In this paper it is proved that the two-point boundary value problem, namely $({d^{(4)}}/d{x^4} + f)y = g,y(0) - {A_1} = y(1) - {A_2} = y''(0) - {B_1} = y''(1) …
In this paper it is proved that the two-point boundary value problem, namely $({d^{(4)}}/d{x^4} + f)y = g,y(0) - {A_1} = y(1) - {A_2} = y''(0) - {B_1} = y''(1) - {B_2} = 0$, has a unique solution provided ${\inf _x}f(x) = - \eta > - {\pi ^4}$. The given boundary value problem is discretized by a finite difference scheme. This numerical approximation is proved to be a second order convergent process by establishing an error bound using the ${L_2}$-norm of a vector.
A new entropy criterion (maximal dissipation condition) for the quasilinear wave equation with generally nonmonotone nonlinearity is introduced and tested on self-similar solutions of the corresponding Riemann problem. It is …
A new entropy criterion (maximal dissipation condition) for the quasilinear wave equation with generally nonmonotone nonlinearity is introduced and tested on self-similar solutions of the corresponding Riemann problem. It is shown that the maximally dissipating solution exists and it is uniquely determined. The relation between the maximal dissipation principle and other entropy criteria is discussed.
In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the …
In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.
Let D contained in/implied by R3 be a bounded domain with a smooth boundary Gamma , - Delta u+q(x)u=0 in D u=f, uN=h on Gamma and q(x) in Linfinity (D). …
Let D contained in/implied by R3 be a bounded domain with a smooth boundary Gamma , - Delta u+q(x)u=0 in D u=f, uN=h on Gamma and q(x) in Linfinity (D). From knowledge of the set (f, h) where f runs through C1( Gamma ) the coefficient q(x) is uniquely recovered. Analytical formulae for q(x) are given. Applications are considered.
We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation \partial_t u + \div(bu)=0 admits a unique bounded solution (in …
We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation \partial_t u + \div(bu)=0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b . As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain non-autonomous vector fields b with bounded divergence.
This article discusses the existence and uniqueness of global solution to a new type of one-phase Stefan problem.In order to attain the global solution,we first translate the Stefan problem into …
This article discusses the existence and uniqueness of global solution to a new type of one-phase Stefan problem.In order to attain the global solution,we first translate the Stefan problem into an equivalent integral equations,then we define a Banach space L_sand a mapping F no it.By proving F being a contraction mapping on a closed subset of L_(s,M),we get the existence and uniqueness of local solution to the integral equation.
We establish a uniqueness result for an overdetermined boundary value problem. We also raise a new question.
We establish a uniqueness result for an overdetermined boundary value problem. We also raise a new question.
This article proves the existence and uniqueness of global solutions to a new type of Stefan problem based on the fact that its local solutions have been proved existent in …
This article proves the existence and uniqueness of global solutions to a new type of Stefan problem based on the fact that its local solutions have been proved existent in other articles.
A uniqueness theorem is established for the scattering of harmonic elastic waves by a body with continuously varying parameters placed in a homogeneous medium.
A uniqueness theorem is established for the scattering of harmonic elastic waves by a body with continuously varying parameters placed in a homogeneous medium.
where a#(#) are analytic functions defined in an open set $C^2 ?and of real-valued if i+j=m.Let £f be an analytic curve defined by <p(x) = (p(x Q \ #°E$ 5 …
where a#(#) are analytic functions defined in an open set $C^2 ?and of real-valued if i+j=m.Let £f be an analytic curve defined by <p(x) = (p(x Q \ #°E$ 5 where <p(oc) is a real-valued analytic function defined in Q.From now on we assume that q> Xl (x 0> )=£Q.Now let us assume that £f is a double characteristic curve of (1) ?that is, Pm(x,v3\*> = 0, Py(x 9 ^\y = 0 for i = l, 2,(2)Pm' j} (x, 0>*)U=£0 for some z,/=l, 2 3 where P m (x, f)= E atjW^l P%\x, S") = dP m (x 9 f)/9ft and P«-»(*,And also we assume that P m (x, cp x } vanishes at the first order on £f, that is,(3)
The present paper is concerned with the Stefan problem for the heat equations, which arises, for instance, in the study of melting of ice adjacent to the heated water.Free boundary …
The present paper is concerned with the Stefan problem for the heat equations, which arises, for instance, in the study of melting of ice adjacent to the heated water.Free boundary problems for the heat equations have been considered for a century.Although in some special cases explicit solutions had happened to be known very early, existence theorems of general nature were proved first only twenty years ago in connection with the 1-dimensional Stefan problem by Rubinstein [1] and Dacev [2].Since then, various papers on the Stefan problem have been published by many authors including Friedman [3], [10], Evans [4], J. Douglas & Gallic [5], Sestini [6], Miranker [7], J. Douglas [8], Kyner [9], I.T. Kolodner [11], Ladyzenskaya [12], Oleinic [13], Brezis p4], Nogi [15] and others.Among their contributions we refer to the existence theorems due to Kyner, Friedman and Oleinic which state the existence of solutions of the problem subject to the Dirichlet or Neumann boundary conditions imposed on the boundary of the heated water, under the assumption that even at the initial moment there does exist some water.As a matter of fact, the absence of water at the initial moment invoke a certain singularity or difficulty of the problem from the mathematical view point, which is rather mild for the case of the Neumann boundary condition but is quite hard to handle for the case of the Dirichlet boundary condition.Indeed, Kyner [9] and Friedman [10] have succeeded in dealing with
A vertical long soil column subject to rainfall filtrating through the upper end of the column has been considered. The mathematical formulation of this physical model, based on the theory …
A vertical long soil column subject to rainfall filtrating through the upper end of the column has been considered. The mathematical formulation of this physical model, based on the theory of fluid flows in a nonsaturated porous medium, leads to a one-phase, nonlinear parabolic free boundary problem, the free boundary being the wetting front. For such a problem, theorems on existence, uniqueness, and continuous dependence on the data and the coefficients appearing in its mathematical equations have been proved.
Purpose – The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, …
Purpose – The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method. Design/methodology/approach – The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem. Findings – The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat. Originality/value – The paper presents a new solution algorithm for the Stefan problem.
CONTENTSIntroductionChapter I. General statement of the Stefan problem and some of its variantsChapter II. The one-dimensional non-stationary problemChapter III. The quasi-stationary many-dimensional problemChapter IV. The method of integral functionals with …
CONTENTSIntroductionChapter I. General statement of the Stefan problem and some of its variantsChapter II. The one-dimensional non-stationary problemChapter III. The quasi-stationary many-dimensional problemChapter IV. The method of integral functionals with a variable domain of integrationChapter V. The many-dimensional non-stationary problemChapter VI. Stability and stabilization. Open questionsReferences
This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of …
This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of water or foodstuff, welding and casting processes. We point out that known solution to standard Stefan problem solutions can be recovered with this formalism, and then propose a simple phase transition estimator that relies only on boundary measurements. Simulations indicate that the estimates converge for noise-free measurements.
We study a free boundary problem which arises as the continuum version of a stochastic particles system in the context of Fourier law. Local existence and uniqueness of the classical …
We study a free boundary problem which arises as the continuum version of a stochastic particles system in the context of Fourier law. Local existence and uniqueness of the classical solution are well known in the literature of free boundary problems. We introduce the notion of generalized solutions (which extends that of classical solutions when the latter exist) and prove global existence and uniqueness of generalized solutions for a large class of initial data. The proof is obtained by characterizing a generalized solution as the unique element which separates suitably defined lower and upper barriers in the sense of mass transport inequalities.