Showing papers on "Free boundary problem published in 1979"
TL;DR: In this article, a technique for developing radiating boundary conditions for artificial computational boundaries is described and applied to a class of problems typical in exploration seismology involving acoustic and elastic wave equations.
Abstract: A technique for developing radiating boundary conditions for artificial computational boundaries is described and applied to a class of problems typical in exploration seismology involving acoustic and elastic wave equations. First, one considers a constant coefficient scalar wave equation where the artificial boundary is one edge of a rectangular domain. By using continued fraction expansions, a systematic sequence of stable highly absorbing boundary conditions with successively better absorbing properties as the order of the boundary conditions increases is obtained. There follows a systematic derivation of a hierarchy of local radiating boundary conditions for the elastic wave equation. A theoretical procedure to guarantee stability at corners of the rectangular domain is worked out. A technique for fitting the discrete radiating boundary conditions directly to the difference scheme itself is proposed.
780 citations
TL;DR: Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed and the resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
Abstract: Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed. The aspects of this method considered include error estimation, adaptive mesh selection, B-spline basis function evaluation, linear system solution and nonlinear problem solution. The resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
514 citations
Journal Article•
400 citations
257 citations
TL;DR: In this article, the plane boundary value problem for the Laplacian with Dirichlet data on one part and Neumann data on the remaining part of the boundary is converted into a system of Fredholm integral equations.
Abstract: Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2 and given Neumann data on the remaining part Γ2 of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ2 and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ1 ∩ Γ2 even for C∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2 providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1 and Γ2 augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).
149 citations
Book•
01 Jan 1979
135 citations
TL;DR: In this paper, the generic properties of nonlinear boundary value problems are discussed and discussed in the context of Partial Differential Equations (PDE) and partial differential equations (PDEs).
Abstract: (1979). Generic properties of nonlinear boundary value problems. Communications in Partial Differential Equations: Vol. 4, No. 3, pp. 293-319.
122 citations
TL;DR: In this paper, a general class of free boundary problems for parabolic equations in one space dimension, dealing with nonlinearities both in the differential equation and in the free boundary condition, is studied.
100 citations
TL;DR: The use of piecewise quadratic polynomial approximations in the boundary integral equation method for the solution of boundary value problems involving Laplace's equation and certain Poisson equations is described in this paper.
87 citations
68 citations
TL;DR: In this paper, the second boundary value problem is considered for the equation in a domain of complicated structure of the form, where is a closed finely partitioned set lying in the domain for all.
Abstract: The second boundary value problem is considered for the equation in a domain of complicated structure of the form , where is a closed finely partitioned set lying in a domain () for all . The asymptotic behavior of a solution of this problem is studied as , when becomes more and more finely divided and is situated in so that the distance from to any point tends to zero. It is proved that under specific conditions converges in to a function that is a solution of a conjugation problem. Sufficient conditions for convergence are formulated.Bibliography: 9 titles.
TL;DR: In this paper, a multi-grid method for the solution of boundary control problems with quadratic cost functions was presented, where the state is a solution of a parabolic initial-boundary value problem.
Abstract: We present a multi-grid method for the solution of boundary control problems with quadratic cost functions, where the state is a solution of a parabolic initial-boundary value problem. The computationalwork is proportional to the work needed for the integration of the parabolic equation, The method can be extended also to nonlinear problems.
TL;DR: In this article, a technique for the solution of the Helmholtz equation together with associated boundary conditions based on a generalization of a method used for the Dirichlet problem of potential theory, in which a dipole distribution is introduced on the boundary of a region to generate the potential inside.
TL;DR: In this paper, the Dirichlet problem for the biharmonic equation is solved using the thirteen-point formula, where the prescribed normal derivative on the boundary is replaced by two classes of boundary approximations in order to define the solution at certain fictitious node points.
TL;DR: In this article, the phase function for a three-term scattering law for radiative transfer in a finite slab was established, which is the basic to the inverse problem of radiative transfers.
Abstract: Elementary considerations are used to establish the phase function for a three-term scattering law basic to the inverse problem for radiative transfer in a finite slab.
TL;DR: In this paper, the basic function in the circumferential direction satisfying the boundary conditions of the radial edges is substituted into the free vibration equation of the curved plate by a suitable transformation, an ordinary differential equation is obtained.
TL;DR: In this paper, the problem of determining a second-order differential operator (in a normal form) on the half-axis from its spectral function is studied, and the solution of problem (i) is constructive and stability of the solution under perturbation of the data, in a well-defined sense.
TL;DR: In this paper, an integral equation method for thin elastic clamped plates of arbitrary plan form was extended to include arbitrary boundary conditions (i.e. clamped, simply-supported and free edges).
TL;DR: In this article, an analytically exact solution to the free boundary problem is found and established, which is expressed in terms of some functions and polynomials of the error integral family and timet.
Abstract: The paper is concerned with the free boundary problem of a semi-infinite body with an arbitrarily prescribed initial condition and an arbitrarily prescribed flux at its face. An analytically exact solution to the problem is found and established, which is expressed in terms of some functions and polynomials of the error integral family and timet. Convergence of the series solution is considered and proved. For illustration, the solution is applied to a special problem, which is discussed in detail. Some remarks about the solution are also included.
TL;DR: In this paper, it was shown that the linear boundary value problem has a solution defined on an interval J = [a, b), x < a < b d + xc;.
Abstract: where A and L are linear operators and M is a continuous, generally nonlinear operator. We want to show that, under suitable hypotheses, the problem (1.1) has a solution defined on an interval J = [a, b), x < a < b d + xc;. The analogous problem for ordinary differential equations on a compact interval has been treated by Scrucca [ 11. The linear boundary value problem has been studied by Cecchi et al. [Z] in the univoque case and by Anichini and Zecca [3,4], in the multivoque one. For a wide bibliography and exposition on boundary value problems for differential equations see Conti [5]. 2. NOTATIONS AND HYPOTHESES
TL;DR: In this article, the authors define a procedure for numerically approximating the solution of one such heat flow problem and present explicit error estimates for the numerical procedure when the data are known only approximately.
Abstract: : In many physical problems in heat conduction, it is impossible to obtain an initial temperature distribution within a material. In many of these cases, in order to obtain approximations of the temperature within the body, one must rely entirely upon data which can be measured at the boundary. An additional problem is that these boundary data are only accurate to within some prescribed measurement errors. The purpose of this paper is to define a procedure for numerically approximating the solution of one such heat flow problem and to present explicit error estimates for the numerical procedure. A priori error estimates are presented when the data are known only approximately.
TL;DR: In this article, the authors survey numerical techniques of order 2, 4 and 6 for the solution of a two-point boundary value problem associated with a fourth-order linear ordinary differential equation.
Abstract: In this report we survey numerical techniques of order 2, 4 and 6 for the solution of a two-point boundary value problem associated with a fourth-order linear ordinary differential equation. A sufficient condition guaranteeing a unique solution of the boundary value problem is also given. Numerical results are tabulated for two typical numerical examples and compared with some known methods including the shooting technique employing the classical fourth-order Runge-Kutta method.
TL;DR: In this article, the minimum time-to-climb problem is formulated as a two-point boundary value problem arising from a general optimal control problem and the linearized zeroth-order boundary layer equations of the problem are derived and solved.
Abstract: Ardema (1974) has formally linearized the two-point boundary value problem arising from a general optimal control problem, and has reviewed the known stability properties of such a linear system. In the present paper, Ardema's results are applied to the minimum time-to-climb problem. The linearized zeroth-order boundary layer equations of the problem are derived and solved.
TL;DR: In this article, a boundary value problem for the Lavrent'ev-Bitsadze differential equation of mixed type is divided into an elliptic problem involving an unusual boundary condition on the parabolic line, and a hyperbolic problem.
Abstract: A boundary value problem for the Lavrent’ev–Bitsadze differential equation of mixed type is divided into an elliptic problem involving an unusual boundary condition on the parabolic line, and a hyperbolic problem. The elliptic problem is then converted into an equivalent variational form and shown to be well posed under certain conditions which always include the characteristic boundary case. The finite element method is applied using triangular elements and a few singular functions near the intersection of the parabolic line and the elliptic boundary. This method is shown to be second order accurate and easy to implement. Once the elliptic problem is solved, thus giving u and its derivatives along the parabolic line, the straightforward Cauchy problem is solved in the hyperbolic region. Numerical verification of these results is presented.