Showing papers on "Free boundary problem published in 1980"
TL;DR: In this article, a sequence of radiating boundary conditions is constructed for wave-like equations, and it is proved that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r exp -m-1/2) for the m-th boundary condition.
Abstract: In the numerical computation of hyperbolic equations it is not practical to use infinite domains; instead, the domain is truncated with an artificial boundary. In the present study, a sequence of radiating boundary conditions is constructed for wave-like equations. It is proved that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r exp -m-1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature of the boundary conditions.
999 citations
TL;DR: In this article, a differentiation with respect to the domain in boundary value problems is presented, where the domain is defined as the domain of the boundary value problem and boundary value is defined.
Abstract: (1980). Differentiation with Respect to the Domain in Boundary Value Problems. Numerical Functional Analysis and Optimization: Vol. 2, No. 7-8, pp. 649-687.
394 citations
TL;DR: In this article, a nonreflecting boundary condition is presented for numerical solution of the time-dependent compressible Navier-Stokes equations when these equations are used to obtain a steady state.
371 citations
TL;DR: In this article, boundary layer equations for the class of non-Newtonian fluids termed pseudoplastic are examined under the classical conditions of uniform flow past a semi-infinite flat plate.
Abstract: The boundary layer equations for the class of non-Newtonian fluids termed pseudoplastic are examined under the classical conditions of uniform flow past a semi-infinite flat plate. The adoption of Crocco variables results in a nonlinear, singular boundary value problem for the shear function which is an interesting and natural generalization of the well known Crocco equation arising from the standard Newtonian fluid case. The uniqueness, existence and analyticity of the solution are established and subsequently an explicit power series solution is exhibited.
355 citations
163 citations
157 citations
TL;DR: In this paper, an unknown non-homogeneous term in a linear PDE is determined from overspecified boundary data, using a linear partial differential equation (LPDE) model.
Abstract: (1980). Determination of an unknown non-homogeneous term in a linear partial differential equation .from overspecified boundary data. Applicable Analysis: Vol. 10, No. 3, pp. 231-242.
144 citations
TL;DR: In this paper, the problem of determining appropriate ABCs to use at a finite point was studied for both linear and non-linear boundary value problems, and a theory for doing this correctly was devised, which consists in determining appropriate asymptotic boundary conditions.
Abstract: To solve boundary value problems posed on semi-infinite intervals the problem is frequently replaced by one on a finite interval. We devise a theory for doing this correctly for both linear and nonlinear problems. In brief the method consists in determining appropriate asymptotic boundary conditions (ABC) to use at a finite point. In the linear case these are “projections” into the subspace of bounded solutions. In the nonlinear case, nonlinear boundary conditions result. They involve possibly unknown projections for the problem linearized about the solution at infinity. A linear eigenvalue problem for the Schrodinger equation and a nonlinear elasticity problem are solved to show the power of the new methods.
138 citations
TL;DR: In this article, a variational energy stability theory for two-dimensional buoyancy-thermocapillary convection in a layer with a free surface is presented. But the authors do not consider the case of planar interfaces.
Abstract: Energy stability theory has been formulated for two-dimensional buoyancy–thermocapillary convection in a layer with a free surface. The theory yields a critical Rayleigh number RE for which R < RE is a sufficient condition for stability of the layer. RE emerges from the variational formulation as an eigenvalue of a nonlinear system of Euler–Lagrange equations. For the case of small capillary number (large mean surface tension) explicit values are obtained for RE. The analogous linear-theory results for this case are obtained in terms of a critical Rayleigh number RL. These are compared. It is found that the existence of the deformable interface can lead to a stabilization relative to the case of a planar interface. This result is explained in physical terms. The energy theory is then generalized to include general flow problems having three-dimensional disturbances, non-Newtonian bulk fluids and general interfacial mechanics such as surface viscosity and elasticity.
135 citations
TL;DR: In this paper, a method for solving the toroidal magnetohydrodynamic equilibrium equation in a coordinate system based on the magnetic field lines is presented for obtaining equilibria to use in tokamak stability and transport calculations.
TL;DR: In this paper, the authors derived the following relations for U, the flux function of the meridian magnetic field: Tu = g(r, u), in QP (0.1) Yu = 0, in Qc, ( 0.3) u is a constant on I(whose value is unknown), and u = 0 on rP.
Abstract: INTRODUCTION THE PROBLEM we study in this paper has its origin in plasma physics. It stems from a model describing the equilibrium of a plasma confined in a toroidal cavity (a “Tokamak machine”). For a detailed presentation of this model the reader is referred to [l, 21 and also to the appendix in [3]. Let 51 denote the meridian cross section of the cavity and Oz the axis of toroidal symmetry, (a n Oz = Qr). The plasma occupies an unknown region C$, c !A; let Ta = ~?a,, and I= 20. The region sZV = R Q, is assumed to be vacuous (in particular, there are no external currents in C$,, cf. [4]). From the Maxwell equations and the magneto-hydrodynamic theory of macroscopic equilibrium in the plasma, one derives the following relations for U, the flux function ofthe meridian magnetic field : Tu = g(r, u), in QP (0.1) Yu = 0, in Qc, (0.2) u = 0, on rP, (0.3) U # 0, in !Z$, (0.4) u is a constant on I(whose value is unknown), (0.5)
TL;DR: In this paper, the Compactness Methods in Free Boundary Problems (CMPs) were used to solve the free boundary problems in partial differential equations (PDE) problems.
Abstract: (1980). Compactness methods in free boundary problems. Communications in Partial Differential Equations: Vol. 5, No. 4, pp. 427-448.
TL;DR: A characterization of all possible supplementary boundary conditions which work, the rate of convergence of the solution of the “finite’ problem to that of the original “infinite” problem as the interval length of the finite problem tends to infinity, and the supplementary boundary Conditions for which this rate is optimal.
Abstract: Boundary value problems for ordinary differential equations on infinite intervals are often solved by restricting the problem to a large but finite interval and imposing certain supplementary boundary conditions at the far end. The success of this procedure depends on the proper choice of these conditions. For a rather general class of problems we give a characterization of all possible supplementary boundary conditions which work, examine the rate of convergence of the solution of the “finite” problem to that of the original “infinite” problem as the interval length of the finite problem tends to infinity, and describe the supplementary boundary conditions for which this rate is optimal.
TL;DR: In this paper, a new numerical method is used to solve stationary free boundary problems for fluid flow through porous media, which also applies to inhomogeneous media, and to cases with a partial unsaturated flow.
Abstract: A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.
TL;DR: In this paper, a method for the solution of boundary value problems governed by a second-order elliptic partial differential equation with variable coefficients is derived by expressing the solution to a particular problem in terms of an integral taken round the boundary of the region under consideration.
Abstract: A method is derived for the solution of boundary value problems governed by a second-order elliptic partial differential equation with variable coefficients. The method is obtained by expressing the solution to a particular problem in terms of an integral taken round the boundary of the region under consideration.
TL;DR: In this article, the existence in time and uniqueness of solutions for the Cauchy problem for the Vlasov-Maxwell system of equations in one dimension was proved and the limiting values of the field ±(x, t) as the space variable x → E ∞ were shown to be uniquely determined by the initial data.
TL;DR: In this paper, the authors proved that the time-optimal controls associated with arbitrary reachable target temperature distributions in boundary control for the heat equation (with bounds on the admissible controls) are "bang-bang".
Abstract: Following previous work by H. O. Fattorini, J. Henry and the present author it is proved that the time-optimal controls associated with arbitrary reachable target temperature distributions in boundary control for the heat equation (with bounds on the admissible controls) are “bang-bang”.
TL;DR: In this paper, a theory is derived which defines classes of appropriate additional boundary conditions, and boundary conditions which produce convergence with the largest expectable order are devised, in the sense that the solutions of the approximate problems converge to the actual solution of the infinite problem as the length of the finite interval tends to infinity.
Abstract: An ad hoc method to solve boundary value problems which are posed on infinite intervals is to reduce the infinite interval to a finite but large one and to impose additional boundary conditions at the far end. These boundary conditions should be posed in a way so that they express the asymptotic behavior of the actual solution well. In this paper a theory is derived which defines classes of appropriate additional boundary conditions. Appropriate is to be understood in the sense that the solutions of the approximate problems converge to the actual solution of the “infinite” problem as the length of the finite interval tends to infinity. Moreover, boundary conditions which produce convergence with the largest expectable order are devised.
TL;DR: In this article, the stability of the initial boundary value problem for the method of lines applied to hyperbolic and parabolic partial differential equations in one space dimension was studied, and the theory was analogous to that developed by Gustafsson, Kreiss and Sundstrom.
01 Apr 1980
TL;DR: In this paper, the authors considered the boundary value problems associated with the d/dt (mu - lambda delta mu)-delta mu=0 in a cylindrical domain.
Abstract: : This report consider the equation d/dt (mu - lambda delta mu)-delta mu=0 in a cylindrical domain. Unlike the heat equation, the positivity of the boundary data is not sufficient to insure that the solution is nonnegative. It is desirable to identify those boundary data for which the above property is true. One reason is that, since the above equation is a model for heat conduction and for fluid flow in fractured porous media, it is of interest to locate those boundary data that make the correspondent physical process meaningful. In this paper several boundary value problems associated with the above equation are studied and necessary and sufficient conditions on the data are given to insure the nonnegativity of the solution.
TL;DR: In this article, a generalized generalized Grad-Shafranov equilibrium equation is derived under the assumption of a general symmetry and an elementary formulation of the boundary conditions is given and the existence of solutions is investigated.
Abstract: Under the assumption of a general symmetry (dependency on two space variables only), a generalized Grad–Shafranov equilibrium equation is derived and discussed. An elementary formulation of the boundary conditions is given and the existence of solutions is investigated. It emerges that from the equilibrium requirements almost no restrictions follow for the two arbitrary functions appearing in the equilibrium equation.
TL;DR: In this paper, a fast iterative process for the numerical solution of this problem was proposed, which can be applied to very special problems (for example, Poisson equation for a rectangle) as well as to general equations (arbitrary dimensions, general region).
Abstract: Elliptic control problems with a quadratic cost functional require the solution of a system of two elliptic boundary-value problems. We propose a fast iterative process for the numerical solution of this problem. The method can be applied to very special problems (for example, Poisson equation for a rectangle) as well as to general equations (arbitrary dimensions, general region). Also, nonlinear problems can be treated. The work required is proportional to the work taken by the numerical solution of a single elliptic equation.
TL;DR: In this paper, a new kind of boundary conditions for computer experiments of thermodynamic systems is introduced, and the merits of using a D-dimensional system as the surface of a (D + 1)-dimensional sphere are discussed in general.
TL;DR: In this article, it was shown that if u is a bounded solution on R + of u″(t) ϵ Au(t + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈Lloc2( R +;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t ) − ω(t)) 0 and u′(t)-ωω ϵ 0 as t → ∞.
TL;DR: The boundary integral equation method (BIEM) is used to solve three-dimensional potential flow problems in porous media as mentioned in this paper, where the problems considered here are time dependent and have a nonlinear boundary condition on the free surface.
Abstract: The boundary integral equation method (BIEM) is used to solve three-dimensional potential flow problems in porous media. The problems considered here are time dependent and have a nonlinear boundary condition on the free surface. The entire boundary, including the moving free surface, discretized into linear finite elements for the purpose of evaluating the boundary integrals. The technique allows transient, three-dimensional problems to be solved with reasonable computational costs. Numerical examples include recharge through rectangular and circular areas and seepage flow from a surface pond. The examples are used to illustrate the method and show the nonlinear effects.