Showing papers on "Free boundary problem published in 2015"
TL;DR: In this paper, the authors studied nonlinear diffusion problems of the form ut = uxx+f(u) with free boundaries and showed that the omega limit set of every bounded positive solution is determined by a stationary solution.
Abstract: We study nonlinear diffusion problems of the form ut = uxx+f(u) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special f(u) of the Fisher-KPP type, the problem was investigated by Du and Lin [8]. Here we consider much more general nonlinear terms. For any f(u) which is C and satisfies f(0) = 0, we show that the omega limit set ω(u) of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter σ in the initial data, we reveal a threshold value σ∗ such that spreading (limt→∞ u = 1) happens when σ > σ∗, vanishing (limt→∞ u = 0) happens when σ < σ ∗, and at the threshold value σ∗, ω(u) is different for the three different types of nonlinearities. When spreading happens, we make use of “semi-waves” to determine the asymptotic spreading speed of the front.
218 citations
Dissertation•
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01 Jul 2015
TL;DR: In this article, the existence and uniqueness of solution to the Stefan problem were investigated. But the results on the uniqueness of the solution were not discussed. And the theoretical development and mathematical formulation of supercooled Stefan problems were not considered.
Abstract: This bachelor thesis deals with the Stefan problem, from its historical background to the existence and uniqueness of solution to the problem. The physical background is presented at the beginning. Afterwards there are presented some results related to the problem, like explicit solutions and an analysis of the technique for obtaining solutions to the problem by perturbation methods. Also the theoretical development and mathematical formulation of supercooled Stefan problems is included in this part. Finally, results on existence and uniqueness for the Stefan problems are shown. In particular, the cases treated here are the ones concerning small and large times, both for Dirichlet boundary conditions, and Neumann boundary conditions for small times.
154 citations
TL;DR: In this article, the local-in-time well-posedness of three-dimensional compressible Euler equations for polytropic gases with a physical vacuum was established by considering the problem as a free boundary problem.
Abstract: An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of a vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only a few mathematical results available near a vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local-in-time well-posedness of three-dimensional compressible Euler equations for polytropic gases with a physical vacuum by considering the problem as a free boundary problem. © 2015 Wiley Periodicals, Inc.
147 citations
TL;DR: In this paper, the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to coupled integral boundary conditions, are investigated.
Abstract: Abstract We investigate the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to coupled integral boundary conditions. The nonsingular and singular cases are studied.
97 citations
TL;DR: In this article, the authors considered a radially symmetric free boundary problem with logistic nonlinear term, where the spatial environment is assumed to be asymptotically periodic at infinity in the radial direction.
82 citations
TL;DR: In this paper, it was shown that stable cones for the one-phase free boundary problem are hyperplanes in dimension 4, and that both one and two-phase energy minimizing hypersurfaces are smooth.
Abstract: We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension 4. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension 4.
81 citations
TL;DR: In this article, the singular limit of a diffuse-interface tumor growth model is characterized, and it is shown that as the coefficient of the reaction term tends to infinity, the solution converges to the solution of a free boundary problem.
Abstract: We consider a diffuse-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coefficient of the reaction term tends to infinity, the solution converges to the solution of a novel free boundary problem. We present numerical simulations which illustrate the convergence of the diffuse-interface model to the identified sharp-interface limit.
81 citations
TL;DR: In this article, Lyapunov-type inequalities are established for a fractional differential equation under mixed boundary conditions, and intervals where certain MittagLeffler functions have no real zeros are obtained.
Abstract: Lyapunov-type inequalities are established for a fractional differential equation under mixed boundary conditions. Using such inequalities, we obtain intervals where certain MittagLeffler functions have no real zeros. Mathematics subject classification (2010): 34A08, 34A40, 26D10, 33E12.
78 citations
TL;DR: In this paper, it is shown that the solution of the coupled nonlinear Schrodinger equation on the half-line can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex k-plane.
Abstract: Initial-boundary value problems for the coupled nonlinear Schrodinger equation on the half-line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrodinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so-called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.
76 citations
TL;DR: In this article, a fractional differential equation with initial and three-point boundary conditions is studied, where the concept of fractional derivative is introduced to obtain Green's function for the linear problem and then the nonlinear differential equation is studied.
Abstract: We study a fractional differential equation using a recent novel concept of fractional derivative with initial and three-point boundary conditions. We first obtain Green's function for the linear problem and then we study the nonlinear differential equation.
76 citations
TL;DR: In this article, the authors established new Lyapunov-type inequalities for a class of fractional boundary value problems and obtained a lower bound for the eigenvalues of corresponding equations.
Abstract: In this paper, we establish new Lyapunov-type inequalities for a class of fractional boundary value problems. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.
TL;DR: The derivation of stability estimates is based on a strong positivity property of the Calderón boundary operators for the Helmholtz and wave equations and uses energy estimates both in time and frequency domain, and yields optimal-order error bounds of the full discretization.
Abstract: The acoustic wave equation on the whole three-dimensional space is considered with initial data and inhomogeneity having support in a bounded domain, which need not be convex. We propose and study a numerical method that approximates the solution using computations only in the interior domain and on its boundary. The transmission conditions between the interior and exterior domain are imposed by a time-dependent boundary integral equation coupled to the wave equation in the interior domain. We give a full discretization by finite elements and leapfrog time-stepping in the interior, and by boundary elements and convolution quadrature on the boundary. The direct coupling becomes stable on adding a stabilization term on the boundary. The derivation of stability estimates is based on a strong positivity property of the Calderon boundary operators for the Helmholtz and wave equations and uses energy estimates both in time and frequency domain. The stability estimates together with bounds of the consistency error yield optimal-order error bounds of the full discretization.
TL;DR: In this article, the authors considered the spreading of a thin two-dimensional droplet on a solid substrate and used a model for viscous fluids where the evolution is governed by Darcy's law.
Abstract: We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy’s law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $${\varepsilon}$$
.
TL;DR: By using the endpoint result for multifunctions, the existence of solutions for a boundary value problem for fractional differential inclusions with sum and integral boundary conditions is investigated.
TL;DR: In this article, a boundary value problem of Caputo type sequential fractional differential equations supplemented with nonlocal Riemann-Liouville fractional integral boundary conditions is investigated and existence results for the given problem are obtained via standard tools of fixed point theory and are well illustrated with the aid of examples.
Abstract: This paper investigates a boundary value problem of Caputo type sequential fractional differential equations supplemented with nonlocal Riemann-Liouville fractional integral boundary conditions. Some existence results for the given problem are obtained via standard tools of fixed point theory and are well illustrated with the aid of examples. Some special cases are also presented.
TL;DR: In this article, the authors proved the nonlinear stability of Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent lies in the stability regime.
Abstract: The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent $\gamma$ lies in the stability regime $(4/3, 2)$, by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. The results obtained in this paper extend those in \cite{LXZ} of the authors for the constant viscosities to the case of density dependent viscosities which are degenerate at vacuum states.
TL;DR: In this article, Wu et al. proposed the local weak form meshless methods for option pricing; especially in this paper, they select and analyze two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI), and they use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system.
Book•
26 Oct 2015
TL;DR: Boundary value problems for systems of Differential, Difference and Fractional Equations: Positive Solutions as discussed by the authors discusses the concept of a differential equation that brings together a set of additional constraints called the boundary conditions.
Abstract: Boundary Value Problems for Systems of Differential, Difference and Fractional Equations: Positive Solutions discusses the concept of a differential equation that brings together a set of additional constraints called the boundary conditions. As boundary value problems arise in several branches of math given the fact that any physical differential equation will have them, this book will provide a timely presentation on the topic. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed. * Explains the systems of second order and higher orders differential equations with integral and multi-point boundary conditions* Discusses second order difference equations with multi-point boundary conditions* Introduces Riemann-Liouville fractional differential equations with uncoupled and coupled integral boundary conditions
TL;DR: Two-point boundary value problems for conservative systems are studied in the context of the least action principle, whereby two-point boundaries value problems are converted to initial value problems via an idempotent convolution of the fundamental solution with a cost function related to the terminal data.
Abstract: Two-point boundary value problems for conservative systems are studied in the context of the least action principle. One obtains a fundamental solution, whereby two-point boundary value problems are converted to initial value problems via an idempotent convolution of the fundamental solution with a cost function related to the terminal data. The classical mass-spring problem is included as a simple example. The $N$-body problem under gravitation is also studied. There, the least action principle optimal control problem is converted to a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. Solutions are obtained as indexed sets of solutions of Riccati equations.
TL;DR: In this paper, a free boundary problem for a spherically symmetric tumor with free boundary r 0 if liminft→∞α(t)>0 was considered and the global asymptotic stability of steady state when μ is sufficiently small was shown.
TL;DR: In this article, the authors approximate a two-phase model by the compressible Navier-Stokes equations with a singular pressure term up to a subsequence, these solutions are shown to converge to a global weak solution of the compressed system with the congestion constraint studied for instance by Lions and Masmoudi.
Abstract: We approximate a two–phase model by the compressible Navier-Stokes equations with a singular pressure term. Up to a subsequence, these solutions are shown to converge to a global weak solution of the compressible system with the congestion constraint studied for instance by Lions and Masmoudi. The paper is an extension of the previous result obtained in one-dimensional setting by Bresch et al. to the multi-dimensional case with heterogeneous barrier for the density.
TL;DR: In this article, the problem of heat transfer in the 3D domain of heating tissue is described by dual-phase lag equation supplemented by adequate boundary and initial conditions, and the general boundary element method is proposed.
Abstract: Heat transfer processes proceeding in the 3D domain of heating tissue are discussed. The problem is described by dual-phase lag equation supplemented by adequate boundary and initial conditions. To solve the problem the general boundary element method is proposed. The examples of computations are presented in the final part of the paper. The efficiency and exactness of the algorithm proposed are discussed and the conclusions are also formulated.
TL;DR: In this article, the inverse boundary value problem for a 3D nonlinear elliptic partial differential equation in a bounded domain is considered and a truncation approach is developed to approximate the solution of the ill-posed problem in a regularization manner.
TL;DR: In this paper, a simplified model of plaque growth involving LDL and HDL cholesterols, macrophages and foam cells is proposed, which satisfy a coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow.
TL;DR: In this paper, the authors proposed an optimum design method for two-dimensional heat conduction problem with heat transfer boundary condition based on the boundary element method (BEM) and the topology optimization method.
Abstract: This paper proposes an optimum design method for two-dimensional heat conduction problem with heat transfer boundary condition based on the boundary element method (BEM) and the topology optimization method. The level set method is used to represent the structural boundaries and the boundary mesh is generated based on iso-surface of the level set function. A major novel aspect of this paper is that the governing equation is solved without ersatz material approach and approximated heat convection boundary condition by using the mesh generation. Additionally, the objective functional is defined also on the design boundaries. First, the topology optimization method and the level set method are briefly discussed. Using the level set based boundary expression, the topology optimization problem for the heat transfer problem with heat transfer boundary condition is formulated. Next, the topological derivative of the objective functional is derived. Finally, several numerical examples are provided to confirm the validity of the derived topological derivative and the proposed optimum design method.
TL;DR: Small perturbations of source-type self-similar solutions for the corresponding classical free boundary problem are studied and a global existence and uniqueness theory within weighted $L^2$-spaces under minimal assumptions is set up.
Abstract: We investigate compactly supported solutions for a thin-film equation with linear mobility in the regime of perfect wetting. This problem has already been addressed by Carrillo and Toscani, proving that the source-type self-similar profile is a global attractor of entropy solutions with compactly supported initial data. Here we study small perturbations of source-type self-similar solutions for the corresponding classical free boundary problem and set up a global existence and uniqueness theory within weighted $L^2$-spaces under minimal assumptions. Furthermore, we derive asymptotics for the evolution of the solution, the free boundary, and the center of mass. As spatial translations are scaled out in our reference frame, the rate of convergence is higher than the one obtained by Carrillo and Toscani.
Posted Content•
TL;DR: In this article, it was shown that for small perturbations of a Lane-Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary provided that $\gamma$ lies in $(4/3,
Abstract: This paper proves the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant $\gamma$ lies in the stability range $(4/3, 2)$. It is shown that for small perturbations of a Lane-Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary provided that $\gamma$ lies in $(4/3, 2)$. The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane-Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary.
Book•
17 Sep 2015
TL;DR: The Fokker-Planck Equation and the Stefan free boundary problem as mentioned in this paper have been used to solve the problem of finding the optimal solution of the problem in the presence of spiking spikes, spots, spikes and pulses.
Abstract: 1.Parabolic Equations in Biology.- 2.Relaxation, Perturbation and Entropy Methods.- 3.Weak Solutions of Parabolic Equations in whole Space.- 4.Traveling Waves.- 5.Spikes, Spots and Pulses.- 6.Blow-up and Extinction of Solutions.- 7.Linear Instability, Turing Instability and Pattern Formation.- 8.The Fokker-Planck Equation.- 9.From Jumps and Scattering to the Fokker-Planck Equation.- 10.Fast Reactions and the Stefan free Boundary Problem.
TL;DR: In this paper, the authors established the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier-Stokes equations for which the smoothness extends all the way to the boundaries.
Abstract: In this paper we establish the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier–Stokes equations for which the smoothness extends all the way to the boundaries. The results obtained in this work include the physical vacuum for which the sound speed is C1/2-Holder continuous near the vacuum boundaries when 1 < γ < 3. The novelty of this result is its global-in-time regularity which is in contrast to the previous main results of global weak solutions in the literature. Moreover, in previous studies of the one-dimensional free boundary problems of compressible Navier–Stokes equations, the Lagrangian mass coordinates method has often been used, but in the present work the particle path (flow trajectory) method is adopted, which has the advantage that the particle paths and, in particular, the free boundaries can be traced.
TL;DR: In this paper, an analytical solution of the phase change problem, known as the Moving Boundary Problem, in a PCM layer (phase change materials) subject to boundary conditions that are variable in time, is presented, in steady periodic regime.