This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid emb...
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This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996;and J. Comput. Phys. 133(2), 1997) and Ditkowski (Ph.D. thesis, Tel Aviv University, 1997), for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the Stefan problem. Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher than 2nd-order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies, that the numerical solution remains consistent and valid for a long integration time. A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2nd and a 4th-order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method.
The initial-Dirichlet and initial-Neumann problems in Lipschitz cylinders are studied forthe general second order parabolic equations of constant coefficients with squarely integrableboundary data. By layer potential ...
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The initial-Dirichlet and initial-Neumann problems in Lipschitz cylinders are studied forthe general second order parabolic equations of constant coefficients with squarely integrableboundary data. By layer potential method developed in the past decade, the author provesthat the double layer potential and the single layer potential operators are invertible and henceobtains the solvability of the initial boundsry valueproblems. Also, the solutions can berepresented by these operators.
Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator are considered. Solutions to these problems are constructed based ...
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Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansions and results on existence and uniqueness are established. To solve the resultant equations, a solution to such kind of non-homogeneous fractional differential equation is also presented.
In this paper we study initialvalueboundaryproblems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove...
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In this paper we study initialvalueboundaryproblems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations Under discussion. (C) 2008 Elsevier Inc. All rights reserved.
While there exist now formulations of initial boundary value problems for Einstein's field equations which are well posed and preserve constraints and gauge conditions, the question of geometric uniqueness remains...
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While there exist now formulations of initial boundary value problems for Einstein's field equations which are well posed and preserve constraints and gauge conditions, the question of geometric uniqueness remains unresolved. For two different approaches we discuss how this difficulty arises under general assumptions. So far it is not known whether it can be overcome without imposing conditions on the geometry of the boundary. We point out a natural and important class of initial boundary value problems which may offer possibilities to arrive at a fully covariant formulation.
In this paper we study initial boundary value problems of three types of two-component shallow water systems on the half line subject to homogeneous Dirichlet boundary conditions. We first prove local well-possedness ...
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In this paper we study initial boundary value problems of three types of two-component shallow water systems on the half line subject to homogeneous Dirichlet boundary conditions. We first prove local well-possedness of the two-component Camassa-Holm system, the modified two-component Camassa-Holm system, and the two-component Degasperis- Procesi system in the Besov spaces. Then, we are able to specify certain conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite lifespan. Moreover, in the case of finite time singularities we are able to describe the precise blow-up scenario for breaking waves. Finally we investigate global weak solutions for the two-component Camassa-Holm system and the modified two-component Camassa-Holm system on the half line, respectively. Our approach is based on sharp extension results for functions on the half line and several symmetry preserving properties of the systems under discussion.
In this paper, we construct a new method of probabilistic representation of a solution of initial boundary value problems for series of evolution equations in a circle based on constructing a special continuation of a...
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In this paper, we construct a new method of probabilistic representation of a solution of initial boundary value problems for series of evolution equations in a circle based on constructing a special continuation of an initial function from the circle to the whole plane.
In this paper we study initial boundary value problems of the Camassa-Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary v...
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In this paper we study initial boundary value problems of the Camassa-Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundaryvalueproblems into Cauchy problems for the Camassa-Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa-Holm equation on a compact interval possesses no nontrivial global classical solutions.
In the last years the study of initial boundary value problems for nonlinear dispersive equations on the half-lines has given attention of many researchers. This turns out to be a rather challenging problem, mainly wh...
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In the last years the study of initial boundary value problems for nonlinear dispersive equations on the half-lines has given attention of many researchers. This turns out to be a rather challenging problem, mainly when studied in low Sobolev regularity. In this note we present a review of the main results about this topic and also introduce interesting open problems which still requires attention from the mathematical point of view.
This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established...
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