerical solution of a two point boundary value problem involving a fourth order linear differential equation this spline method is shown to be closely related to a known fourth order finite difference scheme

the not a knot piecewise interpolatory cubic polynomial in this paper the main idea of the not a knot cubic spline of de boor 8 will be extended to all the interior points knots of the spline interval to obtain a piecewise interpolatory cubic polynomial instead of the original and conventional not a knot cubic spline to construct this piecewise polynomial we do not need to solve a tridiagonal system of linear equations and this provides a shortcut which helps us save the time of doing more computations the order of convergence of this piecewise cubic polynomial is o h 4

end conditions for cubic spline interpolation derived from integration in the literature most of the end conditions for spline functions are obtained from conditions imposed on the spline function its finite differences or its derivatives near the two endpoints in this paper a class of end conditions is derived for cubic spline functions by use of integration these end conditions are given in terms of function values at the knots data and give rise to o h 4 spline approximation also it is shown that some of these end conditions have been obtained before from other constraints

end conditions for improved cubic spline derivative approximations abstract we consider the problem of deriving accurate end conditions for cubic spline interpolation at equally spaced knots in particular we derive a number of end conditions which lead to derivative approximations of high accuracy

diffusive stability of oscillations in reaction diffusion systems we study nonlinear stability of spatially homogeneous oscillations in reaction diffusion systems assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase we prove that spatially localized perturbations decay algebraically with the diffusive rate t n 2 in space dimension n we also compute the leading order term in the asymptotic expansion of the solution and show that it corresponds to a spatially localized modulation of the phase our approach is based on a normal form transformation in the kinetics ode which partially decouples the phase equation at the expense of making the whole system quasilinear stability is then obtained by a global fixed point argument in temporally weighted sobolev spaces corresponding author arnd scheel

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

pacemakers in large arrays of oscillators with nonlocal coupling we model pacemaker effects of an algebraically localized heterogeneity in a 1 dimensional array of oscillators with nonlocal coupling we assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation we concentrate on the case of heterogeneities with positive average and show that steady solutions to the nonlocal problem exist in particular we show that these heterogeneities act as a wave source this effect is not possible in 3 dimensional systems such as the complex ginzburg landau equation where the wavenumber of weak sources decays at infinity to obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation we then use fredholm properties of the laplace operator in kondratiev spaces to obtain solutions to the eikonal equation and by extension to the nonlocal problem

the nonlinear heat equation on w random graphs for systems of coupled differential equations on a sequence of w random graphs we derive the continuum limit in the form of an evolution integral equation we prove that solutions of the initial value problems ivps for the discrete model converge to the solution of the ivp for its continuum limit these results combined with the analysis of nonlocally coupled deterministic networks in medvedev the nonlinear heat equation on dense graphs and graph limits arxiv e prints 2013 justify the continuum thermodynamic limit for a large class of coupled dynamical systems on convergent families of graphs

post processing piecewise cubics for monotonicity monotone interpolation schemes of the fit and modify type are considered algorithms with optimal order error properties are developed numerical examples are given comparing these methods with others the theoretical underpinnings of the algorithms include sufficient conditions for the existence of monotone splines with specified error and smoothness properties and a result concerning approximate projections onto convex sets

numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials the homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones this is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown at the same time the prevailing numerical methods such as concurrent methods require extensive computational efforts a simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous ma terials at small strains the proposed numerical approach comprises three steps at the rst step a representative volume element rve for a given nonlinear heterogeneous material is de ned and a loading space consisting of all the boundary conditions to be imposed on the rve is discretized into a sufficiently large number of points called nodes at the second step the boundary condition corresponding each node is prescribed on the surface of the rve and the resulting nonlinear boundary value problem is solved by the nite element method fem so as to determine the effective response of the heterogeneous material to the loading as sociated to each node of the loading space at the third step the nodal effective responses are interpolated via appropriate interpolation functions so that the effective strain energy stress strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way this leads to a non concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials the rst version of the proposed approach uses mutidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank one tensors to interpolate effective nodal responses and avoid high rank data these two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power law model are involved the numerical results given by our approach are compared with available analytical estimates exact results and full fem or concurrent multilevel fem solutions

the use of spline on spline for the approximation of cauchy principal value integrals in this paper the spline on spline interpolatory polynomials are used to approximate cauchy principal value integrals and their derivatives this technique yields better results than using the derivatives of the traditional single spline functions which has been considered by orsi improved order of convergence for these approximations are discussed

cubic and quintic spline on spline interpolation this paper is concerned with the cubic spline on spline procedure considered by dolezal and tewarson 3 we show that when the knots are equally spaced with spacing h then the order of accuracy of the spline on spline approximations can be better by one power of h than that predicted by the results of 3 we also indicate how the cubic spline on spline process can be extended to quintic splines

on the convergence of some cubic spline interpolation schemes cubic spline interpolation schemes which require no derivative information at the end points are of great practical importance and have been included in several general purpose software libraries in this paper optimal order error estimates are developed for three popular schemes of this derivative free type the approximation of c 1 a b backslash c 2 a b functions by any such derivative free method that reproduces cubics necessarily displays some dependence on the local mesh ratio however for the spline interpolants studied here this dependence is restricted to the first and last subintervals

target patterns in two dimensional heterogeneous oscillatory reaction diffusion systems properties of target patterns created by pacemakers representing local regions with the modified oscillation frequency are studied for twodimensional oscillatory reaction diffusion systems described by the complex ginzburg landau equation an approximate analytical solution based on the phase dynamics approximation is constructed for a circular core and compared with numerical results for circular and square cores the dependence of the wavenumber and frequency of generated waves on the size and frequency shift of the pacemaker is discussed instabilities of target patterns involving repeated creations of ring shaped amplitude defects are further considered c 2006 elsevier b v all rights reserved

uniform convergence of optimal order quadrature rules for cauchy principal value integrals abstract for the numerical evaluation of cauchy principal value integrals of the form e 1 1 f e c s 1 1 we consider a quadrature method based on spline interpolation of odd degree 2 k 1 k n 0 we show that these rules converge uniformly for 1 1 in particular we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded s th derivative if s e s 2 k 1 2 k 2 finally we compare the rule to the well known quadrature rule of elliott and paget which only converges pointwise

finite size scaling study of two dimensional dilute potts models the author uses transfer matrices to calculate the free energy and various thermodynamic quantities of the two and three state potts models with random bond interactions at the critical temperature he verifies the harris criterion for the three state potts model and estimates the new random critical exponents for the ising case the results are consistent with randomness being irrelevant for the critical behaviour

two dimensional scaling in magnetic systems with non integer characteristic lengths we studied the ferromagnetic ising model on two dimensional finite systems with non integer characteristic lengths first we considered very long strips with a finite number i of complete rows and one partially filled row with probability x which is the two dimensional version of a layer by layer growth at fixed temperature when the characteristic length l i x increases the free energy per spin the specific heat and the magnetic susceptibility oscillate attaining relative extremes at integer l the oscillations in the free energy are interpreted as surface corrections related to the oscillations in the mean coordination number finite size scaling relations are not satisfied with continuous l but still hold for fixed x and variable i where the differences of mean widths are integers for fixed x the fits of the free energy give the conformal anomaly with very good accuracy we also studied strips with discretized gaussian distributions of widths with non integer means l and rms deviation in these structures the thermodynamic quantities vary monotonically with continuous l but some methods for calculating critical exponents do not work properly when generalized to continuous l we also obtain with good accuracy in these systems we discuss the possible implications of our results to real systems behaviour

effects of interface width scaling and spatial correlations on ising systems with rough boundaries we studied the ferromagnetic ising model on two dimensional systems with rough boundaries and several thickness distributions first we considered very long strips with discretized gaussian distributions of widths with mean 3 l 12 systems with fixed interface width w and with increasing roughness w l with 0 1 were analysed we also considered strips with columns heights correlations where the difference in the heights of neighbouring columns was 0 or 1 with 0 0 5 free energies per spin fl and initial susceptibilities l were calculated using transfer matrix techniques at the critical temperature of the two dimensional ising model the scaling of fl and of the finite size estimates of the ratio of exponents l was analysed in systems with small and slowly increasing w 0 2 correlated or not extrapolations of fl and l to l give accurate estimates of the critical parameters f and in systems with rapidly increasing roughness the deviations of those estimates from the exact values indicate the presence of large corrections in finite size scaling relations the comparison of systems with nearly the same shows that spatial correlations have no systematic effect on corrections to scaling it suggests the reliability of uncorrelated models or related approximate systems for the description of scaling properties of thin magnetic films and other low dimensional structures

approximation by integro cubic splines years ago after i presented a paper on spline functions at oxford university professor powell criticized us for using most of the time the function values rather than the integral values on constructing of the spline functions his comments and his request became the main motivation for this work in this paper we assume that on each subinterval of the spline interval a b the integral value of the function y y x is known by using these values rather than the function values at the knots we introduce a class of new types of interpolatory cubic splines to approximate the function y y x the selection of the end conditions for our integro cubic splines will be discussed the numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

on elliptic systems in rn on etudie des systemes elliptiques k k d operateurs differentiels dans r n qui peuvent s ecrire sous la forme a a q ou a est un systeme elliptique d operateurs a coefficients constants et q est une perturbation a coefficient variable avec certaines proprietes de decroissance a x

computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis this paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a non concurrent multiscale approach the geometry of the microstructure is random since the non concurrent multiscale approach is based on the use of a tensorial decomposition which is then submitted to the curse of dimensionality we perform an analysis with respect to the stochastic dimension the technique uses a database describing the strain energy density function potential in both the macroscopic cauchy green strain space and the geometrical random parameters domain each value of the potential is numerically computed by means of the finit element method on an elementary cell an interpolation scheme is finall introduced to obtain a continuous explicit form of the potential which by derivation allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations two numerical examples are presented

interpolation by integro quintic splines recently behforooz 1 has introduced a new approach to construct cubic splines by using the integral values rather than the usual function values at the knots also he has established different sets of end conditions for cubic and quintic splines by using the integral values see behforooz 2 4 in this paper we will use the same techniques of 1 to construct integro quintic splines although by using the integral values we expected to face a more complicated process for our construction it turned out that the matrix of the system of linear equations that produces the parameters became a diagonally dominant matrix and the process became very simple the selection of the required end conditions for our integro quintic splines will be discussed the numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation