Showing papers on "Free boundary problem published in 2013"
TL;DR: In this paper, the authors study the diffusive logistic equation with a free boundary in time-periodic environment and show that the spreading-vanishing dichotomy is retained in time periodic environment, and also determine the spreading speed.
151 citations
TL;DR: In this paper, the authors define a finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface PDE posed on the boundary of the bulk domain.
Abstract: In this paper, we define a new finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation and apply these to find the optimal-order error estimates. A numerical experiment is described which demonstrates the order of convergence.
127 citations
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TL;DR: In this article, the authors investigated a free boundary problem for a predator-prey model with double free boundaries in one space dimension, and proved a spreading vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as $t\to \infty$ at both fronts and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state.
Abstract: In this paper we investigate a free boundary problem for a predator-prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading-vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as $t\to \infty$ at both fronts and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.
106 citations
TL;DR: This paper derives some sufficient conditions for exponential boundary observer design using only the information from the boundary control and the boundary conditions by means of Lyapunov based techniques.
88 citations
TL;DR: In this article, the free vibrations of cylindrical shells with non-uniform elastic boundary constraints were investigated using improved Fourier series method, in which each of three displacements of the shell is represented by a Fourier-series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions.
84 citations
TL;DR: In this paper, the authors prove well-posedness results for the solution to an initial and boundary-value problem for an Allen-Cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain.
Abstract: We prove well-posedness results for the solution to an initial and boundary-value problem for an Allen–Cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain. The dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls in Fischer (1997) [13] , Kenzler (2001) [14] . We show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments.
80 citations
TL;DR: In this article, the inverse boundary value problem for the hyperbolic partial differential equation is studied on a bounded and smooth cylindric domain ( − ∞, ∞) × Ω.
Abstract: For the time-dependent vector and scalar potentials (A0, ..., An) and V(t, x) respectively, the inverse boundary value problem for the hyperbolic partial differential equation is studied on a bounded and smooth cylindric domain ( − ∞, ∞) × Ω. Using a geometric optics construction, it is shown that the boundary data allow for the recovery of integrals of the potentials along 'light rays'. The uniqueness of these potentials modulo a gauge transform is also established.
74 citations
TL;DR: In this paper, a Lipschitz-type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values, which is known as the Dirichlet-to-Neumann map.
Abstract: In this paper we study the inverse boundary value problem of determining the potential in the Schrodinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz-type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values.
70 citations
TL;DR: In this article, the unknown boundary function is determined from overposed data in a time-fractional diffusion equation based on the free space fundamental solution, and a representation for the solution f is derived as a nonlinear Volterra integral equation of second kind with a weakly singular kernel.
Abstract: In this article we consider an inverse boundary problem, in which the unknown boundary function is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.
66 citations
TL;DR: In this article, strong hyperbolicity guarantees well-posedness of the initial value problem and symmetric Hyperbolic systems are shown to render the initial boundary value problem wellposed with maximally dissipative boundary conditions.
Abstract: These lecture notes accompany two classes given at the NRHEP2 school In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations I show how strong hyperbolicity guarantees well-posedness of the initial value problem Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions I discuss the Laplace–Fourier method for analyzing the initial boundary value problem Finally, I state how these notions extend to systems that are first-order in time and second-order in space In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable
64 citations
Book•
15 Jul 2013TL;DR: In this paper, the boundary conditions are enforced through determination of the flux contribution at the boundary to the solution residual, and the boundary values converged to the requested boundary condition with approximately second-order accuracy for all of the cases.
Abstract: Several boundary conditions that allow subsonic and supersonic flow into and out of the computational domain are discussed. These boundary conditions are demonstrated in the FUN3D computational fluid dynamics (CFD) code which solves the three-dimensional Navier-Stokes equations on unstructured computational meshes. The boundary conditions are enforced through determination of the flux contribution at the boundary to the solution residual. The boundary conditions are implemented in an implicit form where the Jacobian contribution of the boundary condition is included and is exact. All of the flows are governed by the calorically perfect gas thermodynamic equations. Three problems are used to assess these boundary conditions. Solution residual convergence to machine zero precision occurred for all cases. The converged solution boundary state is compared with the requested boundary state for several levels of mesh densities. The boundary values converged to the requested boundary condition with approximately second-order accuracy for all of the cases.
TL;DR: In this article, the authors consider a free boundary problem for a reaction-diffusion logistic equation with a time-dependent growth rate and present several sharp thresholds for information diffusion that either last forever or suspends in finite time.
TL;DR: In this article, the authors analyzed the global existence of classical solutions to the initial boundary value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons.
Abstract: In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled ...
TL;DR: In this article, a boundary element method (BEM) is developed to calculate the elastic band gaps of two-dimensional (2D) phononic crystals which are composed of square or triangular lattices of solid cylinders in a solid matrix.
Abstract: A boundary element method (BEM) is developed to calculate the elastic band gaps of two-dimensional (2D) phononic crystals which are composed of square or triangular lattices of solid cylinders in a solid matrix. In a unit cell, the boundary integral equations of the matrix and the scatterer are derived, the former of which involves integrals over the boundary of the scatterer and the periodic boundary of the matrix, while the latter only involves the boundary of the scatterer. Constant boundary elements are adopted to discretize the boundary integral equations. Substituting the periodic boundary conditions and the interface conditions, a linear eigenvalue equation dependent on the Bloch wave vector is derived. Some numerical examples are illustrated to discuss the accuracy, efficiency, convergence and the computing speed of the presented method.
TL;DR: In this article, the authors considered the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD) and showed the existence and uniqueness of the solution in suitable anisotropic Sobolev spaces.
Abstract: We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.
Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces. The proof is based on the results proved in the companion paper arXiv:1112.3101, about the well-posedness of the homogeneous linearized problem and the proof of a basic a priori energy estimate. The proof of the resolution of the nonlinear problem given in the present paper follows from the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.
TL;DR: In this article, the existence of solutions for the singular nonlinear fractional boundary value problem was studied by using fixed point results on cones, and they showed that the solution of the problem is solvable.
Abstract: By using fixed point results on cones, we study the existence of solutions for the singular nonlinear fractional boundary value problem
TL;DR: A method for estimating unknown parameters that appear on an avascular, spheric tumor growth model by fitting the numerical solution with real data, obtained via in vitro experiments and medical imaging is presented.
Abstract: In this paper we present a method for estimating unknown parameters that appear on an avascular, spheric tumor growth model. The model for the tumor is based on nutrient driven growth of a continuum of live cells, whose birth and death generate volume changes described by a velocity field. The model consists of a coupled system of partial differential equations whose spatial domain is the tumor, that changes in size over time. Thus, the situation can be formulated as a free boundary problem. After solving the direct problem properly, we use the model for the estimation of parameters by fitting the numerical solution with real data, obtained via in vitro experiments and medical imaging. We define an appropriate functional to compare both the real data and the numerical solution. We use the adjoint method for the minimization of this functional.
TL;DR: In this article, the existence of positive solutions for the nonlinear fractional boundary value problem with a p-Laplacian operator was studied and a positive solution was proposed.
Abstract: In this paper, we study the existence of positive solutions for the nonlinear fractional boundary value problem with a p-Laplacian operator
TL;DR: In this article, the shape of an inclusion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation.
Abstract: Determining the shape of an inclusion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation. We present a solution method for such an inverse boundary value problem with a generalized impedance boundary condition on the inclusion via boundary integral equations. Both the determination of the unknown boundary and the determination of the unknown impedance functions are considered. In addition to describing the reconstruction algorithms and illustrating their feasibility by numerical examples, we also obtain a uniqueness result on determining the impedance coefficients.
TL;DR: In this article, the authors studied the boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary and showed that for a given inlet being a perturbation of an arc centered at the vertex of the nozzle and a given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc, there is a unique continuous subsonic-sonic flow from the given inLET with the angle of the velocity orthogonal to the inlet and the given incoming
Abstract: This paper concerns the well-posedness of a boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary. Such problems arise when studying continuous subsonic–sonic flows in a convergent nozzle with straight solid walls. It is shown that for a given inlet being a perturbation of an arc centered at the vertex of the nozzle and a given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc, there is a unique continuous subsonic–sonic flow from the given inlet with the angle of the velocity orthogonal to the inlet and the given incoming mass flux. Furthermore, the sonic curve of this continuous subsonic–sonic flow is a free boundary, where the flow is singular in the sense that while the speed is C1/2 Holder continuous at the sonic state, the acceleration blows up at the sonic state.
TL;DR: In this paper, the existence of weak solutions for a nonlinear boundary value problem of fractional q-difference equations in Banach space is discussed, which relies on the Monch's fixed-point theorem combined with the technique of weak noncompactness.
Abstract: In this paper, we discuss the existence of weak solutions for a nonlinear boundary value problem of fractional q-difference equations in Banach space. Our analysis relies on the Monch’s fixed-point theorem combined with the technique of measures of weak noncompactness.
TL;DR: In this article, the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD) was considered and the authors proved the well-posedness of the linearized problem in conormal Sobolev spaces.
Abstract: We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface we assume that the total pressure is continuous and that the magnetic field is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. Under a suitable stability condition satisfied at each point of the plasma-vacuum interface, we prove the well-posedness of the linearized problem in conormal Sobolev spaces.
TL;DR: In this article, a new multiple reciprocity formulation is developed to solve the transient heat conduction problem, where the time dependence of the problem is removed temporarily from the equations by the Laplace transform.
Abstract: In this paper, a new multiple reciprocity formulation is developed to solve the transient heat conduction problem. The time dependence of the problem is removed temporarily from the equations by the Laplace transform. The new formulation is derived from the modified Helmholtz equation in Laplace space (LS), in which the higher order fundamental solutions of this equation are firstly derived and used in multiple reciprocity method (MRM). Using the new formulation, the domain integrals can be converted into boundary integrals and several non-integral terms. Thus the main advantage of the boundary integral equations (BIE) method, avoiding the domain discretization, is fully preserved. The convergence speed of these higher order fundamental solutions is high, thus the infinite series of boundary integrals can be truncated by a small number of terms. To get accurate results in the real space with better efficiency, the Gaver-Wynn-Rho method is employed. And to integrate the geometrical modeling and the thermal analysis into a uniform platform, our method is implemented based on the framework of the boundary face method (BFM). Numerical examples show that our method is very efficient for transient heat conduction computation. The obtained results are accurate at both internal and boundary points. Our method outperforms most existing methods, especially concerning the results at early time steps.
TL;DR: In this article, the inverse problem of finding the time-dependent heat source together with the temperature solution of heat equation with nonlocal boundary and integral additional conditions is investigated, and a boundary element method combined with the Tikhonov regularization of various orders is developed in order to obtain a stable solution.
TL;DR: In this article, a unified way to determine the law of the Loewner chain given boundary conditions of the free field and to prove existence of the coupling is provided, which always relies on Hadamard's formula and properties of harmonic functions.
Abstract: The relation between level lines of Gaussian free fields (GFF) and SLE4-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE4 is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.
TL;DR: In this paper, the existence of solutions for a fractional boundary value problem involving Hadamard-type fractional differential inclusions and integral boundary conditions was studied, including the cases for convex as well as non-convex valued maps.
Abstract: In this paper, we study the existence of solutions for a fractional boundary value problem involving Hadamard-type fractional differential inclusions and integral boundary conditions. Our results include the cases for convex as well as non-convex valued maps and are based on standard fixed point theorems for multivalued maps. Some illustrative examples are also presented.
TL;DR: In this article, the existence of at least one positive solution to the semipositone problem with a two-point boundary condition has been studied, where the boundary condition can possibly be both non-local and nonlinear.
Abstract: We consider the existence of at least one positive solution to the discrete fractional equation , where and , equipped with a two-point boundary condition that can possibly be both non-local and nonlinear. Due to the fact that f is allowed to be negative for some values of t and y, we consider here the semipositone problem. In addition to discussing conditions under which this problem is guaranteed to have at least one positive solution for small values of , we provide an example to illustrate the use of our results. Due to the generality of our results, we include many boundary conditions as special cases such as the conjugate- and multipoint-type conditions.
TL;DR: In this article, the authors analyzed the linear stability of rectilinear compressible current-vortex sheets in two-dimensional isentropic magnetohydrodynamics, which is a free boundary problem with the boundary being characteristic.
Abstract: We analyze the linear stability of rectilinear compressible current-vortex sheets in two-dimensional isentropic magnetohydrodynamics, which is a free boundary problem with the boundary being characteristic. In the case when the magnitude of the magnetic field has no jump on the current-vortex sheets, we find a necessary and sufficient condition of linear stability for the rectilinear current-vortex sheets, showing that magnetic fields exert a stabilization effect on compressible vortex sheets. In addition, a loss of regularity with respect to the source terms, both in the interior domain and on the boundary, occurs in a priori estimates of solutions to the linearized problem for a rectilinear current-vortex sheet, as the Kreiss–Lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of frequency spaces. In this study, the construction of symmetrizers for a reduced differential system, which has poles at which the Kreiss–Lopatinskii condition may fail simultaneously, plays a crucial role in the a priori estimates.
TL;DR: This work proposes a new point selection method, based on an overlapping surface decomposition of the boundary, which is implicitly defined by a level set function and can be easily chosen to be locally uniform along a coordinate axis in two space dimensions and locally uniform in three space dimensions, which allows efficient numerical differentiation and boundary reconstruction/representation.
Journal Article•
TL;DR: In this paper, the existence of boundary value solution for fractional differential equation with p-Laplacain operator at resonance was discussed, by using coincidence degree theory and the extension of Mawhin's continuation theorem and constructing suitable continuous projectors.
Abstract: This paper discussed the existence of solution of boundary value for fractional differential equation with p-Laplacain operator at resonance.By using coincidence degree theory and the extension of Mawhin's continuation theorem and constructing suitable continuous projectors,some results of existence of solutions for boundary value problems were obtained,which enrich and support previous results.