Showing papers on "Partial differential equation published in 1972"

Nonlinear Modulation of Gravity Waves

Abstract: Slow modulation of gravity waves on water layer with uniform depth is investigated by using singular perturbation methods. It is found, to the lowest order of perturbation, that the complicated system of equations governing such modulation can be reduced to a simple nonlinear Schrodinger equation. A nonlinear plane wave solution to this equation is found to correspond to the so-called Stokes wave. The linear stability of this plane wave solution is essentially determined by the sign of the product of two coefficients in this equation, yielding Benjamin and Whitham's criterion. The same equation is found to give a weak cnoidal wave derived from the Korteweg-de Vries equation in the shallow-water limit.

Finite-Element Solution of Unbounded Field Problems

Abstract: An unbounded region is divided into local picture-frame regions where a partial differential-equation solution is obtained, with the remaining unbounded region represented by an integral equation. (The method permits the use of free-space Green's functions, and thus special problem-dependent Green's functions need not be found.) The integral equation is formulated as a constraint upon the local picture-frame solutions, whence these local solutions are solved directly by a variational method, using finite elements, in a manner such that the problem of the Green's-function singularity is side-stepped. The technique is applicable where sources and media inhomogeneities and anisotropies are local, and can all be placed within one or several picture frames. It is in these cases that the integral-equation approach is at a particular disadvantage, and the use of a partial differential-equation technique is advisable if not necessary. Examples presented include the static and harmonic fields of a parallel-plate capacitor, a microstrip line on a dielectric substratum, and a radiating antenna with dielectric obstacles.

Nonlinear vibration of cylindrical shells

Abstract: The large amplitude vibrations of a thin-walled cylindrical shell are analyzed using the Donnell's shallow-shell equations. A perturbation method is applied to reduce the nonlinear partial differential equations into a system of linear partial differential equations. The simply-supported boundary condition and the circumferential periodicity condition are satisfied. The resulting solution indicates that in addition to the fundamental modes, the response contains asymmetric modes as well as axisymmetric modes with the frequency twice that of the fundamental modes. In the previous investigations in which the Galerkins procedure was applied, only the additional axisymrnetric modes were assumed. Vibrations involving a single driven mode response are investigated. The results indicate that the nonlinearity is either softening or hardening depending on the mode. The vibrations involving both a driven mode and a companion mode are also investigated. The region where the companion mode participates in the vibration is obtained and the effects due to the participation of the companion mode are studied. An experimental investigation is also conducted. The results are generally in agreement with the theory. "Non-stationary4 response is detected at some frequencies for large amplitude response where the amplitude drifts from one value to another. Various nonlinear phenomena are observed and quantitative comparisons with the theoretical results are made.

Existence and Representation Theorems for a Semilinear Sobolev Equation in Banach Space

Abstract: An existence theory is developed for a semilinear evolution equation in Banach space which is modeled on boundary value problems for partial differential equations of Sobolev type. The operators are assumed to be measurable and to satisfy coercive estimates which are not necessarily uniform in their time dependence, and to satisfy Lipschitz conditions on the nonlinear term. Applications are briefly indicated. 1. Introduction. We shall consider the abstract Cauchy problem for the nonlinear evolution equation ////(t)u'(t) + (t)u(t) f(t, u(t)) in a separable and reflexive Banach space. The linear operators /(t) are assumed to be weakly measurable in and to satisfy nonuniform coercive estimates over the Banach space which permit them to degenerate for certain values of t. The family of linear operators &(t) are assumed to be weakly measurable in t. The nonlinear term f(t, u) is measurable in and Lipschitz in u. Three types ofsolution are considered weak, mild, and strong.A mild solution is (essentially) a weak solution which permits a certain integral representation, and we shall prove that these two notions differ by a measurability assumption. A strong solution is a weak solution for which each term in the equation belongs to a specified Hilbert space for almost every t. The plan of the paper is as follows. Section 2 contains some technical results and notation we shall use. These include measurability of vector- and operator- valued functions, Gronwall's inequality, and an elementary fixed-point theorem for Banach space-valued functions. The weak solution is defined in 3, where we obtain results on uniqueness, local existence and global existence under various hypotheses. These results are used in 4 to construct the linear propagator (which resolves the linear equation withf 0) and thereby to introduce the notion of a mild solution. We prove that mild solutions (local and global) exist with the same hypotheses as used for existence of weak solutions. Strong solutions are introduced in 5. We give sufficient conditions for a mild solution to be strong; these conditions are essentially that the operators ///(t) dominate the operators &(t). Finally we obtain independently a sufficient condition for the existence (and uniqueness) of a strong solution; this condition requires that the function f be dominated by the operators /(t).

Formulation of a Reflector-Design Problem for a Lighting Fixture

Abstract: The problem of designing a reflector to distribute the illumination of a nonisotropic point source on a plane aperture according to a pre-assigned pattern is analyzed. An integral equation and equivalent partial differential equation are derived. The form of the latter reveals this reflector-design problem to be a singular elliptic Monge–Ampere boundary-value problem.

Stationary toroidal equilibria at finite beta.

Abstract: We investigate the effects of plasma flow on axisymmetric, self-consistent equilibria in toroidal geometry. The investigation is of considerable interest in relation to hot plasmas confined in toroidal systems with longitudinal current. On the basis of the one-fluid MHD plasma model, we use a concise formulation to elucidate important features of the equilibrium. In contrast to previous flow calculations which were treated almost exclusively in the low-beta approximation, we retain, together with flow, all beta effects.As in the treatment of the full flow problem at low beta, we find conditions for equilibrium. The form of our description allows a quite general discussion of its nature and existence for finite beta. This shows that there is a close relationship between the solvability conditions for equations arising from integrals of the system, and the nature of the characteristics of the partial differential equation describing the radial force balance. In the case of large aspect-ratio, these considerations lead to a generalized Bennett relation and to an expression for plasma displacement exhibiting beta and flow effects.

Non-local effects in the stability of flow between eccentric rotating cylinders

Abstract: The linear stability of the flow between two long eccentric rotating circular cylinders is considered The problem, which is of interest in lubrication technology, is an extension of the classical Taylor problem for concentric cylinders The basic flow has components in the radial and azimuthal directions and depends on both of these co-ordinates As a consequence the linearized stability equations are partial differential equations rather than ordinary differential equations Thus standard methods of stability theory are not immediately useful By letting the clearance ratio and eccentricity tend to zero in a given way a global solution to the stability problem is obtained

A modified Navier-Stokes equation, and its consequences on sound dispersion

Abstract: Taking into account some considerations developed in the past by Cattaneo we suggest a modified Navier-Stokes equation which includes the inertial property of the momentum flux. The linearized system of hydrodynamic equations where the inertial properties of both momentum and heat flux are taken into account leads to a new dispersion equation which does not exhibit the theoretical disadvantages of the Kirchhoff equation. Also the consistency with the experimental results about sound dispersion in rarefied monoatomic gases is definitely improved.

Existence of Variational Principles for the Navier‐Stokes Equation

Abstract: Frechet differentials are introduced to decide when a classical variational principle exists for a given nonlinear differential equation. The formalism is applied to the steady‐state Navier‐Stokes equation and the continuity equation, and no variational principle exists unless u × (∇ × u) = 0 or u· ∇u = 0. The concept of an adjoint equation is extended to nonlinear equations and a variational principle is derived for the Navier‐Stokes equation and its adjoint.

Scattering Theory for the Acoustic Equation in an Even Number of Space Dimensions

Abstract: : The paper extends the general theory of scattering developed by the authors for hyperbolic systems in an odd number of spacial dimensions to the even dimensional case. As before a key role is played by the incoming and outgoing subspaces and the corresponding translation representations - in this case the Radon transform. For the even dimensional case these two subspaces are no longer orthogonal. This eliminates from the previous theory the associated semigroup of operators which was so useful in characterizing the poles of the scattering matrix. (Modified author abstract)

Optimal filtering in linear distributed-parameter systems†

Abstract: The optimal filter of Kalman type is derived for linear distributed-parameter systems, which are subjected to both white gaussian distributed noise and boundary noise. The measurement data are also corrupted by white gaussian noise. It is assumed that the number of measuring instruments is finite; therefore, the measurement data are taken at several points of spatial domain. The Wiener—Hopf equation is obtained by applying the calculus of variations technique, from which the filter partial differential equations for the optimal estimate and its covariance matrix can be derived. The optimal estimate and its covariance are expanded into the series of eigenfunctions of the homogeneous partial differential equation with the homogeneous boundary condition. Thus a system of ordinary differential equations for coefficient functions of the series is derived

Computer Solutions to the Schrödinger Equation

Abstract: Numerical solution to the Schrodinger equation can be introduced to undergraduate students at the junior level. Numerov process is discussed in detail for the homogeneous differential equation and an extension of the method to inhomogeneous equations and self-consistent field calculations is briefly mentioned. An application to the simple harmonic oscillator is given as an example.

A theory of supercritical wing sections,: With computer programs and examples

Abstract: Mathematical methods for the design of supercritical wings, which depend on the numerical solution of the partial differential equations of two-dimensional gas dynamics, are developed. The main contribution is a computer program for the design of shockless transonic airfoils using the hodograph transformation and analytic continuation into the complex domain. The mathematical theory is described, and a manual for users of the programs is provided. Numerical examples are given and computational results are discussed, and the computer programs themselves are listed. The analysis routine can be used to ascertain whether the profiles behave well at off-design conditions, or to smooth coordinates and obtain a desirable shape more quickly when perfectly shockless flow is not essential.

General population theory in the age-time continuum

Abstract: A linear first-order partial differential equation is derived for the age density k(x,t) of a biological population. It contains factors which are time-dependent functions defining attrition and immigration. Emigration may be included in the attrition function. The general solution k(x,t) of the differential equation contains an arbitrary function that is identified as the birth rate B(t) of the community. A Volterra integral equation for B(t) arises if the total population N of the community is planned to be a prescribed function of time. Also, projection of the population on the basis of a given net maternity function and a given immigration function requires the solution B(t) of a linear integral equation. This equation can be transformed to a Volterra type, but another form in which the limits of integration are constant bounds of the fertility period has some advantages. Some properties of solutions of the integral equation are discussed, and effects of stepwise discontinuities in the kernel are considered.

Moment Equations for Waves Propagated in Random Media

Abstract: We show that the partial differential equations obtained by Tatarski, Beran, and Ho for the first, second, and fourth statistical moments of a wave propagating in a random medium are special cases of a general partial differential equation satisfied by the mth moment when the refractive-index inhomogeneities are sufficiently weak. In addition, we derive moment equations that apply regardless of the strength of the inhomogeneities. The results obtained in this case are in the form of difference relations satisfied for the moments.

On the Optimal Search for a Randomly Moving Target

Abstract: The problem involved in the search for a randomly moving target is considered in the case where the probability density function of the location of the target satisfies an equation of type (1). The searching effort is expressed in terms of a time-dependent search density function, and a necessary condition for the optimality of the search density function is derived. In the case of a stationary target this condition becomes the familiar one of Koopman [4]. Since most applications would involve the solution of a partial differential equation of parabolic type with appropriate initial and boundary conditions, the application of the optimality condition is difficult. Obviously a special numerical technique needs to be introduced, a task which we shall not attempt in the present paper.

Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems

Abstract: Introductory remarks.- A summary of some results on controls of hyperbolic partial differential equations.- The optimal control of vibrating beams.- Optimal control theory for thin plates.- Classification of the boundary conditions in optimal control theory of beams and thin plates.

On the inverse problem for a hyperbolic dispersive partial differential equation. II

Abstract: The inverse problem for a two‐dimensional (space‐time) hyperbolic partial differential equation, with coefficients, functions of the spatial variable only, is considered. Exterior to a region of compact support in the spatial variable, the equation reduces to the wave equation, and, from knowledge of the solution in the exterior region (namely in terms of reflected and transmitted waves for a prescribed incident wave), the problem is to deduce the coefficients in the interior region. This is achieved by treating the problem as a Cauchy initial value problem and using the Riemann function to deduce a dual set of integral equations. The coefficients or linear combinations of them are deduced from the solutions of the integral equations. The question of uniqueness is partially answered, by estimating the domain of convergence of the Neumann series. The application of the analysis to electromagnetic scattering from a slab of varying conductivity and permitivity is indicated.