One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise line...One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.展开更多
In this paper, we prove the existence theorems of locbal or global classical solutions to Stefan problems with various kinetic conditions at the free boundary.
A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
In this paper we consider an evolutionary continuous casting problem with convection partial derivative + b(y) chi - Delta kappa(u) + (v) over right arrow .del u = 0 coupled with Stokes equation in the liquid phase pa...In this paper we consider an evolutionary continuous casting problem with convection partial derivative + b(y) chi - Delta kappa(u) + (v) over right arrow .del u = 0 coupled with Stokes equation in the liquid phase partial derivative(t) (v) over right arrow - v Delta (v) over right arrow + del p = (f) over right arrow(u) The mixed boundary condition is put on temprature. The existence of a weak solution is obtained by using methods of regularization, temperature dependent penalty form in the Stokes equation and compact arguments.展开更多
We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé...We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815;2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .展开更多
In this paper,a meshless regularization method of fundamental solutions is proposed for a two-dimensional,two-phase linear inverse Stefan problem.The numerical implementation and analysis are challenging since one nee...In this paper,a meshless regularization method of fundamental solutions is proposed for a two-dimensional,two-phase linear inverse Stefan problem.The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions.Furthermore,the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution.Therefore,regularization is necessary in order to obtain a stable solution.Numerical results for several benchmark test examples are presented and discussed.展开更多
A Stefan problem with nonlinear boundary flux and internal convection of a material are considered. The existence, uniqueness and continuous dependence of globally weak solution of this problem are obtained. This pape...A Stefan problem with nonlinear boundary flux and internal convection of a material are considered. The existence, uniqueness and continuous dependence of globally weak solution of this problem are obtained. This paper extends the results of Fahuai Yi and T.M.Shih, relaxes restrictions that does not be to accord with reality very much on internal convection and boundary conditions in their articles.展开更多
The existence of a local classical solution to the Mullins-Sekerka problem and the convergence to the two-phase quasi-stationary Stefan problem are proved when surface tension approaches zero. This convergence gives a...The existence of a local classical solution to the Mullins-Sekerka problem and the convergence to the two-phase quasi-stationary Stefan problem are proved when surface tension approaches zero. This convergence gives a proof of the existence of a local classical solution of quasi-stationary Stefan problem. The methods work in all dimensions.展开更多
This paper considers the quasi-stationary Stefan problem:△u(x,t)=0 in space-time domain,u=0 and Vv+δu/δv=0 on the free boundary. under the natural conditions the existence of classical solution locally in time is p...This paper considers the quasi-stationary Stefan problem:△u(x,t)=0 in space-time domain,u=0 and Vv+δu/δv=0 on the free boundary. under the natural conditions the existence of classical solution locally in time is proved by making use of the property of Frechet derivative operator and fixed point theorem. For the sake of simplicity only the one-phase problem is dealt with. In fact two-phase problem can be dealt with in a similar way with more complicated calculation.展开更多
In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is pre...In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.展开更多
基金supported by the Fundamental Research Funds for the Central Universities(Grant 2015XKMS014)
文摘One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.
文摘In this paper, we prove the existence theorems of locbal or global classical solutions to Stefan problems with various kinetic conditions at the free boundary.
文摘A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
文摘In this paper we consider an evolutionary continuous casting problem with convection partial derivative + b(y) chi - Delta kappa(u) + (v) over right arrow .del u = 0 coupled with Stokes equation in the liquid phase partial derivative(t) (v) over right arrow - v Delta (v) over right arrow + del p = (f) over right arrow(u) The mixed boundary condition is put on temprature. The existence of a weak solution is obtained by using methods of regularization, temperature dependent penalty form in the Stokes equation and compact arguments.
文摘We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815;2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .
基金T.Reeve would like to acknowledge the financial support received from the EPSRC.
文摘In this paper,a meshless regularization method of fundamental solutions is proposed for a two-dimensional,two-phase linear inverse Stefan problem.The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions.Furthermore,the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution.Therefore,regularization is necessary in order to obtain a stable solution.Numerical results for several benchmark test examples are presented and discussed.
基金Supported by National Natural Science Foundation of China (90410011)the Natural Science Foundation of the Education Department of Anhui Province (2005KJ316ZC).
文摘A Stefan problem with nonlinear boundary flux and internal convection of a material are considered. The existence, uniqueness and continuous dependence of globally weak solution of this problem are obtained. This paper extends the results of Fahuai Yi and T.M.Shih, relaxes restrictions that does not be to accord with reality very much on internal convection and boundary conditions in their articles.
文摘The existence of a local classical solution to the Mullins-Sekerka problem and the convergence to the two-phase quasi-stationary Stefan problem are proved when surface tension approaches zero. This convergence gives a proof of the existence of a local classical solution of quasi-stationary Stefan problem. The methods work in all dimensions.
文摘This paper considers the quasi-stationary Stefan problem:△u(x,t)=0 in space-time domain,u=0 and Vv+δu/δv=0 on the free boundary. under the natural conditions the existence of classical solution locally in time is proved by making use of the property of Frechet derivative operator and fixed point theorem. For the sake of simplicity only the one-phase problem is dealt with. In fact two-phase problem can be dealt with in a similar way with more complicated calculation.
基金supported by the National Natural Science Foundation (10871179) of China
文摘In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.
