Showing papers on "Free boundary problem published in 2017"

A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries

Abstract: In this paper we investigate a free boundary problem for the classical Lotka–Volterra type 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 $$t\rightarrow \infty $$ at both fronts and survives in the new environment, or it spreads within a bounded area 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.

A Roadmap to Well Posed and Stable Problems in Computational Physics

Abstract: All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier---Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

On a Frankl-type problem for a mixed parabolic-hyperbolic equation

Abstract: We state a new nonlocal boundary value problem for a mixed parabolic-hyperbolic equation. The equation is of the first kind, i.e., the curve on which the equation changes type is not a characteristic. The nonlocal condition involves points in hyperbolic and parabolic parts of the domain. This problem is a generalization of the well-known Frankl-type problems. Unlike other close publications, the hyperbolic part of the domain agrees with a characteristic triangle. We prove unique solvability of this problem in the sense of classical and strong solutions.

Fuzzy transform to approximate solution of two-point boundary value problems

Abstract: We present a numerical method based on the fuzzy transform for solving second-order differential equations with boundary conditions. We demonstrate the effectiveness of the method by some examples. Copyright © 2016 John Wiley & Sons, Ltd.

Boundary Estimation of Boundary Parameters for Linear Hyperbolic PDEs

Abstract: We propose an adaptive observer scheme to estimate boundary parameters in first-order hyperbolic systems of Partial Differential Equations (PDE). The considered systems feature an arbitrary number of states traveling in one direction and one counter-convecting state. Uncertainties in the boundary reflection coefficients and boundary additive errors are estimated relying on a pre-existing observer design and a novel Lyapunov-based adaptation law.

Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel

Abstract: This article examines questions of unique solvability for an inverse boundary value problem to recover the coefficient and boundary regime of a nonlinear integro-differential equation with degenerate kernel We propose a novel method of degenerate kernel for the case of inverse boundary value problem for the considered ordinary integro-differential equation of second order By the aid of denotation, the integro-differential equation is reduced to a system of algebraic equations Solving this system and using additional conditions, we obtained a system of two nonlinear equations with respect to the first two unknown quantities and a formula for determining the third unknown quantity We proved the single-value solvability of this system using the method of successive approximations

A novel derivation of the boundary term for the action in Lanczos–Lovelock gravity

Abstract: We present a novel derivation of the boundary term for the action in Lanczos–Lovelock gravity, starting from the boundary contribution in the variation of the Lanczos–Lovelock action. The derivation presented here is straightforward, i.e., one starts from the Lanczos–Lovelock action principle and the action itself dictates the boundary structure and hence the boundary term one needs to add to the action to make it well-posed. It also gives the full structure of the contribution at the boundary of the complete action, enabling us to read off the degrees of freedom to be fixed at the boundary, their corresponding conjugate momenta and the total derivative contribution on the boundary. We also provide a separate derivation of the Gauss–Bonnet case.

Existence of solutions for a mixed fractional boundary value problem

Abstract: In this paper, we prove the existence of solutions for a boundary value problem involving both left Riemann-Liouville and right Caputo-type fractional derivatives. For this, we convert the posed problem to a sum of two integral operators, then we apply Krasnoselskii’s fixed point theorem to conclude the existence of nontrivial solutions.

The motion of the rigid body in the viscous fluid including collisions. Global solvability result

Abstract: We shall consider the problem of the motion of a rigid body in an incompressible viscous fluid filling a bounded domain. This problem was studied by several authors. They mostly considered classical non-slip boundary conditions, which gave them very paradoxical result of no collisions of the body with the boundary of the domain. Only recently there are results when the Navier type of boundary is considered. In our paper we shall consider the Navier condition on the boundary of the body and the non-slip condition on the boundary of the domain. This case admits collisions of the body with the boundary of the domain. We shall prove the global existence of weak solution of the problem.

Hyers–Ulam Stability to a Class of Fractional Differential Equations with Boundary Conditions

Abstract: In this article, we study existence and uniqueness of a class of highly non-linear boundary value problem of fractional order differential equations. The concerned problem is investigated by means of classical fixed point theorem for the mentioned requirements. The Ulam–Hyer’s stability is also established for the class of fractional differential equations. Appropriate example is also provided which demonstrate the applicability of our results.

Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions

Abstract: In this study, we consider a new type boundary value problem consisting of a Sturm-Liouville equation on two disjoint intervals together with interaction conditions and with eigenvalue parameter in the boundary conditions. We suggest a special technique to reduce the considered problem into an integral equation by the use of which we define a new concept, the so-called weak eigenfunction for the considered problem. Then we construct some Hilbert spaces and define some self-adjoint compact operators in these spaces in such a way that the considered problem can be interpreted as a self-adjoint operator-pencil equation. Finally, it is shown that the spectrum is discrete and the set of weak eigenfunctions form a Riesz basis of the suitable Hilbert space.

Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem

Abstract: We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classication of free boundary points, the regularity of the regular set and the structure of the singular set.

On the Motion of Free Interface in Ideal Incompressible MHD

Abstract: For the free boundary problem of the plasma–vacuum interface to 3D ideal incompressible magnetohydrodynamics, the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div–curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangential to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the “fictitious particle” endowed with a fictitious velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The L2-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.

Slip Boundary Conditions for the Compressible Navier-Stokes Equations

Abstract: The slip boundary conditions for the compressible Navier–Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman–Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar–Gross–Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.

Analysis of a system modelling the motion of a piston in a viscous gas

Abstract: We study a free boundary problem modelling the motion of a piston in a viscous gas. The gas-piston system fills a cylinder with fixed extremities, which possibly allow gas from the exterior to penetrate inside the cylinder. The gas is modeled by the 1D compressible Navier–Stokes system and the piston motion is described by the second Newton’s law. We prove the existence and uniqueness of global in time strong solutions. The main novelty brought in by our results is that they include the case of nonhomogeneous boundary conditions which, as far as we know, have not been studied in this context. Moreover, even for homogeneous boundary conditions, our results require less regularity of the initial data than those obtained in previous works.

Relationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions

Abstract: We obtain for the two-phase Lame-Clapeyron-Stefan problem for a semi-infinite material an equivalence between the temperature and convective boundary conditions at the fixed face in the case that an inequality for the convective transfer coefficient is satisfied. Moreover, an inequality for the coefficient which characterizes the solid-liquid interface of the classical Neumann solution is also obtained. This inequality must be satisfied for data of any phase-change material, and as a consequence the result given in Tarzia, Quart. Appl. Math., 39 (1981), 491-497 is also recovered when a heat flux condition was imposed at the fixed face.

Constructing a basis from systems of eigenfunctions of one not strengthened regular boundary value problem

Abstract: We investigate a nonlocal boundary value spectral problem for an ordinary differential equation in an interval. Such problems arise in solving the nonlocal boundary value for partial equations by the Fourier method of variable separation. For example, they arise in solving nonstationary problems of diffusion with boundary conditions of Samarskii-Ionkin type. Or they arise in solving problems with stationary diffusion with opposite flows on a part of the interval. The boundary conditions of this problem are regular but not strengthened regular. The principal difference of this problem is: the system of eigenfunctions is comlplete but not forming a basis. Therefore the direct applying of the Fourier method is impossible. Based on these eigenfunctions there is constructed a special system of functions that already forms the basis. However the obtained system is not already the system of the eigenfunctions of the problem. We demonstrate how this new system of functions can be used for solving a nonlocal boundary value equation on the example of the Laplace equation.

Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model

Abstract: We give a complete description of the long-time asymptotic profile of the solution to a free boundary model considered recently in [ 10 ]. This model describes the spreading of an invasive species in an environment which shifts with a constant speed, and the research of [ 10 ] indicates that the species may vanish, or spread successfully, or fall in a borderline case.In the case of successful spreading, the long-time behavior of the population is not completely understood in [ 10 ].Here we show that the spreading of the species is governed by two traveling waves, one has the speed of the shifting environment, giving the profile of the retreating tail of the population, while the other has a faster speed determined by a semi-wave, representing the profile of the advancing front of the population.

Decomposition and Fictitious Domains Methods for Elliptic Boundary Value Problems

Abstract: Boundary value problems for elliptic second order equations in three-dimensional domains with piecewise smooth boundaries are considered. Discretization of the problem is performed using a conventional version of the nite element method with piecewise linear basis functions. The main purpose of the paper is the construction of a preconditioning operator for the resulting system of grid equations. The method is based on two approaches: decomposition of the domain into subdomains and using a new version of the method of ctitious domains. The rate of convergence of the corresponding conjugate gradient method is independent of both the grid size and the number of subdomains.

A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions

Abstract: In this paper, we introduce a new concept of coupled non-separated boundary conditions and solve a coupled system of fractional differential equations supplemented with these conditions. The existence results obtained in the given configuration are not only new but also yield some new special results corresponding to particular values of the parameters involved in the problem. For the illustration of the existence and uniqueness result, an example is constructed.

Infinitely many solutions for a class of fourth-order partially sublinear elliptic problem

Abstract: In this paper, we study the existence of infinitely many solutions for a class of fourth-order partially sublinear elliptic problem with Navier boundary value condition by using an extension of Clark’s theorem.