Showing papers on "Partial differential equation published in 1971"

The application of integral equation methods to the numerical solution of some exterior boundary-value problems

Abstract: The application of integral equation methods to exterior boundary-value problems for Laplace’s equation and for the Helmholtz (or reduced wave) equation is discussed. In the latter case the straightforward formulation in terms of a single integral equation may give rise to difficulties of non-uniqueness; it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first. Finally, an outline is given of methods for transforming the integral operators with strongly singular kernels which occur in the second equation.

Monotonicity Methods in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations

Abstract: Publisher Summary This chapter discusses the monotonicity methods in Hilbert spaces and presents some applications to nonlinear partial differential equations. It describes classical properties of maximal monotone operators in Hilbert spaces. It focuses on a particular class of monotone operators, namely those that are gradients of convex functions. The chapter also highlights their specific properties that do not hold for general monotone operators. Evolution equations associated with gradients of convex functions: smoothing effect on the initial data, behavior at infinity, and so on are discussed in the chapter along with some applications to nonlinear partial differential equations.

Parallel Propagation of Nonlinear Low‐Frequency Waves in High‐β Plasma

Abstract: A pair of coupled, nonlinear, partial differential equations which describe the evolution of low‐frequency, large‐scale‐length perturbations propagating parallel, or nearly parallel, to the equilibrium magnetic field in high‐β plasma have been obtained. The equations account for irreversible resonant particle effects. In the regime of small but finite propagation angles, the pair of equations collapses into a single Korteweg‐de Vries equation (neglecting irreversible terms) which agrees with known results.

Geometric Approach to Invariance Groups and Solution of Partial Differential Systems

Abstract: Methods are discussed for discovery of physically or mathematically special families of exact solutions of systems of partial differential equations. Such systems are described geometrically using equivalent sets of differential forms, and the theory derived for obtaining the generators of their invariance groups‐vector fields in the space of forms. These isovectors then lead naturally to all the special solutions discussed, and it appears that other special ansatze must similarly be capable of geometric description. Application is made to the one‐dimensional heat equation, the vacuum Maxwell equations, the Korteweg‐de Vries equation, one‐dimensional compressible fluid dynamics, the Lambropoulos equation, and the cylindrically symmetric Einstein‐Maxwell equations.

Optimal limited state variable feedback controllers for linear systems

Abstract: This paper considers the problem of designing compensators, the dimensions of which are fixed a priori, for linear systems. Two types of compensators are considered: first, static (gain only) compensators which operate directly upon the output signals to generate the controls, and second, dynamic compensators of fixed dimension. The equations that define the parameters of such compensators are developed.

Alternating-direction galerkin methods on rectangles

Abstract: Publisher Summary This chapter presents an overview of alternating-direction Galerkin methods on rectangles. Alternating-direction methods in several forms have proved to be very valuable in the approximate solution of partial differential equations problems involving several space variables by finite differences. The methods have been applied to transient problems directly and to stationary problems as iterative procedures. The chapter presents highly efficient procedures for the numerical solution of second-order parabolic and hyperbolic problems in two or more space variables and for the iterative solution of the algebraic equations arising from the Galerkin treatment of elliptic problems. The results presented are limited to rectangular domains. The chapter presents heat equation on a rectangle and extensions to variable coefficients and nonlinear parabolic equations and systems. It describes an iterative procedure for elliptic equations.

Finite-element solution of 2-dimensional exterior-field problems

Abstract: A new method, based on the finite-element method of discretisation and compatible with existing finite-element techniques, is described for the solution of field problems in which the region of prime interest is embedded in an infinitely extending region where Laplace's equation holds. The essence of this method lies in representing the infinitely extending region by a single finite element, which may be included in an element assembly descriptive of the finite region of major interest. No iterative computation is necessary, and the computing effort required for solution is essentially the same as that required for solving the interiorfield problem alone.

Overland Flow on an Infiltrating Surface

Abstract: The partial differential equation for vertical, one-phase, unsaturated moisture flow in soils is employed as a mathematical model for infiltration rate. Solution of this equation for the rainfall-ponding upper boundary condition is proposed as a sensitive means to describe infiltration rate as a dependent upper boundary condition. A nonlinear Crank-Nicholson implicit finite difference scheme is used to develop a solution to this equation that predicts infiltration under realistic upper boundary and soil matrix conditions. The kinematic wave approximation to the equations of unsteady overland flow on cascaded planes is solved by a second order explicit difference scheme. The difference equations of infiltration and overland flow are then combined into a model for a simple watershed that employs computational logic so that boundary conditions match at the soil surface. The mathematical model is tested by comparison with data from a 40-foot laboratory soil flume fitted with a rainfall simulator and with data from the USDA Agricultural Research Service experimental watershed at Hastings, Nebraska. Good agreement is obtained between measured and predicted hydrographs, although there are some differences in recession lengths. The results indicate that a theoretically based model can be used to describe simple watershed response when appropriate physical parameters can be obtained.

Motion of the Seawater Interface in Coastal Aquifers: A Numerical Solution

Abstract: The partial differential equations that describe the motion of the seawater interface and the free surface in a phreatic coastal aquifer (or the freshwater head replacing the latter, in the confined case) are presented. They are based on the Dupuit approximation and take into consideration the geometry of the vertical section through the aquifer, in whose plane the flow takes place, as well as the spatial variation of properties of the porous medium and the spatial and temporal distributions of accretion, recharge, and pumping. An implicit numerical scheme is presented to solve the set of simultaneous partial differential equations. The scheme is based on a linearization of the equations and employs a grid with one spacing over the intrusion length and a different spacing in the remainder of the field. Efficient solution of the resulting set of simultaneous linear equations for each time step is achieved by arranging them in a way that results in a 7 diagonal coefficient matrix. Examples are presented, for which the numerical solutions are compared with analytical solutions or laboratory experiments.

An integral equation for unsteady surface waves and a comment on the Boussinesq equation

Abstract: An integral equation for unsteady inviscid surface waves has been obtained. Existing known approximations are all derived from the one equation. The Boussinesq equation is obtained and criticized.

A Finite Difference Method for Unsteady Flow in Variably Saturated Porous Media: Application to a Single Pumping Well

Abstract: Finite difference equations were derived by using the divergence theorem to convert the nonlinear partial differential equation (which approximately describes liquid flow in a variably saturated, elastic porous medium) to an integral equation, and then to integrate around individual mesh volume elements. Original nonlinearity of the differential equation was preserved by keeping saturations and relative conductivities current with hydraulic heads during the iterative matrix solution method. The problem of axisymmetric flow to a water well that penetrates one or more elastic rock units, the upper one of which is unconfmed, provides a convenient vehicle for analysis of the procedural and theoretical study of unconfined and semiconfined flow. Of the three methods tried to solve the matrix equation that resulted from the finite difference equations (which included a form of the direct alternating direction implicit method, the iterative alternating direction implicit method, and the line successive overrelaxation method), the line successive overrelaxation method was the fastest and was selected for use in a general computer program. A comparison with analytical solutions that use Boulton's convolution integral as a velocity boundary condition at the water table for a single aquifer and an aquifer-aquitard system demonstrates close correspondence of the numerical and analytical solutions, even for a case where the water table is lowered appreciably.

Non-linear wave-number interaction in near-critical two-dimensional flows

Abstract: This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Benard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

New Difference Schemes for the Neutron Transport Equation

Abstract: Three new difference schemes for solving the neutron transport equation are presented. In some situations these new methods give smoother results than the diamond difference scheme, without sacrifi...

Inversion of the telegraph equation and the synthesis of nonuniform lines

Abstract: The synthesis of nonuniform Iossless transmission lines, an important problem arising in fields such as waveguide design, acoustics, electrical circuit design, and scattering theory, is discussed entirely in the time domain. It is shown that corresponding to every (impulse-response) function satisfying certain regularity conditions and the passivity condition there is a lossless line whose taper is simply related to the impulse response through an integral equation. In particular, this correspondence is unique for lines along which the velocity is constant. The proofs use only elementary results from functional analysis, the theory of partial differential equations, and some well-known physical aspects of lossless lines.

An exact analysis of low peclet number thermal entry region heat transfer in transversely nonuniform velocity fields

Abstract: A theoretical method is described for obtaining the exact solution to the problem of thermal entry region heat transfer which takes into account both transverse non-uniformity in the velocity field and axial conduction. To allow for the effect of upstream conduction, the fluid temperature was taken to be uniform at × = – ∞, and the first 20 eigenvalues and the corresponding eigenfunctions were determined separately for the heated and adiabatic regions. Both the temperatures and temperature gradients were then matched at × = 0 by constructing a pair of orthonormal functions from the nonorthogonal eigenfunctions. Nusselt numbers calculated for pipe flow subject to the boundary condition of uniform wall heat flux show virtually perfect agreement with those reported recently by Hennecke, who solved the governing partial differential equation numerically. To illustrate its general applicability, the present method was further employed to analyze the corresponding problem in parallel-plate channel flow, for which no solution has hitherto been reported.

A Global Theorem for Nonlinear Eigenvalue Problems and Applications

Abstract: Publisher Summary This chapter describes a global theorem for nonlinear eigenvalue problems and applications. Equations of the form (0.1) occur in many parts of mathematical physics, particularly in fluid dynamics and in elasticity theory. Thus, the nature of the structure of the set of their solution is an important question. A nonlinear extension of a linear Strum-Liouville theorem for second order ordinary differential equations is presented in the chapter where nodal properties play an important role. Existence results for positive solutions of quasi-linear elliptic partial differential equations are also obtained. The homotopy methods, which go into the proof of main result, can be used to treat (0. 1) under other conditions on G and in particular in situations where bifurcation is not involved. Such a case is provided by requiring that G (0, u) = 0. The solution set of (0. 1) contains a component containing (0,0), which is unbounded both in IR + X E and IR - X E. The chapter provides a simple proof and some applications to elliptic and hyperbolic partial differential equations and extensions of many of the results.

Quasi-Geometric Optics of Media with Inhomogeneous Index of Refraction*

Abstract: A small-wavelength, quasi-geometric optics is developed. The approach is based on an amplitude wave function that represents both the probability amplitude of the geometrical rays and the physical-optics disturbance. All ray paths contribute phases to the amplitude function proportional to their optical paths, measured in units of normalized wavelength. However, the dominant contributions are due to those rays that satisfy Fermat’s principle. This formulation leads to an integral equation for the wave-amplitude function with a kernel function that is a path integral. The integral equation reduces, in weakly focusing paraxial media, to a solution of a Schrodinger type of partial differential equation. The classical ray variables are operators with expectation values satisfying the geometric-optics equations. The second moments of the ray operators, with minimum-uncertainty initial wave packet, define second-order optical parameters. A special case of the quasi-geometric-optics integral equation is the quasi-optical laser-mode equation, allowing the interpretation of the stationary modes as probability amplitudes. For media with arbitrary inhomogeneous index of refraction, a matrix wave equation with a two-component wave function is derived. The matrix ray operators also satisfy a correspondence principle. A physical interpretation of the new wave function is presented.

Kinetic Equation Approach to Phase Transitions

Abstract: An exact mathematical discussion of the linearized Enskog-Vlasov equation is given. A criterion for the occurrence of the linear instability is related to a criterion for the occurrence of the bifurcation of the equilibrium stationary solution to the nonlinear Enskog-Vlasov equation. Mathematical results are interpreted physically in connection with phase transitions.

Nonlinear Functional Analysis and Nonlinear Integral Equations of Hammerstein and Urysohn Type

Abstract: Publisher Summary This chapter discusses nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. The theory of nonlinear integral equations of Hammerstein type has been, since its inception in the paper of Hammerstein, one of the most important domains of application of the ideas and techniques of nonlinear functional analysis, second only to the theory of solutions of boundary value problems for nonlinear partial differential equations. The development of the fixed point and degree theory for compact nonlinear mappings in Banach spaces was strongly influenced, in its form, by the theory of nonlinear integral equations and was directly applied to this domain and many others. The chapter presents a unified development of the theory of the Hammerstein equation using the theory of the topological degree for mappings of the form I - C with C compact as well as the basic theory of monotone nonlinear mappings from X to X*.

Approximate kinetic equations in rarefied gas theory

Abstract: The fundamental kinetic equation of gas theory, the Boltzniann equation, is a complex integrodiffcrential equation. The difficulties associated with its solution are the result not only of the large number of independent variables, seven in the general case, but also of the very complicated structure of the collision integral. However, for the mechanics of rarefied gases the primary interest lies not in the distribution function itself, which satisfies the Boltzmann equation, but rather in its first few moments, i.e., the averaged characteristics. This circumstance suggests the possibility of obtaining the averaged quantities by a simpler way than the direct method of direct solution of the Boltzmann equation with subsequent calculation of the integrals. It is well known that if a distribution function satisfies the Boltzmann equation, then its moments satisfy an infinite system of moment equations. Consequently, if we wish to obtain with satisfactory accuracy some number of first moments, then we must require that these moments satisfy the exact system of moment equations. However, this does not mean that to determine the moments of interest to us we must solve this system, particularly since the system of moment equations is not closed. The closure of the system by specifying the form of the distribution function (method of moments) can be considered only as a rough approximate method of solving problems. First, in this case it is not possible to satisfy all the equations and we must limit ourselves to certain of the equations; second, generally speaking, we do not know which equation the selected distribution function satisfies, and, consequently, we do not know to what degree it has the properties of the distribution function which satisfies the Boltzmann equation. A more reliable technique for solving the problems of rarefied gasdynamics is that based on the approximation of the Boltzmann equation, more precisely, the approximation of the collision integral. The idea of replacing the collision integral by a simpler expression is not new [1–4]. The kinetic equations obtained as a result of this replacement are usually termed model equations, since their derivation is usually based on physical arguments and not on the direct use of the properties of the Boltzmann collision integral. In this connection we do not know to what degree the solutions of the Boltzmann equation and the model equations are close, particularly since the latter do not yield the possibility of refining the solution. Exceptions are the kinetic model for the linearized Boltzmann equation [5] and the sequence of model equations of [6], constructed by a method which is to some degree analogous with that of [5]. In the present paper we suggest for the simplification of the solution of rarefied gas mechanics problems a technique for constructing a sequence of approximate kinetic equations which is based on an approximation of the collision integral. For each approximate equation (i.e., equation with an approximate collision operator) the first few moment equations coincide with the exact moment equations. It is assumed that the accuracy of the approximate equation increases with increase of the number of exact moment equations. Concretely, the approximation for the collision integral consists of a suitable approximation of the reverse collision integral and the collision frequency. The reverse collision integral is represented in the form of the product of the collision frequency and a function which characterizes the molecular velocity distribution resulting from the collisions, where the latter is selected in the form of a locally Maxwellian function multiplied by a polynomial in terms of the components of the molecular proper velocities. The collision frequency is approximated by a suitable expression which depends on the problem conditions. For the majority of problems it may obviously be taken equal to the collision frequency calculated from the locally Maxwellian distribution function; if necessary the error resulting from the inexact calculation of the collision frequency may be reduced by iterations. To illustrate the method, we solve the simplest problem of rarefied gas theory-the problem on the relaxation of an initially homogeneous and isotropic distribution in an unbounded space to an equilibrium distribution.

Numerical design of transonic airfoils

Abstract: Publisher Summary This chapter discusses numerical design of transonic airfoils. It describes an inverse method of computing plane transonic flows past air foils that are not only free of shocks but also have adverse pressure gradients so moderate that no separation of the turbulent boundary layer should take place. Up-to-date existence and uniqueness theorems combine with the experimental evidence to assure that these flows are physically realistic and will occur in practice. The chapter presents the partial differential equations governing steady two-dimensional flow of an inviscid compressible fluid by numerical analysis of characteristic initial value problems for the analytic continuation of the solution into the complex domain. The finite difference scheme presented in the chapter was originally introduced to describe the detached shock wave in front of a blunt body but is actually better suited to the inverse problem of shaping air foils so as to achieve shock-free transonic flow. It is related to Bergman's integral operator method and does exploit simplifications associated with the linearity of the equations of motion in the hodograph plane.