Solution Of Free Boundary Problem Research Articles

A high-order deferred correction method for the solution of free boundary problems using penalty iteration, with an application to American option pricing

This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved on a fixed computational grid, the order of the solution is low due to the discontinuity in the solution at the free boundary, even if a high-order method is used. Using a detailed error analysis, we observe that the order of convergence of the solution can be increased to fourth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. The corrections are applied solely to the right-hand side, and leave the finite difference matrix unchanged. The penalty iterations converge quickly given a good initial guess. We demonstrate the accuracy and efficiency of our algorithm using several examples. Numerical results show that our algorithm gives fourth-order convergence for both the solution and the free boundary location. We also test our algorithm on the challenging American put option pricing problem. Our algorithm gives the expected high-order convergence.

Existence of solution of a free boundary problem for reaction-diffusion systems

In this paper, we prove the existence of solution of a novel free boundary problem for reaction-diffusion systems describing growth of biological tissues due to cell influx and proliferation. For this aim, we transform it into a problem with fixed boundary, through a change of variables. The new problem thus obtained has space and time dependent coeffcients with nonlinear terms. We then prove the existence of solution for the corresponding linear problem, and deduce the existence of solution for the nonlinear problem using the xed point theorem. Finally, we return to the problem with free boundary to conclude the existence of its solution.

On the uniqueness and monotonicity of solutions of free boundary problems

For any smooth and bounded domain Ω⊂RN, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on Ω in a neat interval depending only by the best constant of the Sobolev embedding H01(Ω)↪L2p(Ω), p∈[1,NN−2) and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for p>1, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for p=1. The threshold, which is sharp for p=1, yields a new condition which guarantees that there is no free boundary inside Ω. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes.

On the dynamics of thin layers of viscous flows inside another viscous fluid

In this work we will study the dynamics of a thin layer of a viscous fluid which is embedded in the interior of another viscous fluid. The resulting flow can be approximated by means of the solutions of a free boundary problem for the Stokes equation in which one of the unknowns is the shape of a curve which approximates the geometry of the thin layer of fluid. We also derive the equation yielding the thickness of this fluid. This model, that will be termed as the Geometric Free Boundary Problem, will be derived using matched asymptotic expansions. We will prove that the Geometric Free Boundary Problem is well posed and the solutions of the thickness equation are well defined (in particular they do not yield breaking of fluid layers) as long as the solutions of the Geometric Free Boundary Problem exist.

Optimal stopping of random walk with exponential utility function

The article provides solutions for optimal stopping problems for finite and infinite horizons in the case of random walk with exponential utility function. For these problems we found: i) explicit solution of the corresponding Bellman equation; ii) solution of free boundary problem; iii) optimal stopping rule.

New universal estimates for free boundary problems arising in plasma physics

For $\Omega\subset \mathbb{R}^2$ a smooth and bounded domain, we derive a sharp universal energy estimate for non-negative solutions of free boundary problems on $\Omega$ arising in plasma physics. As a consequence, we are able to deduce new universal estimates for this class of problems. We first come up with a sharp positivity threshold which guarantees that there is no free boundary inside $\Omega$ or either, equivalently, with a sharp necessary condition for the existence of a free boundary in the interior of $\Omega$. Then we derive an explicit bound for the $L^{\infty}$-norm of non-negative solutions and also obtain explicit estimates for the thresholds relative to other neat density boundary values. At least to our knowledge, these are the first explicit estimates of this sort in the superlinear case.

Asymptotic Behavior of Solutions of Free Boundary Problems for Fisher-KPP Equation

We study a free boundary problem for Fisher-KPP equation: $$u_t=u_{xx}+f(u)$$ ( $$g(t)< x < h(t)$$ ) with free boundary conditions $$h'(t)=-u_x(t,h(t))-\beta $$ and $$g'(t)=-u_x(t,g(t))-\alpha $$ for $$\alpha >0$$ and $$\beta \in \mathbb {R}$$ . Such a free boundary problem can model the spreading of a biological or chemical species affected by the boundary environment. $$\beta >0$$ means that there is a “resistance force” with strength $$\beta $$ at boundary $$x=h(t)$$ . $$\beta <0$$ (resp. $$\alpha >0$$ ) means that there is an enhancing force with strength $$\beta $$ (resp. $$\alpha $$ ) at the boundary $$x=h(t)$$ (resp. g(t)). There are many parts of $$(\alpha ,\beta )$$ . In different parts, the asymptotic behavior of solutions are different. In the first part, we have a spreading-transition-vanishing result: either spreading happens (the solution converges to 1 in the moving frame), or in the transition case (the solution will converge to the compactly supported traveling wave), or vanishing happens (the solution converges to 0 within a finite time). In the second part, we also have a trichotomy result, but in transition case the solution will converge to the non-monotonous traveling semi-wave, and the vanishing case has three different types. For the third part, only spreading happens for any solution. In the fourth part ( $$\alpha $$ or $$\beta $$ large), any solution will vanish, also there are three types of vanishing. For the case $$\alpha = \beta $$ , we have two different trichotomy results and a dichotomy result.

Existence and uniqueness of solution of free boundary problem with partially degenerate diffusion

In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.

Existence and uniqueness of solutions of free boundary problems in heterogeneous environments

In this short paper we study the existence and uniqueness of solutions of free boundary problems coming from ecology in heterogeneous environments.

Asymptotic behaviour of solutions of free boundary problems for Fisher-KPP equation

We study a free boundary problem for the Fisher-KPP equation: ut = uxx + f(u) (g(t) &lt; x &lt; h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) − α and g′(t) = −ux(t, g(t)) + β for 0 &lt; β &lt; α. This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We investigate the asymptotic behaviour of bounded solutions. There are two parameters α0 and α* with 0 &lt; α0 &lt; α* which play key roles in the dynamics. More precisely, (i) in case 0 &lt; β &lt; α0 and 0 &lt; α &lt; α*, we obtain a trichotomy result: (i-1) spreading, i.e., h(t) − g(t) → +∞ and u(t, ⋅ + ct) → 1 with c ∈ (cL, cR), where cL and cR are the asymptotic spreading speed of g(t) and h(t), respectively, (cR &gt; 0 &gt; cL when 0 &lt; β &lt; α &lt; α0; cR = 0 &gt;cL when 0 &lt; β &lt; α = α0; 0 &gt; cR &gt; cL when α0 &lt; α &lt; α* and 0 &lt; β &lt; α0); (i-2) vanishing, i.e., limt→Th(t) = limt→Tg(t) and limt→T u(t, x) = 0, where T is some positive constant; (i-3) transition, i.e., g(t) → −∞, h(t) → −∞, 0 &lt; limt→∞[h(t) − g(t)] &lt; +∞ and u(t, x) → V*(x − c*t) with c* &lt; 0, where V*(x − c*t) is a travelling wave with compact support and which satisfies the free boundary conditions. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

Hölder space solutions of free boundary problems that arise in combustion theory

Hölder space solutions of free boundary problems that arise in combustion theory

Effect of Current Efficiency on Electrochemical Micromachining by Moving Electrode

In this work, the effect of current efficiency on the electrochemical micromachining by moving electrode is studied theoretically. The Laplace equation for the electric potential and the equation of workpiece surface evolution are used as the mathematical model of the process. A new scheme of solution of free boundary problem for steady-state electrochemical micromachining is proposed. According to the scheme, the initial approximation of the workpiece surface is prescribed. In the course of modeling, the workpiece surface moves in the normal direction at a rate proportional to the discrepancy of the steady-state condition. The effect of various dependences of current efficiency on the local current density is analyzed. As a result of simulation, the dependences of the shape and sizes of machined surface on the current efficiency and the machining parameters are obtained.

Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction–diffusion equations

We discuss free boundary problems modeling the diffusion of invasive or new species in a multi-dimensional ball or annulus, where unknown functions are population density and the outer boundary representing the spreading front of the species. We define spreading and vanishing to describe the asymptotic behaviors of radially symmetric solutions. The main purpose is to study the underlying principle to determine spreading and vanishing for the free boundary problems of general reaction–diffusion equations. We also focus on the problems with a logistic or bistable reaction term to show dichotomy results, vanishing speed and sufficient conditions for spreading or vanishing.

Asymptotic Behavior of Solutions of Free Boundary Problem with Logistic Reaction Term

We study a free boundary problem for a reaction diffusion equation modeling the spreading of a biological or chemical species. In this model, the free boundary represents the spreading front of the species. We discuss the asymptotic behavior of bounded solutions and obtain a trichotomy result: spreading (the free boundary tends to+∞and the solution converges to a stationary solution defined on[0+∞)), transition (the free boundary stays in a bounded interval and the solution converges to a stationary solution with positive compact support), and vanishing (the free boundary converges to 0 and the solution tends to 0 within afinitetime).

Global Uniqueness of Transonic Shocks in Two-Dimensional Steady Compressible Euler Flows

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C 1 smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing a point where the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L ∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of the shock polar applied on the (maybe curved) shock-fronts.

Using adjoint error estimation techniques for elastohydrodynamic lubrication line contact problems

The use of an adjoint technique for goal-based error estimation described by Hartit et al. (Int. J. Numer. Meth. Fluids 2005; 47:1069–1074) is extended to the numerical solution of free boundary problems that arise in elastohydrodynamic lubrication (EHL). EHL systems are highly nonlinear and consist of a thin-film approximation of the flow of a non-Newtonian lubricant which separates two bodies that are forced together by an applied load, coupled with a linear elastic model for the deformation of the bodies. A finite difference discretization of the line contact flow problem is presented, along with the numerical evaluation of an exact solution for the elastic deformation, and a moving grid representation of the free boundary that models cavitation at the outflow in this one-dimensional case. The application of a goal-based error estimate for this problem is then described. This estimate relies on the solution of an adjoint problem; its effectiveness is demonstrated for the physically important goal of the total friction through the contact. Finally, the application of this error estimate to drive local mesh refinement is demonstrated. Copyright © 2010 John Wiley & Sons, Ltd.

Mathematical model of an incompressible viscoelastic Maxwell medium

Nonstationary motions of incompressible viscoelastic Maxwell continuum with a constant relaxation time are considered. Because in an incompressible continuous medium, pressure is not a thermodynamic variable but coincides with the stress-tensor trace to within a factor, it follows that, separating the spherical part from this tensor, one can assume that the remaining part of the stress tensor has zero trace. In the case of an incompressible medium, the equations for the velocity, pressure, and stress tensor form a closed system of first-order equations which has both real and complex characteristics, which complicates the formulation of the initial-boundary-value problem. Nevertheless, the resolvability of the Cauchy problem can be proved in the class of analytic functions. Unique resolvability of the linearized problem was established in the classes of functions of finite smoothness. The class of effectively one-dimensional motions for which the subsystem of three equations is a hyperbolic one was studied. The results of an asymptotic analysis of the latter imply the possible formation of discontinuities during the evolution of the solution. The general system of equations of motion admits an infinite-dimensional Lie pseudo-group which contains an extended Galilean group. The theorem of the invariance of the conditions on the a priori unknown free boundary was proved to obtain exact solutions of free-boundary problems. The problem of deformation of a viscoelastic strip subjected to tangential stresses applied to the free boundary is considered as an example of application of this theorem. In this problem, a scale effect of short-wave instability caused by the absence of diagonal dominance of the stress tensor deviator was found.

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial–boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.

Perpetual convertible bonds with credit risk

A convertible bond is a security that the holder can convert into a specified number of underlying shares. We enrich the standard model by introducing some default risk of the issuer. Once default has occured payments stop immediately. In the context of a reduced form model with infinite time horizon driven by a Brownian motion, analytical formulae for the no-arbitrage price of this American contingent claim are obtained and characterised in terms of solutions of free boundary problems. It turns out that the default risk changes the structure of the optimal stopping strategy essentially. Especially, the continuation region may become a disconnected subset of the state space.

The Solution of a Free Boundary Problem Related to Environmental Management Systems

Explicit solutions of free boundary problems are notoriously difficult to find. In this article, we consider two log-normal diffusions. One represents the level of pollution, or degradation, in some environmental area. The second models the social, political, or financial cost of the pollution. A single control parameter is considered that reduces the rate of pollution. The optimal time to implement the change in the parameter is found by explicitly solving a free boundary problem. The novelty is that the smooth pasting conditions, which are difficult to justify, are not used in the derivation.