Showing papers on "Free boundary problem published in 1981"
765 citations
TL;DR: A survey of methods for imposition of radiation boundary conditions in numerical schemes is presented in this article, where combining of absorbing boundary conditions with damping (in particular, sponge filters) and with wave-speed modification are shown to offer significant improvements over earlier methods.
625 citations
01 Jan 1981
TL;DR: In this paper, nonlinear two point boundary value problems will lead readers to love reading starting from now, and they may not be able to make readers love reading, but nonlinear boundary value problem will lead them to read more books.
Abstract: We may not be able to make you love reading, but nonlinear two point boundary value problems will lead you to love reading starting from now. Book is the window to open the new world. The world that you want is in the better stage and level. World will always guide you to even the prestige stage of the life. You know, this is some of how reading will give you the kindness. In this case, more books you read more knowledge you know, but it can mean also the bore is full.
282 citations
255 citations
TL;DR: In this article, the SEUDODIFFEI%ENTIAL OPERATORS i.i.d. PSEUDODifFEI percentential operators i.e.
Abstract: II. PSEUDODIFFEI%ENTIAL OPERATORS i. Symbol spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2. Operators on open sets . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3. Definition on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4. Kernels and adjoints . . . . . . . . . . . . . . ~ . . . . . . . . . . . . 176 5. Boundary values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6. Symbols and residual operators . . . . . . . . . . . . . . . . . . . . . . 187 7. Composition and el]iptieity . . . . . . . . . . . . . . . . . . . . . . . . 194 8. Wavefront set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9. Normal regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 i0. L ~ estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
213 citations
TL;DR: Silliman's analysis of slot coating is extended to accommodate film flows with highly bent menisci, as in slide and curtain coating, by combining polar and Cartesian coordinate parametrizations of meniscus shape as discussed by the authors.
205 citations
TL;DR: In this paper, the energy decay rates of the wave equation in a domain where boundary damping is present were studied and a regulator problem was also formally discussed by the synthesis method, where the authors generalize the geometrical conditions obtained earlier in (J. Purer Appl., 58 (1979), pp. 249 and 273) by using some more general multipliers of Strauss (Comm. Pure Appl., 28 (1975), pp 265 and278).
Abstract: We study the energy decay rates of the wave equation in a domain where boundary damping is present. We generalize the geometrical conditions obtained earlier in (J. Math. Purer Appl., 58 (1979), pp. 249–273) by using some more general multipliers of Strauss (Comm. Pure Appl. Math., 28 (1975), pp. 265–278). The interaction between distributed damping and boundary damping is discussed. A regulator problem is also formally discussed by the synthesis method.
134 citations
TL;DR: In this article, the nonlinear heat conduction equation was transformed into Laplace's equation using Kirchhoff's transform, and the remaining boundary conditions of first and second kind, became linear.
104 citations
TL;DR: In this paper, a Schrodinger equation for a well potential with varying width is studied and generalized canonical transformations are shown to transform the problem into a time-dependent harmonic oscillator problem submitted to fixed boundary conditions.
Abstract: A Schrodinger equation for a well potential with varying width is studied. Generalized canonical transformations are shown to transform the problem into a time‐dependent harmonic oscillator problem submitted to fixed boundary conditions. This transformed problem is solved by a perturbation technique and gives the evolution of the average energy of the system according to the motion of the well. Motions corresponding to a renormalization or compaction group are shown to be solvable by separation of variables.
90 citations
TL;DR: In this paper, the variable time step method introduced by Douglas and Gallie for solving a one-dimensional Stefan problem with constant heat flux at the fixed end is extended to cover a more general boundary condition.
84 citations
TL;DR: In this paper, a method for solving boundary value problems for thin plate flexure is proposed, which leads to a system of boundary integral equations involving values of deflection, slope, bending moment and transverse shear force along the edge.
Abstract: A method for solving boundary value problems for thin plate flexure as described by Kirchhoff's theory is proposed. An integral formulation leads to a system of boundary integral equations involving values of deflection, slope, bending moment and transverse shear force along the edge. A discretization leading to a matrix formulation is proposed. To solve problems with inner conditions in the plate domain, an elimination of boundary unknowns proves successful. The degenerate case where the boundary is free (which leads to a non-invertible matrix) is investigated. Three examples illustrate the efficiency of the method.
TL;DR: In this paper, the generalized integral equation for the electric potential governed by a quasi-harmonic equation can be derived via a variational formulation, and numerical solutions of the general heterogeneous problem can be obtained with the reciprocal averaging technique.
Abstract: The generalized integral equation for the electric potential governed by a quasi-harmonic equation can be derived via a variational formulation. For surface current distributions it is not always a Fredholm integral equation of the second kind. Numerical solutions of the general heterogeneous problem can be obtained with the “reciprocal averaging technique”, where the solution is obtained a second time after exchange of source and field points.
TL;DR: In this paper, the authors use energy invariants to study the growth and decay estimates for solutions of the wave equation in a domain with moving boundary, and sufficient conditions are formulated which insure the exact (distributedparameter) controllability of wave equation.
Abstract: We use energy invariants to study the growth and decay estimates for solutions of the wave equation in a domain with moving boundary. Sufficient conditions are formulated which insure the exact (distributed-parameter) controllability of the wave equation.
TL;DR: The boundary regularity results of H. Lewy and W. Jager for area minimizing minimal surfaces with a free boundary are shown to be true also for minimal surfaces which are only stationary points of the Dirichlet integral with respect to a given boundary configuration as mentioned in this paper.
Abstract: The well known boundary regularity results of H. Lewy and W. Jager for area minimizing minimal surfaces with a free boundary are shown to be true also for minimal surfaces which are only stationary points of the Dirichlet integral with respect to a given boundary configuration.
TL;DR: The solution of the Fokker–Planck equation is considered for the case of steady, one‐dimensional flow with prescribed flux at the outer boundary and complete absorption at the inner boundary using a modified version of the bimodal Maxwellian moment method due to Lees.
Abstract: We consider the solution of the Fokker–Planck equation for the case of steady, one‐dimensional flow with prescribed flux at the outer boundary and complete absorption at the inner boundary. An approximate solution is obtained using a modified version of the bimodal Maxwellian moment method due to Lees, which has been used previously with success in treating boundary layer problems in the context of the Boltzmann equation. We obtain explicit results for the density in the physical space which is characterized by a boundary layer of order (inverse velocity relaxation time/thermal velocity)−1 and a Milne extrapolation length (distance beyond the boundary at which the extrapolated asymptotic value of the density is zero) of 1.44 in appropriately normalized units. This latter value compares surprisingly well with recent analytical–numerical results which find a value of 1.46 for this quantity.
TL;DR: In this paper, a continuous random walk procedure for solving some elliptic partial differential equations at at a single point is generalized to estimate the solution everywhere, where the only error is the statistical sampling error that tends to zero as the sample size increases.
TL;DR: In this article, a combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems and its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension, electrochemical machining and Hele-Shaw flow in two dimensions, and a Stefan and ablation problem in three dimensions.
Abstract: A combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems. Its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension, electrochemical machining and Hele–Shaw flow in two dimensions, and a Stefan and ablation problem in three dimensions.
TL;DR: In this paper, a numerical integration of the system of governing equations which define a boundary value problem written down in the form of a coupled system of first-order ordinary differential equations is shown to be a powerful technique.
Abstract: Numerical integration of the system of governing equations which define a boundary value problem written down in the form of a coupled system of first-order ordinary differential equations is shown to be a powerful technique. After presenting the basic approach the paper critically examines the numerical schemes available for situations when the boundary value problem so defined has boundary layer characteristics. One such method which is originally due to Goldberg, Setlur and Alspaugh3 is described in detail, with documentation in the form of a flow diagram and a FORTRAN listing of a working subroutine. The method is shown to be computationally efficient and reliable for the solution of a class of problems in the field of solid mechanics. Potential use of the method for the solution of magnetostatic problems is indicated.
01 Jan 1981
TL;DR: In contrast to boundary value problems, initial value problems for elliptic differential equations are not properly posed as a rule (cf. Example 1.14), and they arise, among other places, in the areas of fluid dynamics, electrodynamics, stationary heat and mass transport (diffusion), statics, and reactor physics (neutron transport).
Abstract: Boundary value problems for elliptic differential equations are of great significance in physics and engineering. They arise, among other places, in the areas of fluid dynamics, electrodynamics, stationary heat and mass transport (diffusion), statics, and reactor physics (neutron transport). In contrast to boundary value problems, initial value problems for elliptic differential equations are not properly posed as a rule (cf. Example 1.14).
TL;DR: In this paper, a cubic spline method is described for the numerical solution of a two-point boundary value problem involving a fourth order linear differential equation, which is closely related to a known fourth order finite difference scheme.
01 Jan 1981
TL;DR: In this paper, the authors consider an error analysis of boundary integral equations that only takes into account the effects of the first three kinds of discretization, and they consider so-called asymptotic estimates.
Abstract: The numerical treatment and corresponding error analysis of boundary integral equations hinges on the type of discretization due to the shape and type of trial functions used for the approximation of the unknown functions, due to the type of test functionals replacing the integral equations — which hold everywhere on the boundary — by a finite number of equations and due to the numerical integration. In reality further errors accumulate from round off effects. Here we are concerned with an error analysis which only takes into account the effects of the first three kinds. Since it seems to be a too pretentious task to find computable error bounds we consider so called asymptotic estimates.
TL;DR: Theorem 3.4 as discussed by the authors establishes the existence of a unique solution of (1.1) tending to zero as 1x1 tends to co. Theorem 4.1 shows that a solution which tends to zero is unique.
01 Jan 1981
TL;DR: The boundary element method was used to determine the natural frequencies and mode shapes of thin elastic plates of uniform thickness with arbitrary boundary conditions and arbitrarily shaped edges as mentioned in this paper, and the boundary elements are taken as straight lines or as circular arcs, and unknown boundary functions are assumed to vary linearly along the elements.
Abstract: The boundary element method is used to determine the natural frequencies and mode shapes of thin elastic plates of uniform thickness with arbitrary boundary conditions and arbitrarily shaped edges. The boundary element formulation leads to a pair of integral equations that contain the boundary condition variables of displacement, normal slope, bending moment, and effective shear, providing an effective means of solving plate problems with arbitrary boundary conditions. The boundary elements are taken as straight lines or as circular arcs, and unknown boundary functions are assumed to vary linearly along the elements.