Free Boundary Formulation Research Articles

Free boundary formulations for two extended Blasius problems

In this paper, we have defined the free boundary formulation for two extended Blasius problems. These problems are of interest in boundary layer theory and are deduced from the governing partial differential equations by using appropriate similarity variables. The computed results, for the so-called missing initial condition, are favorably compared with recent results available in the literature.

A Note on the Pricing of American Capped Power Put Option

We give an explicit solution to the perpetual American capped power put option pricing problem in the BlackScholes-Merton Model. The approach is mainly based on free-boundary formulation and verification. For completeness we also give an explicit solution to the perpetual American standard power (_ 1) option pricing problem.

Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation Methods

This paper is concerned with two examples on the application of the free boundary formulation to BVPs on a semi-infinite interval. In both cases we are able to provide the exact solution of both the BVP and its free boundary formulation. Therefore, these problems can be used as benchmarks for the numerical methods applied to BVPs on a semi-infinite interval and to free BVPs. Moreover, we emphasize how for two classes of free BVPs, we can define non-iterative initial value methods, whereas BVPs are usually solved iteratively. These non-iterative methods can be deduced within Lie's group invariance theory. Then, we show how to apply the non-iterative methods to the two introduced free boundary formulations in order to obtain meaningful numerical results. Finally, we indicate several problems from the literature where our non-iterative transformation methods can be applied.

Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals

The classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary. A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach. On the other hand, the free boundary approach overcomes the need for a priori definition of the truncated boundary. In fact, in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution.In this paper we consider a different way to overcome the introduction of a truncated boundary, namely non-standard finite difference schemes defined on quasi-uniform grids. A quasi-uniform grid allows us to describe the infinite domain by a finite number of intervals. The last node of such grid is placed on infinity so that right boundary conditions are taken into account exactly. We apply the proposed approach to the Falkner–Skan model and to a problem of interest in foundation engineering. The obtained numerical results are found in good agreement with those available in literature. Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson’s extrapolation. Finally, we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line.

Existence and uniqueness of weak solutions for precipitation fronts: A novel hyperbolic free boundary problem in several space variables

AbstractThe determination of the large‐scale boundaries between moist and dry regions is an important problem in contemporary meteorology. These phenomena have been addressed recently in a simplified tropical climate model through a novel hyperbolic free boundary formulation yielding three families (drying, slow moistening, and fast moistening) of precipitation fronts. The last two wave types violate Lax's shock inequalities yet are robustly realized. This formal hyperbolic free boundary problem is given here a rigorous mathematical basis by establishing the existence and uniqueness of suitable weak solutions arising in the zero relaxation limit. A new L2‐contraction estimate is also established at positive relaxation values. © 2010 Wiley Periodicals, Inc.

On the moving boundary formulation for parabolic problems on unbounded domains

The aim of this paper is to propose an original numerical approach for parabolic problems whose governing equations are defined on unbounded domains. We are interested in studying the class of problems admitting invariance property to Lie group of scalings. Thanks to similarity analysis the parabolic problem can be transformed into an equivalent boundary value problem governed by an ordinary differential equation and defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [R. Fazio, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal. 33 (1996), pp. 1473–1483]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either a noniterative, or an iterative method. Finally, the solution of the parabolic problem is retrieved by applying the inverse map of similarity.

Optimal Stopping of Integral Functionals and a “No-Loss” Free Boundary Formulation

This paper is concerned with a modification of the classical formulation of the free boundary problem for the optimal stopping of integral functionals of one-dimensional diffusions with, possibly, irregular coefficients. This modification was introduced in [L. Ruschendorf and M. A. Urusov, Ann. Appl. Probab., 18 (2008), pp. 847–878]. As a main result of that paper a verification theorem was established. Solutions of the modified free boundary problem imply solutions of the optimal stopping problem. The main contribution of this paper is to establish the converse direction. Solutions of the optimal stopping problem necessarily also solve the modified free boundary problem. Thus the modified free boundary problem is also necessary and does not “lose” solutions. In particular, we prove smooth fit in our situation. In the final part of this paper we discuss related questions for the viscosity approach and describe an advantage of the modified free boundary formulation.

An iterative method for solving the Falkner–Skan equation

We propose a new iterative method to solve the boundary value problems (BVPs) of the Falkner–Skan equation over a semi-infinite interval. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method to transform the BVP into initial value problems (IVPs). In order to find the “shooting angle” and the unknown free boundary, a modification of the classical Newton’s method is used where the Jacobian matrix can be accurately obtained by solving another two IVPs. To illustrate the effectiveness of our method, we compare our numerical results with those obtained by previous methods under various instances of the Falkner–Skan equation.

Optimal stopping of integral functionals and a “no-loss” free boundary formulation

Optimal stopping of integral functionals and a “no-loss” free boundary formulation

A FREE-BOUNDARY MODEL FOR ANAEROBIOSIS IN SATURATED SOIL AGGREGATES

A new model for the diffusion of oxygen in spherical aggregates of soils with simultaneous consumption at constant, linear, and nonlinear rate is proposed. To evaluate the effects of the dynamics of soil domain in which the diffusion and consumption takes place, a free-boundary formulation is studied. The three resultant free-boundary models are solved by a modification of the Constrained Integral Method. The obtained results are consistent with those presented by earlier authors who have solved the problems using fixed domains. The results obtained by our model for diffusion with constant consumption slightly overpredict the experimental values with respect to the results obtained by Currie (J Currie. Gaseous diffusion in the aeration of aggregated soil. Soil Sci. 1961;92:40-45) in fixed domains. Moreover, the results obtained by our model for diffusion with simultaneous nonlinear consumption overpredict the experimental values, partially improving the underpredicted results of Sierra et al. (J Sierra, P Renault, and V Valles. Anaerobiosis in saturated soil aggregates: modelling and experiment. Eur J Soil Sci. 1995;46:519-531) in fixed domains. Thus, our model could represent an alternative method to study and to generalize existing models.

Numerical Tests of a Phase Field Model with Second Order Accuracy

Numerical computations are performed for a recently derived phase field model for the interface between two phases. The rigorous results indicate that solutions to this new phase field model should converge more rapidly than traditional ones to solutions of the corresponding sharp interface (free boundary) formulation for sufficiently small values of the approximation parameter $\varepsilon $ representing the thickness of the interfacial region. In particular, the distance between the sharp interface of the limiting model and the zero level set of the phase function in the phase field model is of order $\varepsilon^2$ rather than $\varepsilon$. Numerical computations within a three-dimensional spherically symmetric setting compare the computed solutions of this new model with the known exact solutions for the limiting free boundary problem and confirm the second order accuracy predictions of the theory for sufficiently small $\varepsilon$. The sets of parameters include those of succinonitrile used in dendritic experiments.

On the moving boundary formulation for parabolic problems on unbounded domains

AbstractA parabolic problem defined on unbounded domains and admitting invariance property to Lie group of scalings can be transformed into an equivalent boundary value problem governed by an ODE defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [Fazio, SIAM J. Numer. Anal., 33 (1996), pp. 1473‐1484]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either an iterative or a non‐iterative method. Finally, the solution of the parabolic problem is retrieved applying the inverse map of similarity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Obstacle problems for a coupling of quasilinear hyperbolic-parabolic equations

set in a parabolic area Qp = ]0, T [ × (Ω Ωh), complementary to the former, and for suitable conditions across the interface between the two regions Qp and Qh. The geometrical configuration is such that:Ω = Ωh∪Ωp;Ωh andΩp are two disjoint bounded domains with Lipschitz boundaries denoted Γl , for l in {h, p}. In addition, the interface Γhp = Γh ∩ Γp is Lipschitz and such that Hn−1(Γ hp ∩ Γl Γhp) = 0, where for q in [0, n + 1], H denotes the q-dimensional Hausdorff measure. For a given threshold θ , the (bilateral) obstacle problem for Th and Tp may be formally written in the free boundary formulation: find a bounded measurable function u on Q ≡ ]0, T [ ×Ω such

Optimal stopping and American options with discrete dividends and exogenous risk

In this paper we analyze some problems arising in the evaluation of American options when the underlying security pays discrete dividends. To this aim, we study the problem of maximizing the expected gain process over stopping times taking values in the union of disjoint, real compact sets. The results we obtain can be applied to evaluate options with restrictions on exercise periods, but are also useful for the evaluation of American options on assets that pay discrete dividends. In particular, we generalize the evaluation formula for American call options due to Whaley [Journal of Financial Economics 9 (1981) 207], allowing for a stochastic jump of the underlying security at the ex-dividend date and discuss the existence of the optimal stopping time. In the same framework, we analyze American put options, justifying the procedure used in Meyer [Journal of Computational Finance 5 (2) (2002)] to account for the presence of discrete dividends in the free boundary formulation from the perspective of optimal stopping theory.

A variational approach to nonlinear dynamics of nanoscale surface modulations

In this paper, we propose a variational formulation to study the singular evolution equations that govern the dynamics of surface modulations on crystals below the roughening temperature. The basic idea of the formulation is to expand the surface shape in terms of a complete set of basis functions and to use a variational principle equivalent to the continuum evolution equations to obtain coupled nonlinear ordinary differential equations for the expansion coefficients. Unlike several earlier approaches that rely on ad hoc regularization procedures to handle the singularities in the evolution equations, the only inputs required in the present approach are the orientation dependent surface energies and the diffusion constants. The method is applied to study the morphological equilibration of patterned unidirectional and bidirectional sinusoidal modulations through surface diffusion. In the case of bidirectional modulations, particular attention is given to the analysis of the profile decay as a function of ratio of the modulating wavelengths in the coordinate directions. A key question that we resolve is whether the one-dimensional decay behavior is recovered as one of the modulating wavelengths of the two-dimensional profiles diverges, or whether one-dimensional decay has qualitatively distinct features that cannot be described as a limiting case of the two-dimensional behavior. In contrast to some earlier suggestions, our analytical and numerical studies clearly show that the former situation is true; we find that the one-dimensional profiles, like the highly elongated two-dimensional profiles, decay with formation of facets. While our results for the morphological equilibration of symmetric one-dimensional profiles are in agreement with the free-boundary formulation of Spohn, the present approach can also be used to study the evolution of asymmetric profile shapes where the free-boundary approach is difficult to apply. The variational method is also used to analyze the decay of unidirectional modulations in the presence of steps that arise in most experimental studies due to a small misorientation from the singular surface.

A survey on free boundary identification of the truncated boundary in numerical BVPs on infinite intervals

A free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in Fazio (SIAM J. Numer. Anal. 33 (1996) 1473). We consider here a survey on recent developments related to the free boundary identification of the truncated boundary. The goals of this survey are: to recall the reasoning for a free boundary identification of the truncated boundary, to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature, and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line.

A numerical study of chemical front propagation in a Hele-Shaw flow under buoyancy effects

We consider the propagation of chemical fronts in a Hele-Shaw flow where the front is assumed to propagate with a curvature dependent velocity. The motivation is to model some recent experiments that employ aqueous autocatalytic chemical reactions in such a device. The density change across the front in such experiments is quite small so the Boussinesq approximation can be used, and the flow field generated is exclusively due to buoyancy effects. We derive a free boundary formulation based on Darcy’s law and potential theory, and describe the evolution in terms of the θ−L formulation, in which the tangent angle and the perimeter of the closed front are followed in time. Numerical solutions are obtained for this formulation with a rising and expanding bubble. As observed in the experiments, a fingering phenomenon which is different from the surface tension associated phenomenon appears in our calculations. The mechanisms that control the wavelength selection of the fingers, and a comparison with the result of a linear stability analysis for flat fronts are discussed.

Oscillatory instabilities in rapid directional solidification

The rapid directional solidification of a binary alloy is studied with the phase-field model. An oscillatory instability is evidenced which originates alternating low and high concentration solute bands. This instability, which is related to the banded structure observed in rapid solidification experiments, is predicted by previous studies conducted within the free-boundary formulation of the problem, incorporating nonequilibrium effects as solute trapping. In the present paper the governing equations of the model are numerically solved to show under what conditions (i.e., isotherm velocity and temperature gradient) the banded structure can be observed.

A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite Intervals

The classical numerical treatment of two-point boundary value problems on infinite intervals is based on the introduction of a truncated boundary (instead of infinity) where appropriate boundary conditions are imposed. Then, the truncated boundary allowing for a satisfactory accuracy is computed by trial. Motivated by several problems of interest in boundary layer theory, here we consider boundary value problems on infinite intervals governed by a third-order ordinary differential equation. We highlight a novel approach to define the truncated boundary. The main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero. In the proposed formulation, the truncated boundary is an unknown free boundary and has to be determined as part of the solution. For the numerical solution of the free boundary formulation, a noniterative and an iterative transformation method are introduced. Furthermore, we characterize the class of free boundary value problems that can be solved noniteratively. A nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved. Moreover, the Falkner–Skan equation with relevant boundary conditions is considered and representative results, obtained by the iterative transformation method, are listed for the Homann flow. All the obtained numerical results clearly indicate the effectiveness of our approach. Finally, we discuss the possible extensions of the proposed approach and for the question of a priori error analysis.

The falkneer-skan equation: Numerical solutions within group invariance theory

The iterative transformation method, defined within the framework of the group invariance theory, is applied to the numerical solution of the Falkner-Skan equation with relevant boundary conditions. In this problem a boundary condition at infinity is imposed which is not suitable for a numerical use. In order to overcome this difficulty we introduce a free boundary formulation of the problem, and we define the iterative transformation method that reducess the free boundary formulation to a sequence of initial value problems. Moreover, as far as the value of the wall shear stress is concerned we propose a numerical test of convergence. The usefulness of our approach is illustrated by considering the wall shear stress for the classical Homann and Hiemenz flows. In the Homann's case we apply the proposed numerical test of convergence, and meaningful numerical results are listed. Moreover, for both cases we compared our results with those reported in literature.