Showing papers on "Boundary value problem published in 1969"
TL;DR: In this paper, a numerical method for the dynamic analysis of infinite continuous systems is developed, applicable to systems for which all exciting forces and geometrical irregularities are confined to a limited region and is applicable to both transient and steady state problems.
Abstract: A numerical method for the dynamic analysis of infinite continuous systems is developed. The method is applicable to systems for which all exciting forces and geometrical irregularities are confined to a limited region and is applicable to both transient and steady state problems. The infinite system is replaced by a system consisting of a finite region subjected to a boundary condition which simulates an energy absorbing boundary. The resulting systems may be analyzed by the finite element method. Examples applying the method to foundation vibration problems are presented. Good agreement with existing solutions is found and new results for embedded footings are presented.
2,172 citations
Book•
01 Jan 1969
TL;DR: In this article, elementary differential equations and boundary value problems are studied in the context of boundary value problem, where boundary value is defined as a function of the boundary value of the elementary differential equation.
Abstract: Elementary differential equations and boundary value problems , Elementary differential equations and boundary value problems , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی
1,575 citations
TL;DR: In this article, the limitations of classical laminated plate theory are investigated by comparing solutions of several specific boundary value problems in this theory to the corresponding theory of elasticity solutions, and it is shown that conventional plate theory leads to a very poor description of laminate response at low span-to-depth ratios.
Abstract: Limitations of classical laminated plate theory are investigated by comparing solutions of several specific boundary value problems in this theory to the corresponding theory of elasticity solutions. The general class of problems treated involves the geometric configuration of any number of isotropic or orthotropic layers bonded together and subjected to cylindrical bending. In general it is found that conventional plate theory leads to a very poor description of laminate response at low span-to-depth ratios, but converges to the exact solution as this ratio increases. The analysis presented is also valid in the study of sandwich plates under cylindrical bending.
1,194 citations
TL;DR: In this article, a linearized formulation of the surface is obtained by idealizing the surface as a set of lifting elements which are short line segments of accelerationpotential doub? lets, and the normal velocity induced by an element of unit strength is given by an integral of the subsonic kernel function.
Abstract: Approximate solutions from the linearized formulation are obtained by idealizing the surface as a set of lifting elements which are short line segments of acceleration-potential doub? lets. The normal velocity induced by an element of unit strength is given by an integral of the subsonic kernel function. The load on each element is determined, by, satisfying normal velocity boundary conditions at a set of points oil the surface. It is seen a posteriori
893 citations
TL;DR: In this article, the transverse displacement of a three-layer sandwich beam with a viscoelastic core is derived in terms of the transversal displacement, w, for a 3D beam.
785 citations
TL;DR: The results not only provide a basis for estimating electrotonic length, L = [unk]/lambda, but also provide a new basis for estimates the steady-state ratio, rho, of cylinder input conductance to soma membrane conductance.
638 citations
01 Jan 1969
536 citations
TL;DR: In this paper, a numerical solution capability is developed for the solution of problems in three dimensional elastostatics, which utilizes singular integral equations which can be solved numerically for the unknown surface tractions and displacements for the fully mixed boundary value problem.
521 citations
TL;DR: In this paper, a rational theory is developed to explain the initial pressure rise and consequent separation of a laminar boundary layer when it interacts with a moderately strong shock, where the region of interest is divided into three parts: the major part of the boundary layer, which is shown to change under largely inviscid forces, the supersonic main stream just adjacent to the boundary surface, and a region close to the wall, on boundary-layer scale, in which the relative variation of the velocity is controlled by the incompressible boundary layer equations.
Abstract: A rational theory is developed to explain the initial pressure rise and consequent separation of a laminar boundary layer when it interacts with a moderately strong shock. In this theory, which is firmly based on the linearized theory of Lighthill (1953), the region of interest is divided into three parts: the major part of the boundary layer, which is shown to change under largely inviscid forces, the supersonic main stream just adjacent to the boundary layer in which the pressure variation is small; and a region close to the wall, on boundary-layer scale, in which the relative variation of the velocity is large but is controlled by the incompressible boundary-layer equations, together with novel boundary conditions. We find that the first two parts can be handled in a straightforward way and the problem of self-induced separation reduces, in its essentials, to the solution of a single problem in the theory of incompressible boundary layers. It is found that this problem has three solutions, one of which corresponds to undisturbed flow and another describes a boundary layer which, spontaneously, generates an adverse pressure gradient and a decreasing skin friction which eventually vanishes and then downstream a reversed flow is set up. The third solution generates a favourable pressure gradient and is not relevant to the present study. Although there has hitherto been no valid numerical method of integrating a boundary layer with reversed flow, we find that an ad hoc method seems to lead to a stable solution which has a number of the properties to be expected of a separated boundary layer. Comparison with experiment gives qualitatively good agreement, but quantitatively errors of the order of 20% are found. It is believed that these errors arise because the Reynolds numbers at which the experiments were carried out are too small.
504 citations
08 May 1969
TL;DR: In this article, the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order is investigated. But the results are restricted to the case where the curvatures of the boundaries of the underlying domains are constrained.
Abstract: This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of necessary conditions and sufficient conditions for this problem to be solvable in a given domain with arbitrarily assigned smooth boundary data. A central position in the discussion is played by the concept of global barrier functions and by certain fundamental invariants of the equation. With the help of these invariants we are able to distinguish an important class of ‘ regularly elliptic5 equations which, as far as the Dirichlet problem is concerned, behave comparably to uniformly elliptic equations. For equations which are not regularly elliptic it is necessary to impose significant restrictions on the curvatures of the boundaries of the underlying domains in order for the Dirichlet problem to be generally solvable; the determination of the precise form of these restrictions constitutes a second primary aim of the paper. By maintaining a high level of generality throughout, we are able to treat as special examples the minimal surface equation, the equation for surfaces having prescribed mean curvature, and a number of other non-uniformly elliptic equations of classical interest.
461 citations
Book•
[...]
01 Jan 1969
TL;DR: Weyl-Kodaira theory: elliptical differential operators on an interval of R boundary conditions self-adjoint operators associated with a linear differential equation the case of second order equations example -second order equations with periodic-coefficients example -Gelfand-Levitan equations as mentioned in this paper.
Abstract: Part 1 Weyl-Kodaira theory: elliptical differential operators on an interval of R boundary conditions self-adjoint operators associated with a linear differential equation the case of second order equations example - second order equations with periodic-coefficients example - Gelfand-Levitan equations. Part 2 Multilayer potentials: symbols of rational type the case of hyperplane multilayers general case. Part 3 Fine boundary value problems for elliptical differential operators: the Calderon operator elliptic boundary value problems ellipticity criteria the spaces Hs,r (U+) Hs,r - spaces and P-potentials regularity on the boundary coercive problems generalized Green's formula fine problems associated with coercive problems examples extension to some non-Hermitian operators case of second-order operators Neumann's problem the maximum principle. Part 4 Parabolic equations: construction of one-sided local resolvent the one-sided global Cauchy problem traces and eigenvalues. Part 5 Evolution distributions - the wave equation: generalized Cauchy problem propagation and domain of influence signals, waves and rays. Part 6 Strictly hyperbolic equations: preliminary results construction of a local approximate resolvent examples and variations the Cauchy problem for strictly hyperbolic differential operators existence and local uniqueness global problems extension to manifolds. Part 7 Application to the spectrum of a Hermitian elliptic operator.
TL;DR: In this paper, it was shown that plating which load the substrate can reduce the velocity of straight-crested surface waves, while plating with no load can increase the velocities of the surface wave.
Abstract: An isotropic, elastic solid permits the propagation of one straight‐crested, nondispersive surface wave, called a Rayleigh wave. If a thin film of a different material is deposited on the substrate, an infinite number of straight‐crested surface waves are possible, all of which are dispersive, including the one corresponding to the Rayleigh wave. Since the solution of the three‐dimensional equations of elasticity is extremely unwieldy and the films may be considered thin in the frequency range of interest, the approximate equations of low‐frequency extension and flexure of thin plates are employed in the solution to account for the motion of the platings. These approximate equations enable the entire effect of the plating to be treated as a boundary condition at the surface of the substrate. The accuracy of the approximation is shown to be excellent in the frequency range of interest. It is shown that platings which load the substrate reduce the velocity of straight‐crested surface waves, while platings w...
TL;DR: In this paper, the problem of electromagnetic radiation and scattering from perfectly conducting bodies of revolution of arbitrary shape is considered, and the mathematical formulation is an integro-differential equation, obtained from the potential integrals plus boundary conditions at the body.
Abstract: The problem of electromagnetic radiation and scattering from perfectly conducting bodies of revolution of arbitrary shape is considered The mathematical formulation is an integro-differential equation, obtained from the potential integrals plus boundary conditions at the body A solution is effected by the method of moments, and the results are expressed in terms of generalized network parameters The expansion functions chosen for the solution are harmonic in o (azimuth angle) and subsectional in t (contour length variable) Because of rotational symmetry, the solution becomes a Fourier series in o, each term of which is uncoupled to every other term Illustrative computations are given for radiation from apertures and plane wave scattering from bodies of revolution The impedance elements, currents, radiation patterns, and scattering patterns for a conducting sphere are computed both from the general solution and from the classical eigenfunction solution The agreement obtained serves to check the general solution Similar computations for a cone-sphere illustrate the application of the general solution to problems not solvable by classical methods
TL;DR: In this article, a variational principle is used in conjunction with the finite element method to solve the initial boundary value problem of flow in a saturated porous elastic medium, which results in a powerful solution technique for the determination of stress and displacement history, both for the solid and the liquid phases, for arbitrary boundary conditions and within complex geometrical configurations.
Abstract: A variational principle is used in conjunction with the finite-element method to solve the initial boundary value problem of flow in a saturated porous elastic medium. This results in a powerful solution technique for the determination of stress and displacement history, both for the solid and the liquid phases, for arbitrary boundary conditions and within complex geometrical configurations. Direct application is to problems of consolidation and drainage of saturated soils under load. Linear theory of the coupled fields is treated but extension to nonlinear problems is possible through use of incremental procedures.
TL;DR: In this article, the effect of various electromagnetic boundary conditions on the propagation of surface waves in piezo-electrics is considered and an analytic expression for this impedance is found in the case of weak piezoelectric coupling which permits the calculation of the phase velocity for arbitrary values of the impedance.
Abstract: The effect of various electromagnetic boundary conditions on the propagation of surface waves in piezo‐electrics is considered. Basic for the analysis is the introduction of an electric ``surface impedance'' which relates the electric potential to the normal component of electric displacement in the surface. An analytic expression for this impedance is found in the case of weak piezoelectric coupling which permits the calculation of the phase velocity for arbitrary values of the impedance. It is also shown that within the weak coupling approximation the surface impedance contains the information on material parameters necessary for calculating the power transferred to surface waves from an impressed electric current.
TL;DR: In this article, the analysis of the powers of an elliptic pseudo-differential operator was extended to boundary value problems, where the authors considered a q X q system A = > aaDa of Cdifferential operators.
Abstract: This paper and its successor, [10], extend to boundary value problems the analysis of the powers of an elliptic operator given in [9]. (Similar methods are applied in [3], [5], and [6] for the case without boundary; [5a] announces boundary results.) Although [9] treated general elliptic pseudo-differential operators on compact manifolds, we restrict ourselves here to differential operators, except in the final section, where we correct some (fortunately inconsequential) errors in the proof given in [9]. We consider a q X q system A = > aaDa of Cdifferential operators
Book•
31 Dec 1969
TL;DR: This edition of the expanded version of Zill's "A First Course in Differential Equations with Modeling Applications", places greater emphasis on modelling and the use of technology in problem solving and features more everyday applications as mentioned in this paper.
Abstract: This edition of the expanded version of Zill's "A First Course in Differential Equations with Modeling Applications", places greater emphasis on modelling and the use of technology in problem solving and features more everyday applications. Both Zill texts are identical through the first nine chapters, but this version includes six, additional chapters that provide in-depth coverage of boundary-value problem-solving and partial differential equations, subjects introduced in the first nine chapters. Understandable, step-by-step solutions are provided for every example.
TL;DR: In this paper, the convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis was investigated for the special case of infinite Prandtl number and free boundary conditions.
Abstract: The convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis is investigated for the special case of infinite Prandtl number and free boundary conditions. For slightly supercritical Rayleigh numbers the solutions of the non-linear steady-state equations are derived approximately by an amplitude expansion. A stability calculation shows that no stable steady-state convective flow exists if the Taylor number exceeds the critical value 2285.
TL;DR: In this article, for two model boundary value problems (a set of ordinary differential equations and an elliptic equation), methods of solution on a network with variable steps and an evaluation of the error which is homogeneous in the small parameter, but with few constraints on the smoothness of the coefficients.
Abstract: THE solutions of a number of value problems for differential equations with small parameters having higher derivatives possess singularities of the boundary layer type [1, 2]. For the solution of such problems by finite-difference methods the integration step near the boundary must be substantially less than the thickness of the boundary layer which is a characteristic dimension of the problem. In the case of constant steps throughout the whole region of integration this circumstance leads to a considerable increase in the volume of calculations when the parameters are reduced with higher derivatives. An exception may be only the so-called “quasiclassical approximations” of [3], adapted specially for the solution of problems with small parameters having higher derivatives, but these can only be written for isolated classes of ordinary differential equations. The use of asymptotic methods of solution [2, 4] requires these parameters to be rather small, and their coefficients to be very smooth. The same often applies to the use of numerical methods. Below we construct, for two model boundary value problems (a set of ordinary differential equations and an elliptic equation), methods of solution on a network with variable steps and an evaluation of the error which is homogeneous in the small parameter, but with few constraints on the smoothness of the coefficients.
TL;DR: In this article, the force per unit volume which a flowing fluid exerts on a porous structure, and the divergence of the local volume-averaged extra stress tensor are examined with the aid of the principle of material indifference.
Abstract: The local volume average of the equation of motion is taken for an incompressible fluid flowing through a porous structure under conditions such that inertial effects may be neglected. The result has two terms beyond a pressure gradient: g, the force per unit volume which a flowing fluid exerts on a porous structure, and the divergence of the local volume-averaged extra stress tensor (viscous portion of the stress tensor).
Constitutive equations for g are examined with the aid of the principle of material indifference. When g is assumed to be a function of the velocity of the fluid relative to the solid as well as various scalars, the usual results for a nonoriented (isotropic) porous structure are obtained. When g is assumed to be a function of the local porosity gradient as well, we derive a new expression for g applicable to oriented (anisotropic) porous structures.
For a Newtonian fluid with a constant viscosity, the divergence of the local volume-averaged extra stress tensor is proportional to the Laplacian of the averaged velocity vector. Boundary conditions for the averaged velocity vector are discussed. Three problems are solved for the flow of an incompressible Newtonian fluid in a nonoriented permeable medium. These solutions, as well as an order-of-magnitude analysis, suggest that we may often neglect both the Laplacian of average velocity and the boundary conditions for the tangential components of averaged velocity at an impermeable wall.
Two specific constitutive equations for g are proposed for the flow of incompressible Noll simple fluids in nonoriented porous structures. Flow through a porous medium bounded by an impermeable cylindrical surface is solved for these two constitutive equations, and the results are compared with previously available experimental data.
TL;DR: In this paper, a new form of dynamic orbital is proposed for the collision problem, which automatically achieves complete orthonormality, by making allowances for the reluctance of an electronic charge distribution to follow rapid rotation of an internuclear axis.
Abstract: Requirements are outlined for a suitable set of dynamic orbitals for theoretical studies in collision problems. The effect of these upon the wave function, dynamic energy correction, and effective internuclear potential are all considered. It is shown that earlier forms suggested for this type of problem do not meet all required dynamic boundary conditions, principally because of their failure to recognize that physically, for moderate speed collisions, the electron at times "belongs" to the "molecule" proper and not to either atom individually. The earlier orbitals also fail to make allowances for the reluctance of an electronic charge distribution to follow rapid rotation of an internuclear axis. These considerations suggest a new form of dynamic orbital which by remedying these deficiencies automatically achieves complete orthonormality. The results of preliminary charge transfer calculations with the new orbital basis are presented.
TL;DR: In this paper, the authors extend to boundary problems the analysis of the powers of an elliptic system of operators given in [6] and this paper is a continuation of that one.
Abstract: This paper extends to boundary problems the analysis of the powers of an elliptic system of operators given in [6]. The results depend on the analysis of the resolvent given in [7], and this paper is a continuation of that one. We will think of that earlier paper as part I of this one, and not repeat definitions, theorems, notations, and so on here, but simply refer to the earlier paper. For instance, formula 1(29) refers to formula 29 of [7],
TL;DR: In this paper, formal solutions of inverse scattering problems for scattering from a potential, a variable index of refraction, and a soft boundary are developed using a method devised by Jost and Kohn.
Abstract: Formal solutions of inverse scattering problems for scattering from a potential, a variable index of refraction, and a soft boundary are developed using a method devised by Jost and Kohn.
TL;DR: In this paper, the authors considered the traction boundary value problem in the general case where O(U,, ti) may be nonlinear in both u,, zi,.
TL;DR: Finite element solution of incompressible lubrication problem by minimum principle for transient incompressibility Reynolds equation with boundary conditions as discussed by the authors, where the minimum principle is used to solve the problem.
Abstract: Finite element solution of incompressible lubrication problem by minimum principle for transient incompressible Reynolds equation with boundary conditions
TL;DR: In this paper, a general computer program for determining sets of propagating modes and cutoff frequencies of arbitrarily shaped waveguides is described, which uses a new method of analysis based on approximate extremization of a functional whose Euler equation is the scalar Helmholtz equation, subject to homogeneous boundary conditions.
Abstract: A very general computer program for determining sets of propagating modes and cutoff frequencies of arbitrarily shaped waveguides is described. The program uses a new method of analysis based on approximate extremization of a functional whose Euler equation is the scalar Helmholtz equation, subject to homogeneous boundary conditions. Subdividing the guide cross section into triangular regions and assuming the solution to be representable by a polynomial in each region, the variational problem is approximated by a matrix eigenvalue problem, which is solved by Householder tridiagonalization and Sturm sequences. For reasonably simple convex polygonal guide shapes, the dominant eigenfrequencies are obtained to 5-6 significant figures; for nonconvex or complicated shapes, the accuracy may fall to 3 significant figures. Use of the program is illustrated by calculating the propagating modes of a class of degenerate mode guides of current interest, for which experimental data are available. Numerical studies of convergence rate and discretization error are also described. It is believed that the new program produces waveguide analyses of higher accuracy than any general program previously available.
TL;DR: In this article, the authors proposed a minimum fuel multiple impulse orbital rendezvous for fixed transfer time near circular orbits with a fixed transfer period near circular orbit and fixed transfer times near circular positions.
Abstract: Minimum fuel multiple impulse orbital rendezvous for fixed transfer time near circular orbits