a quintic b spline finite elements scheme for the kdvb equation abstract an algorithm for solving the korteweg de vries burgers kdvb equation based on the collocation method with quintic b spline finite elements is set up to simulate the solutions of the kdv burgers and the kdvb equations the migration of solitary waves and temporal evolution of a maxwellian initial pulse are studied for the kdv equation burgers equation is solved for different values of reynolds number the time evaluation of the solutions of the kdvb equation with different values for the diffusion and dispersion coefficients is studied invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm a linear stability analysis shows the scheme to be unconditionally stable

space time finite element methods for surface diffusion with applications to the theory of the stability of cylinders a family of space time finite element approximation schemes is presented for the nonlinear partial differential equations governing diffusion in the surface of a body of revolution the schemes share with the partial differential equations properties of conservation of volume and decrease of area numerical experiments are described showing that the result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory examples are given of cases in which longitudinal perturbations with high wave number spectra grow in amplitude after an initial rapid decay followed by a long incubation period and result in break up of the body into a necklace of beads the results of finite element calculations are compared with the predictions of a perturbation analysis

2d parallel and stable group explicit finite difference method for solution of diffusion equation abstract recently various versions of alternating group explicit or alternating group explicit implicit methods were proposed for solution of diffusion equation the main benefits of these methods are good stability accuracy and parallelizing but these methods were developed for 1d case and stability and accuracy were investigated for 1d case too in the present study we extend the new group explicit method r tavakoli p davami new stable group explicit finite difference method for solution of diffusion equation appl math comput 181 2006 1379 1386 to 2d with operator splitting method the implementation of method is discussed in details our numerical experiment shows that such 2d extension is unconditionally stable and it is more accurate that traditional unconditional stable explicit method

computational fluid flow solution over endothelial cells inside the capillary vascular system computational blood flow analysis through glycocalyx located over the endothelial cells in human capillary vascular system is performed stable numerical schemes are developed for obtaining approximate solutions to the mixed problem to the partial differential equation with variable space operator to model the blood flow numerical techniques are developed by applying a procedure of the solution to the linear difference equation with matrix coefficients the flow equations inside the core flow region and porous region are established discretization is done and the flow velocities in both regions are calculated the effect of the flow over the glycocalyx is investigated the wall shear stresses and the drag force over the glycocalyx are calculated copyright 2007 john wiley sons ltd

mean square stability of stochastic volterra integro differential equations abstract the mean square stability of a non linear stochastic volterra integro differential equation is studied non convolution volterra terms arise in both the drift and the dispersion term moreover for the convolution case we determine the rate of convergence in terms of an integrability condition on the volterra kernels

coupled surface and grain boundary motion a travelling wave solution abstract existence and uniqueness are proven for a travelling wave solution for a problem in which motion by mean curvature is coupled with surface diffusion this problem pertains to a bicrystal in a quarter loop geometry in which one grain grows at the expense of the other and the internal grain boundary between the two crystals contacts the exterior surface at a groove root or tri junction where various balance laws hold far in front and behind the groove root the overall height of the bicrystal is assumed to be unperturbed whereas in a previous paper acta mater 51 2003 1981 a partially linearized formulation was considered for which explicit solutions could be found here we treat the fully nonlinear problem employing an angle formulation and a scaled arc length parameterization we reduce the problem to the solution of a third order ode with a jump condition at the origin existence is proven if m the ratio of the exterior surface energy to the surface energy of the grain boundary is less than about 0 92 uniqueness of these solutions is demonstrated within the class of single valued solutions a numerical comparison is made with the solution of the partially linearized formulation found earlier for the sake of illustration

on exact rates of decay of solutions of linear systems of volterra equations with delay this paper considers the resolvent of a finite dimensional linear convolution volterra integral equation the main results give conditions which ensure that the exact rate of decay of the resolvent can be determined using a positive weight function related to the kernel the decay rates can be exponential or subexponential many other related results on exact rates of exponential and subexponential decay of solutions of volterra integro differential equations are given we also present an application to a linear compartmental system with discrete and continuous lags

long run behaviour of the autocovariance function of arch models abstract the asymptotic properties of the memory structure of arch equations are investigated this asymptotic analysis is achieved by expressing the autocovariance function of arch equations as the solution of a linear volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear volterra operators it is shown that the autocovariance function decays subexponentially or geometrically if and only if the kernel of the resolvent equation has the same decay property it is also shown that upper subexponential bounds on the autocovariance function result if and only if similar bounds apply to the kernel

variational discretization of axisymmetric curvature flows we present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail although numerical methods for geometric flows have been used frequently in axisymmetric settings numerical analysis results so far are rare in this paper we present stability equidistribution existence and uniqueness results for the introduced approximations numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one dimensional problem has to be solved the good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries

finite difference methods for solution of nonlocal boundary value problems the article is devoted to the construction and investigation of various types of difference schemes relevant to nonlocal boundary value problems stated for multidimensional elliptic equations suitable tools for performing the investigation are developed including difference analogue of green s formula and inequalities corresponding to nonlocal boundary conditions

unconditional stability of parallel alternating difference schemes for semilinear parabolic systems the general alternating schemes with intrinsic parallelism for semilinear parabolic systems are studied first we prove the a priori estimates in the discrete h 1 space of the difference solution for these schemes then the existence of the difference solution for these schemes follows from the fixed point principle finally the unconditional stability of the general alternating schemes is proved the alternating group explicit scheme the alternating segment explicit implicit scheme and the alternating segment crank nicolson scheme are the special cases of the general alternating schemes

a finite element method for surface diffusion the parametric case surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for parametric surfaces with or without boundaries the method is semi implicit requires no explicit parametrization and yields a linear system of elliptic pde to solve at each time step we next develop a finite element method propose a schur complement approach to solve the resulting linear systems and show several significant simulations some with pinch off in finite time we introduce a mesh regularization algorithm which helps prevent mesh distortion and discuss the use of time and space adaptivity to increase accuracy while reducing complexity

axisymmetric surface diffusion dynamics and stability of self similar pinchoff the dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface laplacian of its mean curvature this flow conserves the volume enclosed inside the surface while minimizing its surface area we review the axisymmetric equilibria the cylinder sphere and the delaunay unduloid the sphere is stable while the cylinder is long wave unstable a subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions we present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite time pinchoff we examine the structure of the pinchoff showing it has self similar structure using asymptotic numerical and analytical methods in addition to a previously known similarity solution we find a countable set of similarity solutions each with a different asymptotic cone angle we develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution we also consider similarity solutions describing the dynamics after the topological transition

the d yakonov fully explicit variant of the iterative decomposition method abstract in this paper a new iterative alternating decomposition iade scheme of 4 2 order of accuracy is developed to solve the one dimensional parabolic problem it is based on the two stage fractional splitting strategy suggested by d yakonov and found to be generally more accurate than the recently developed 2 2 accurate alternating group explicit age method of peaceman rachford variant as the method is fully explicit its feature can be fully utilized for parallelization by means of a domain decomposition strategy

a non separable solution of the diffusion equation based on the galerkin s method using cubic splines abstract the two dimensional diffusion equation of the form 2 u x 2 2 u y 2 1 d u t is considered in this paper we try a bi cubic spline function of the form i j 0 n n c i j t b i x b j y as its solution the initial coefficients c i j 0 are computed simply by applying a collocation method c i j f x i y j where f x y u x y 0 is the given initial condition then the coefficients c i j t are computed by x t e tq x 0 where x t c 0 1 c 0 1 c 0 2 c 0 n c 1 0 c n n is a one dimensional array and the square matrix q is derived from applying the galerkin s method to the diffusion equation note that this expression provides a solution that is not necessarily separable in space coordinates x y the results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included

new stable group explicit finite difference method for solution of diffusion equation a new group explicit method for solution of diffusion equation is presented this method is based on domain decomposition concept and using asymmetric saul yev schemes for internal nodes of each sub domain and alternating group explicit method for sub domain s interfacial nodes this new method has several advantages such as good parallelism unconditional stability fully explicit nature and better accuracy than original saul yev schemes the details of implementation and proving stability are briefly discussed numerical experiments on stability and accuracy are also presented

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

an unconditionally stable parallel difference scheme for parabolic equations an unconditionally stable difference scheme with intrinsic parallelism is constructed in this paper taking the parabolic equation u t u x x as an example the main ideas are described as follows we decompose the domain w into some overlapping subdomains take values of the last time layer as values of the time layer on inner boundary points of subdomains solve it with the fully implicit scheme inside each subdomain and then take correspondent values of its neighbor subdomains as their values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points its unconditional stability and convergence are proved though its truncation error is o 1 we will still prove that the estimate order is improved to o t h

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

an accurate numerical solver on second order pdes with variable coefficients in three dimensions in the present paper a numerical technique is developed for the approximate solution of second order partial differential equations pdes with variable coefficients in three dimensions with the temporary introduction of two unknown auxiliary functions of the coordinate system the initial equation is separated into three parts that are reduced to ordinary differential equations one for each dimension that are discretized with a finite difference scheme the use of suitable manipulations and the elimination of the unknown auxiliary functions gives finally a linear system of algebraic equations where the matrix of the coefficients of the unknowns is diagonally dominant a prerequisite for the rapid convergence of the iterative procedure the efficiency and accuracy of the proposed numerical scheme is validated by its application to two test problems of fluid mechanics which have exact solutions the numerical results based on the present technique are more accurate than those obtained by either the standard relaxation treatment with central differences or the adi method when the contribution of the first derivative terms in the initial equation is dominant in all cases the comparison of the numerical results with those of the analytical solution demonstrates the reliability of the presented numerical code

stability of cylindrical bodies in the theory of surface diffusion abstract perturbation arguments and finite element calculations are employed to study the nonlinear partial differential equations governing morphological changes induced by curvature driven diffusion of mass in the surface of an axially symmetric body isotropy of surface properties is assumed second and higher order perturbation analyses indicate that the familiar result of the linear theory of small amplitude longitudinal perturbations of a cylinder to the effect that a long cylinder is stable against all perturbations with spatial fourier spectra containing only wavelengths less than the circumference of the cylinder does not hold in the full nonlinear theory the perturbation analyses yield criteria for determining when longitudinal perturbations with high wave number spectra grow in amplitude after an initial decay followed by an incubation time and result in break up of the body into a necklace of beads the principal conclusions of the formal perturbation analyses are found to be in good accord with numerical solutions obtained by finite element methods

heatlet approach to diffusion equation on unbounded domains we develop heatlets the fundamental solutions of heat equation using wavelets for numerically solving inhomogeneous and homogeneous initial value problems of diffusion equation on unbounded domains unlike finite difference and finite element methods diffusion into an infinite medium is satisfied analytically avoiding the need for artificial boundary conditions on a finite computational domain the approach is applied to a number of examples and the numerical results confirm the theoretical findings

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

implicit explicit parallel asynchronous solver of parabolic pdes this paper presents a method for the solution of parabolic pdes on parallel computers which is a combination of implicit and explicit finite difference schemes based on a domain decomposition dd strategy moreover this method is asynchronous i e no explicit synchronization is required among processors we determine the values at subdomains boundaries by our new high order asynchronous explicit schemes then any known high order implicit finite difference scheme can be applied within each subdomain we present a technique for derivation of appropriate asynchronous explicit schemes based on green s functions synchronous versions of these schemes are obtained as special cases the applicability of this method is also demonstrated for a family of nonlinear problems our new explicit schemes are of high order and yet stable for a large time step as established in our analysis of their numerical properties moreover these schemes provide attractive properties for parallel implementation being asynchronous they allow local time stepping thus eliminating the need for a global synchronized time step moreover our asynchronous computation is time stabilizing in the sense that the calculation implicitly prevents a growing time gap between neighboring subdomains the locality property due to the exponential decay of green s functions implies that communication is needed only between neighboring processors hence this method which is designed to minimize the overhead associated with the synchronization of the multiple processors is specifically suitable for parallel computers having a high synchronization cost or highly varying load even in cases in which some processors have persistent speed differences furthermore the implementation of different resolution in each subdomain e g irregular or unstructured grid makes it valuable as an adaptive algorithm the above schemes were implemented and tested on the shared memory multi user cray j90 and sequent balance machines these implementations prove high accuracy and high degree of parallelism this work is complementary to our previous work on asynchronous schemes comput math appl 24 1992 pp 33 53 appl numer math 12 1993 pp 27 45 numer algorithms 6 1994 pp 275 296 numer algorithms 12 1996 pp 159 192

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

on the numerical solution of the diffusion equation with variable space operator in present paper numerical schemes are developed for obtaining approximate solutions to the mixed problem for one dimensional diffusion equation with variable space operator the method of lines semidiscretization approach is used to transform the model partial differential equation into a system of first order linear ordinary differential equations the partial derivative with respect to the space variable is approximated by the first and second order finite difference approximation for the solution of the resulting system of first order ordinary differential equations we apply the first and second order of accuracy difference schemes stability estimates for the solution of these difference schemes are established numerical techniques are developed by applying a procedure of the solution of first order linear difference equation with matrix coefficients the algorithms are tested on a model problem in biofluid mechanics two regions are considered close to the endothelial cells ec which can be modeled taking the mixed problem for one dimensional diffusion equation with variable space operator

a tangent plane marker particle method for the computation of three dimensional solid surfaces evolving by surface diffusion on a substrate we introduce a marker particle method for the computation of three dimensional solid surface morphologies evolving by surface diffusion the method does not use gridding of surfaces or numerical differentiation and applies to surfaces with finite slopes and overhangs we demonstrate the method by computing the evolution of perturbed cylindrical wires on a substrate we show that computed growth rates at early times agree with those predicted by the linear stability analysis furthermore when the marker particles are redistributed periodically to maintain even spacing the method can follow breakup of the wire

a galerkin procedure for the diffusion equation subject to the specification of mass continuous and discrete galerkin procedures are derived and analyzed for the numerical solution of the diffusion equation u 1 u xx 0 x 1 0 t leqq t u x 0 f x u x 1 t g t and the specification of the mass int 0 b u x t dx m t 0 b 1 some numerical experiments are also presented

stability of difference schemes for hyperbolic parabolic equations the stable difference schemes approximately solving the nonlocal boundary value problem for hyperbolic parabolic equationd 2u t dt 2 au t f t 0 t 1 u 1 au m bu l fdu t dt au t g t 1 t 0 a b 1 0 m l 1in a hilbert space h with self adjoint positive definite operator a are presented the stability estimates for the solutions of the difference schemes of the mixed type boundary value problems for hyperbolic parabolic equations are obtained the theoretical statements for the solution of these difference schemes for hyperbolic parabolic equation are supported by the results of numerical experiments

long memory in a linear stochastic volterra differential equation in this paper we consider a linear stochastic volterra equation which has a stationary solution we show that when the kernel of the fundamental solution is regularly varying at infinity with a log convex tail integral then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined moreover it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential under certain conditions upon the kernel even arbitrarily slow decay rates of the autocovariance function can be achieved analogous results are obtained for the corresponding discrete equation

on wavelet fundamental solutions to the heat equation heatlets abstract we present an application of wavelet theory in partial differential equations we study the wavelet fundamental solutions to the heat equation the heat evolution of an initial wavelet state is called a heatlet like wavelets for the l2 space heatlets are atomic heat evolutions in the sense that any general heat evolution can be assembled from a heatlet according to some simple rules we study the basic properties and algorithms of heatlets and related functions

error analysis of a semidiscrete numerical scheme for diffusion in axially symmetric surfaces we analyze a semidiscrete numerical scheme for approximating the evolution of axially symmetric surfaces by surface diffusion the fourth order equation is split into two coupled second order problems which are approximated by linear finite elements we prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis

local analyticity in weighted spaces and applications to stability problems for volterra equations we study the qualitative properties of the solutions of linear convolution equations such as x x t f x a x f and x t f we are especially concerned with finding conditions which ensure that these equations have resolvents which belong to or are determined up to a term belonging to certain weighted l1 spaces our results are obtained as consequences of more general banach algebra results on functions that are locally analytic with respect to the elements of a weighted l1 space in particular we derive a proposition of wiener levy type for weighted l1 spaces which underlies all subsequent results our method applies equally well to equations more general than those mentioned above we unify and sharpen the results of several recent papers on the asymptotic behavior of volterra convolution equations of the types mentioned above and indicate how many of them can be extended to the fredholm case in addition we give necessary and sufficient conditions on the perturbation term f for the existence of bounded or integrable solutions x in some critical cases when the corresponding limit equations have nontrivial solutions

efficient numerical schemes for gradient vector flow since its publication more than 10 years ago the gradient vector flow gvf technique has been used and adapted to various models and problems its effectiveness has greatly contributed to its popularity the main drawback of gvf and its generalisation however is their expensive computation load and its consequence on the capture range in this work we propose and compare different efficient numerical schemes to solve the gvf and its generalisations

b spline galerkin methods for numerical solutions of the burgers equation the first order splitting method is applied to the burgers equation and two numerical algorithms are set up for this time splitted burgers equation using both quadratic and cubic b spline galerkin finite element techniques the unknowns in the equation is resolved with the b splines in space and standart crank nicolson differencing schemes is used to advance solution in time results of three numerical test problems are used to compare each schemes with available some results

fully discrete finite element approximation for anisotropic surface diffusion of graphs we analyze a fully discrete numerical scheme for approximating the evolution of graphs for surfaces evolving by anisotropic surface diffusion the scheme is based on the idea of second order operator splitting for the nonlinear geometric fourth order equation this yields two coupled spatially second order problems which are approximated by linear finite elements the time discretization is semi implicit we prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis and illustrate surface diffusion

unconditional stability of alternating difference schemes with intrinsic parallelism for two dimensional fourth order diffusion equation in this paper the parallel difference schemes for parabolic equation are studied the general alternating difference schemes with intrinsic parallelism for two dimensional fourth order diffusion equation are presented and proved to be unconditionally stable by the energy method two kinds of alternating band explicit abe1 abe2 schemes alternating band explicit implicit abe i scheme and alternating band crank nicolson abc n scheme are the special cases of the general schemes constructed here numerical experiments are performed to examine the accuracy and unconditional stability