a meshless method for one dimensional stefan problems the paper presents a new meshless numerical technique for solving one dimensional problems with moving boundaries including the stefan problems the technique presented is based on the use of the delta shaped functions and the method of approximate fundamental solutions mafs firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries the numerical examples are presented and the results are compared with analytical solutions the comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness

asymptotic upper curvature bounds in coarse geometry we define a notion of an asymptotic upper curvature bound for gromov hyperbolic metric spaces that is invariant under rough isometries and examine the basic properties of this concept

irreducibility criterion for germs of analytic functions of two complex variables let s x y be the germ of an analytic function of two complex variables near the origin to decide whether f is irreducible or not we can apply a succession of blowing ups to desingularize f and see if the proper transform off ever bifurcates and then use the fact that f is irreducible iff such a bifurcation never takes place alternatively we can use newton s theorem also called puiseux s theorem to completely factor s into linear factors in y by allowing fractional power series in x and then collate these factors into conjugacy classes and now use the fact that fis irreducible iff there is only one conjugacy class in a recent conversation t c kuo of the university of sydney in australia asked me the following question

asymptotic numerical and approximate techniques for a free boundary problem arising in the diffusion of glassy polymers this paper considers approximate solution methods for a one dimensional stefan problem describing solvent diffusion in glassy polymers similar to the classic stefan problem the region initially has zero thickness and so must be analysed carefully before performing a numerical computation a small time analysis gives the correct starting solution which is then incorporated into the second order accurate keller box finite difference scheme we also consider a detailed analysis of small and large time expansions as well as the large control parameter limit and show that our generalised approach enables us to obtain higher order terms than given in previous studies finally we apply the combined integral method cim to this problem which is a refinement of the popular heat balance integral method hbim and compare both the cim and asymptotic solutions to the numerical results

a comparison of numerical models for one dimensional stefan problems in this paper we present a critical comparison of the suitability of several numerical methods level set moving grid and phase field model to address two well known stefan problems in phase transformation studies melting of a pure phase and diffusional solid state phase transformations in a binary system similarity solutions are applied to verify the numerical results the comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques finally it is shown both numerically and analytically that the solid solid phase transformation is a limiting case of the solid liquid transformation

an integral equation method for spherical stefan problems abstract many applications in solidification or melting are described by a two phase stefan problem with spherical symmetry using green s representation theorem the heat equation is reformulated as a system of integral equations for the unknown fluxes on the interface the unknown radius of the solid liquid interface is determined from the stefan condition the integral equation is discretized with the nystrom method for which special singularity corrected quadrature rules are discussed numerical examples are presented that demonstrate the effectiveness of the method

an accurate finite difference method for ablation type stefan problems a recently derived numerical algorithm for one dimensional time dependent stefan problems is extended for the purpose of solving one phase ablation type moving boundary problems in tandem with the keller box finite difference scheme the so called boundary immobilization method is used an important component of the work is the use of variable transformations that must be built into the numerical algorithm in order to preserve second order accuracy in both time and space the analysis also determines that the ablation front initially moves as the time raised to the power 3 2 hence it evolves considerably more slowly than the phase change front in the classical stefan problem with isothermal cooling

a numerical solution of the stefan problem with a neumann type boundary condition by enthalpy method in this paper the enthalpy method based on suitable finite difference approximations has been applied to the one dimensional moving boundary problem with a neumann type boundary condition known as the stefan problem the numerical results obtained by the hopscotch technique are compared with the exact solution of the problem it is shown that all results are found to be in very good agreement with each other and the numerical solution displays the expected convergence to the exact one as the mesh size is refined

nonlinear moving boundary problems and a keller box scheme a modification of the original kelley box scheme is used to approximate nonlinear parabolic partial differential equations with moving boundaries the proposed scheme solves two systems of linear algebraic equations at each time step to produce second order approximations to the solution its derivative the flux and the position of the moving boundary the scheme is applied to a physical problem which has a known similarity solution a comparison of numerical results with the similarity solution gives evidence of stability and convergence

approximation of multivariate functions and evaluation of particular solutions using chebyshev polynomial and trigonometric basis functions summary a two stage numerical procedure using chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation the purpose of such numerical schemes is crucial for the evaluation of particular solutions of a large class of partial differential equations our proposed scheme provides a highly efficient and accurate approximation of multivariate functions and particular solution of certain partial differential equations simultaneously numerical results on the approximation of eight two dimensional test functions and their derivatives are given to demonstrate that the scheme for the approximation of functions can be easily extended to evaluate the particular solution of certain partial differential equations we solve a modified helmholtz equation near machine precision can be achieved for all these test problems copyright 2006 john wiley sons ltd

single boundary control of the two phase stefan system abstract this paper presents the control design of the two phase stefan problem the two phase stefan problem is a representative model of liquid solid phase transition by describing the time evolutions of the temperature profile which is divided by subdomains of liquid and solid phases as the liquid solid moving interface position the mathematical formulation is given by two diffusion partial differential equations pdes defined on a time varying spatial domain described by an ordinary differential equation ode driven by the neumann boundary values of both pde states resulting in a nonlinear coupled pde ode pde system we design a state feedback control law by means of energy shaping to stabilize the interface position to a desired setpoint by using single boundary heat input we prove that the closed loop system under the control law ensures some conditions for model validity and the global exponential stability estimate is shown in the spatial l 2 norm furthermore the robustness of the closed loop stability with respect to the uncertainties of the physical parameters is shown numerical simulation is provided to illustrate the desired performance of the proposed control law in comparison to the control design for the one phase stefan problem

finite difference methods with increased accuracy and correct initialization for one dimensional stefan problems although the numerical solution of one dimensional phase change or stefan problems is well documented a review of the most recent literature indicates that there are still unresolved issues regarding the start up of a computation for a region that initially has zero thickness as well as how to determine the location of the moving boundary thereafter this paper considers the so called boundary immobilization method for four benchmark melting problems in tandem with three finite difference discretization schemes we demonstrate a combined analytical and numerical approach that eliminates completely the ad hoc treatment of the starting solution that is often used and is numerically second order accurate in both time and space a point that has been consistently overlooked for this type of moving boundary problem

applying the combined integral method to two phase stefan problems with delayed onset of phase change in this paper the combined integral method cim is applied to non classical two phase stefan problems with delayed onset of phase change this can occur if the phase change is caused by a heat flux or robin boundary condition the method requires choosing an approximating function typically a polynomial but it is not clear what should be used as the exponent in the highest order term previous studies have determined exponents either from exact solutions or from expansions valid over short time scales neither approach is satisfactory and can be very inaccurate for larger times we combined the heat balance and refined integral methods to determine this exponent as part of the solution process allowing the exponent to be time dependent from comparing the approximate solutions with numerical and exact analytical solutions whenever possible we show that the cim greatly improves the accuracy on standard heat balance integral methods without detracting from its simplicity

an isotherm migration formulation for one phase stefan problem with a time dependent neumann condition in this paper we present a numerical scheme based on an isotherm migration formulation for one dimensional one phase stefan problem with a time dependent neumann condition on the fixed boundary and a constant dirichlet condition on the moving boundary the numerical results obtained by the present method have been compared with exact one and also those obtained by earlier authors and are found to be in very good agreement with each other it is also shown that the numerical solution displays the expected convergence to the exact one as the mesh size is refined

on the numerical solution of a stefan problem with finite extinction time in many phase change problems of practical interest it is important to know when a phase is depleted a quantity referred to as the extinction time however there are no numerical schemes that are able to compute this with any degree of rigour or formal accuracy in this paper we develop such a scheme for the one dimensional time dependent problem of an evaporating spherical droplet the keller box finite difference scheme is used in tandem with the so called boundary immobilization method an important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second order accuracy in both time and space in particular as regards resolving a square root singularity in the droplet radius as the extinction time is approached

characterizing complete operatorname cat kappa spaces kappa 0 with geodesic boundary we investigate the bourdon and hamenstadt boundaries of complete cat spaces 0 and characterize those with geodesic hamenstadt boundary up to isometry

analysis of box schemes for reactive flow problems key properties of the box scheme are shown to be advantageous for reactive flow problems unconditional stability and compact conservation are shown by a detailed modified equation analysis to enable the scheme to reflect exactly the reduced speed enhanced diffusion and dispersion which are typical of such hyperbolic conservation laws with relaxation a novel modified equation analysis is also used to show how the spurious checkerboard mode behaves and can be controlled numerical experiments for some nonlinear one dimensional problems and a two dimensional problem demonstrate that the behavior of the scheme deduced from a simple model problem has general validity

an accurate numerical solution for the transient heating of an evaporating spherical droplet a recently derived numerical algorithm for one dimensional time dependent stefan problems is extended for the purposes of solving a moving boundary problem for the transient heating of an evaporating spherical droplet the keller box finite difference scheme is used in tandem with the so called boundary immobilization method an important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second order accuracy in both time and space an issue not previously discussed in relation to this widely used scheme in addition we demonstrate that our solution is in close agreement with the solution obtained using an alternative numerical scheme that employs an analytic solution of the heat conduction equation inside the droplet for which the droplet radius was assumed to be a piecewise linear function of time the advantages of the new method are discussed

an exponential heat balance integral method implementations of the heat balance integral method are discussed in which exponential functions are used in place of the familiar polynomial approximants the rationale is based upon that of least squares in that the use of appropriate basis functions can enhance solution accuracy whilst this is true in principle it is shown that considerable skill must be exercised when deviating from polynomial approximants the discussions are illustrated by application to a familiar single phase stefan problem that is typical of heat transfer problems exhibiting decay like spatial solution profiles

a galerkin method for stefan problems stefan problems are described by a parabolic partial differential equation along with two boundary conditions on a moving boundary which is to be determined as part of the solution the purpose of this paper is to develop a finite element method for the solution of a one dimensional stefan problem first a coordinate transformation is used to transform the changing physical domain into a fixed computational domain then the weak or galerkin formulation of the initial boundary value problem is used to reduce it to a system of initial value problems in ordinary differential equations several finite difference marching methods are used to solve the resulting initial value problems the method is developed and illustrated using the stefan problem concerning the heat transfer in an ice water medium the computational results are in very good agreement with the results produced earlier by several authors

a computational method for inverse free boundary determination problem based on the method of fundamental solutions and discrepancy principle for the choice of location for source points we extend in this paper the application of the computational method to determine an unknown free boundary of a cauchy problem of parabolic type equation from measured dirichlet and neumann data with noises the standard tikhonov regularization technique with the l curve method for an optimal regularized parameter is adopted for solving the resultant highly ill conditioned system of linear equations both one dimensional and two dimensional numerical examples are given to verify the efficiency and accuracy of the proposed computational method copyright 2007 john wiley sons ltd

behavior of the solution of a stefan problem by changing thermal coefficients of the substance abstract we consider a one dimensional one phase stefan problem for a semi infinite substance we suppose that there is a transient heat flux at the fixed face and the thermal coefficients are constant the goal of this paper is to determine the behavior of the free boundary and the temperature by changing the thermal coefficients we use the maximum principle in order to obtain properties of monotony with respect to the latent heat of fusion the specific heat and the mass density we compute approximate solutions through the quasi stationary the goodman s heat balance integral and the biot s variational methods and a numerical solution through a finite difference scheme we show that the solution is not monotone with respect to the thermal conductivity the results obtained are important in technological applications

a boundary meshless method using chebyshev interpolation and trigonometric basis function for solving heat conduction problems a boundary meshless method has been developed to solve the heat conduction equations through the use of a newly established two stage approximation scheme and a trigonometric series expansion scheme to approximate the particular solution and fundamental solution respectively as a result no fundamental solution is required and the closed form of approximate particular solution is easy to obtain the effectiveness of the proposed computational scheme is demonstrated by several examples in 2d and 3d we also compare our proposed method with the finite difference method and the other meshless method showed in sarler and vertnik comput math appl 2006 51 1269 1282 excellent numerical results have been observed copyright 2007 john wiley sons ltd

variable time step method with coordinate transformation abstract the variable time step methods for solving moving boundary problems are presented by transforming the variable space domain this results in dissociating the mode of advancement of the boundary from the size of the space mesh that is a small movement of the moving boundary may be chosen for computing the time interval while the space domain is subdivided into larger space meshes as a consequence an enormous amount of saving in computer time may be achieved by using the proposed method two sample problems are selected for the illustration of the method

finite element analysis of diffusion with reaction at a moving boundary abstract we consider the model problem of mass transport by diffusion with chemical reaction at a moving boundary the objective is to model the chemical process by predicting the location of the boundary as the reaction proceeds in time a finite element analysis is formulated in one dimension using moving elements to follow the boundary motion and an iterative linearization method is developed to uncouple and solve the diffusion and reaction equations the semidiscrete system for the transport process is integrated using a predictor corrector scheme and its accuracy and stability are examined in numerical experiments numerical results are compared with an analytic solution for the quasi steady case in which diffusion is very rapid in comparison to reaction and to a perturbation solution for the unsteady case other supporting numerical studies are made to examine the sensitivity of the solution and boundary motion to changes in diffusion and reaction rate for both linear and nonlinear reactions

the numerical solution of one phase classical stefan problem in this paper variable space grid and boundary immobilisation techniques based on the explicit finite difference are applied to the one phase classical stefan problem it is shown that all the results obtained by the two methods are in good agreement with the exact solution and exhibit the expected convergence as the mesh size is refined

application of standard and refined heat balance integral methods to one dimensional stefan problems the work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems these include standard test problems both with one and two phase changes which have exact solutions that enable us to test the accuracy of the approximate solutions we also consider situations where no analytical solution is available and compare these to numerical solutions it is popular to use a quadratic profile as an approximation of the temperature but we show that a cubic profile seldom considered in the literature is far more accurate in most circumstances in addition the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases we assess which integral method is better for various problems showing that it is largely dependent on the specified boundary conditions

a variable time step galerkin method for a one dimensional stefan problem a variable time step vts method is presented for the solution of a one dimensional stefan problem describing the evaporation of droplets the method uses a simple coordinate transformation and transforms the moving boundary problem into a problem on a fixed domain the weak or galerkin formulation of the resulting initial boundary value problem is then used to derive a system of initial value problems in ordinary differential equations the solution is advanced in time by an implicit marching technique coupled with an iterative computation of the time step for a given advancement of the moving boundary an integral relationship is used to correct the time step in each iteration the numerical results obtained by the present method exhibit very good agreement with those obtained by previous methods

control and state estimation of the one phase stefan problem via backstepping design this paper develops a control and estimation design for the one phase stefan problem the stefan problem represents a liquid solid phase transition as time evolution of a temperature profile in a liquid solid material and its moving interface this physical process is mathematically formulated as a diffusion partial differential equation pde evolving on a time varying spatial domain described by an ordinary differential equation ode the state dependency of the moving interface makes the coupled pde ode system a nonlinear and challenging problem we propose a full state feedback control law an observer design and the associated output feedback control law of both neumann and dirichlet boundary actuations via the backstepping method also the state feedback control law is provided when a robin boundary input is considered the designed observer allows the estimation of the temperature profile based on the available measurements of liquid phase length and the heat flux at the interface the associated output feedback controller ensures the global exponential stability of the estimation errors the mathcal h 1 norm of the distributed temperature and the moving interface at the desired setpoint under some explicitly given restrictions on the setpoint and observer gain

on the accurate numerical solution of a two phase stefan problem with phase formation and depletion in this paper the keller box finite difference scheme is employed in tandem with the so called boundary immobilization method for the purposes of solving a two phase stefan problem that has both phase formation and phase depletion an important component of the work is the use of variable transformations that must be built into the numerical algorithm in order to resolve the boundary condition discontinuities that are associated with the onset of phase formation and depletion in particular this allows the depletion time to be determined and the solution to be computed after depletion the method gives second order accuracy in both time and space for all variables throughout the entire computation

numerical schemes for one dimensional stefan like problems with a forcing term variable space grid and boundary immobilisation schemes based on the explicit finite difference method are applied to the one phase stefan like problems with a forcing term in order to evaluate the temperature distribution and the interface movement location and speed the numerical results obtained by the two schemes have been compared with the exact values and are found to be in good agreement with each other it is shown that the numerical solutions exhibit the expected convergence to the exact one as the mesh size is reduced a von neumann stability analysis of each scheme is also investigated

on the numerical solution of two phase stefan problems with heat flux boundary conditions a recently derived numerical algorithm for one dimensional one phase stefan problems is extended for the purpose of two phase moving boundary problems in which the second phase first appears only after a finite delay time this can occur if the phase change is caused by a heat flux boundary condition in tandem with the keller box finite difference scheme the so called boundary immobilization method is used an important component of the work is the use of variable transformations that must be built into the numerical algorithm to resolve the boundary condition discontinuity that is associated with the onset of phase change this allows the delay time until solidification begins to be determined and gives second order accuracy in both time and space

a fixed grid local method for 1 d stefan problems abstract this note is concerned with a fixed grid local method for the solution of 1 d free boundary problems the method uses a coordinate transformation after the coordinate transformation it is then possible to obtain exact solutions for the resulting equations in terms of the local variables these exact solutions are in terms of constants of integration that are unknown by imposing the given boundary conditions and some smoothness requirements it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration a number of examples are used to study the applicability of the method

the method of approximate fundamental solutions mafs for stefan problems for the sphere the paper presents a new meshless in space numerical method the method of approximate fundamental solutions mafs for solving problems with moving boundaries of the stefan type this technique first suggested for solving the stefan problems in plane geometry is now extended onto the cases of inward and outward spherical problems computed results for the inward and outward spherical problems are numerically evaluated for comparison with the results obtained by other authors the comparison shows that the method presented provides a very high precision in determining the position of the moving boundary and the results are in a good agreement with the previous studies