Showing papers on "Partial differential equation published in 1978"
Book•
01 Jan 1978TL;DR: In this article, the authors give a systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate.
Abstract: This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.
4,869 citations
TL;DR: In this paper, a method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested, based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equation with their symmetries by using symplectic operators.
Abstract: A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators. It leads to a simple and sufficiently general model of integrable Hamiltonian equation, of which the Korteweg–de Vries equation, the modified Korteweg–de Vries equation, the nonlinear Schrodinger equation and the so‐called Harry Dym equation turn out to be particular examples.
1,432 citations
TL;DR: A new iterative method for the solution of systems of linear equations has been recently proposed by Meijerink and van der Vorst and has been applied to real laser fusion problems taken from typical runs of the laser fusion simulation code LASNEX.
908 citations
IBM1
TL;DR: In this paper, the phase shift method was used to solve the migration of zero-offset seismic records with laterally invariant velocities, and the migration process was solved very accurately by the phase-shift method.
Abstract: Accurate methods for the solution of the migration of zero-offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase-shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than finite-difference methods in general.
885 citations
Book•
01 Jan 1978TL;DR: In this paper, a mathematical framework in terms of semigroups is developed which enables the generalisation of the finite-dimensional results to infinite dimensions, and which includes partial differential equations and delay equations as special cases.
Abstract: The paper considers some control problems for systems described on infinite-dimensional spaces. A mathematical framework in terms of semigroups is developed which enables the generalisation of the finite-dimensional results to infinite dimensions, and which includes partial differential equations and delay equations as special cases. After first surveying some finite-dimensional results, abstract dynamical systems are introduced and the systems theory concepts of stability, controllability, observabiity and the linear quadratic problem are analysed. Throughout the paper the abstract theory is applied to a number of examples to obtain specific results.
793 citations
TL;DR: In this paper, the Korteweg-de Vries equation with the isospectral property is considered and rational solutions are obtained by performing an appropriate limiting procedure on the soliton solutions obtained by direct methods.
Abstract: Rational solutions of certain nonlinear evolution equations are obtained by performing an appropriate limiting procedure on the soliton solutions obtained by direct methods. In this note specific attention is directed at the Korteweg–de Vries equation. However, the methods used are quite general and apply to most nonlinear evolution equations with the isospectral property, including certain multidimensional equations. In the latter case, nonsingular, algebraically decaying, soliton solutions can be constructed.
348 citations
TL;DR: In this article, a new and simpler construction of the family of rational solutions of the Korteweg-deVries equation is given, related to a factorization of the Sturm-Liouville operators into first order operators and a new deformation problem for the latter.
Abstract: A new and simpler construction of the family of rational solutions of the Korteweg-deVries equation is given. This construction is related to a factorization of the Sturm-Liouville operators into first order operators and a new deformation problem for the latter. In the final section the spectral representation for the corresponding complex potentials is discussed.
331 citations
TL;DR: In this paper, the method of collocation using two finite element techniques is applied to the solution of the general population balance equation for particulate systems, and numerical solutions by both techniques are obtained in six cases for which analytical or asymptotic solutions are available.
313 citations
TL;DR: In this article, it was shown that the Poincar6 recurrence theorem is also true for arbitrary k-commuting transformations and that the same result holds for any k-commodity transformations.
Abstract: The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n > 0,/z (T-'A n A) > 0. In [1] this was extended to multiple recurrence: the transformations T, T2, ..., T k have a common power satisfying /x (A n T-hA n ... n T-k"A)> 0 for a set A of positive measure. We also showed that this result implies Szemer6di's theorem stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In [2] a topological analogue of this is proved: if T is a homeomorphism of a compact metric space X, for any e >0 and k = 1,2,3,..-, there is a point x E X and a common power of T, T 2, 9 9 9 T k such that d(x, Tnx) < e, d(x, T2"x) < e,. 9 d(x, Tk~x) < e. This (weaker) result, in turn, implies van der Waerden's theorem on arithmetic progressions for partitions of the integers. Now in this case a virtually identical argument shows that the topological result is true for any k commuting transformations. This would lead one to expect that the measure theoretic result is also true for arbitrary commuting transformations. (It is easy to give a counterexample with noncommuting transformations.) We prove this in what follows. A corollary is the multidimensional extension of Szemer6di's theorem: Theorem B. Let S C Z" be a subset with positive upper density and let F C Z" be any finite configuration. Then there exists an integer d and a vector n E Z" such that n+dFCS.
303 citations
214 citations
TL;DR: In this article, it was shown that the sum of the boundary values of K-finite functions converges in the sense of analytic functionals in the case that the rank of the symmetric space is equal to one.
Abstract: In his paper [14], S. Helgason conjectured that any joint-eigenfunction of all invariant differential operators on a symmetric space can be given by the Poisson integral. The purpose of this paper is to prove this conjecture (see corollary to the theorem in the Section 5). There have appeared several papers dealing with the conjecture in the case that the rank of the symmetric space is equal to one ([12], [13], [15], [17], [30], [34], [35], [36]). The proofs given in these papers follow an idea due to S. Helgason [14] and may be explained as follows. In the rank one case the algebra of all invariant differential operators is generated by the Laplace-Beltrami operator. First, one expands any eigenfunction of the laplacian into K-finite functions. Then these K-finite functions are also eigenfunctions of the laplacian and have boundary values in the natural way. The radial component of the laplacian gives rise to hypergeometric differential equations. Thanks to the classical results on hypergeometric functions, one can estimate the asymptotic behavior of the solution near the boundary. This enables us to prove that the sum of the boundary values of K-finite functions converges in the sense of analytic functionals. In the higher rank case, however, the radial components of invariant differential operators are not ordinary differential operators anymore, so that one is unable to apply the classical results. In the meanwhile, some of the present authors have recently generalized the notion of "regular singularity" for the ordinary differential equation to that for the system of partial differential equations. The essential point is that the system of invariant differential equations has regular singularity along the Martin boundary which assures the existence of the boundary values of a solution as "hyperfunctions." In the method mentioned above, one encounters the crucial difficulty in proving the exist-
TL;DR: In this article, the mixed initial-boundary value problem is considered and the energy method is used to derive well-posed boundary conditions such that, when ε tends to zero, the reduced problem is also well posed.
Abstract: Some partial differential equations encountered in physical applications are of incompletely parabolic type; the Navier–Stokes equations in fluid dynamics are a typical example. In this paper we analyze such systems; in particular we treat the mixed initial-boundary value problem. In many applications there is a small parameter $\varepsilon $ multiplying the coefficient for the highest derivative. The energy method is used to derive well-posed boundary conditions such that, when $\varepsilon $ tends to zero, the reduced problem is also well posed.
TL;DR: In this article, a review of existence theorems for critical points of real-valued functions on a real Banach space is presented and applied to elliptic and hyperbolic partial differential equations.
Abstract: Publisher Summary This chapter reviews some existence theorems for critical points of a real-valued function on a real Banach space and to apply these results to elliptic and hyperbolic partial differential equations The abstract results on critical points are obtained using minimax arguments Applications to elliptic equations are thereafter provided for the same A new proof is given for a recent result of Ahmad et al , as well as some variants of their result The work on abstract results on critical points is applied to hyperbolic problems
TL;DR: In this paper, a simple, highly idealized hydrodynamic model stream is examined to illustrate the fundamental processes involved in the three-dimensional dynamics of stream evolution, and it is found that spatial variations in the rotational stream interaction mechanism produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into relative rarefaction.
Abstract: The paper is concerned with the development of the theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 Rs where the flow is supersonic) is approximately described by the nonlinear, single-fluid, polytropic magnetohydrodynamic or hydrodynamic equations. Efficient numerical techniques are outlined for solving this complex system of coupled, hyperbolic partial differential equations. The present formulation is inviscid and nonmagnetic, but the methods used allow for the potential inclusion of both features with only modest modifications. A simple, highly idealized hydrodynamic model stream is examined to illustrate the fundamental processes involved in the three-dimensional dynamics of stream evolution. It is found that spatial variations in the rotational stream interaction mechanism produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction. Comparison with simpler models demonstrates the essential nonlinear, multidimensional nature of the interplanetary dynamics.
TL;DR: In this article, some of the previous algorithms are generalized to higher orders of accuracy, and a more detailed consideration of the second order algorithm is given which leads naturally to methods of orders three and four.
Abstract: In Part I of this paper (SIAM J Numer Anal, 15 (1978), pp 1212–1224), novel algorithms were introduced for solving parabolic differential equations in which high-frequency components occurred in the solution and for which $A_0 $-stable methods, exemplified by the classical Crank-Nicolson method, were less than satisfactory The algorithms presented were based on a simple extrapolation of the Backward Euler method which produced $L_0 $ -stability In all cases the algorithms presented were second order accurate in time In the present paper some of the previous algorithms are generalized to higher orders of accuracy In particular, a more detailed consideration of the second order algorithm is given which leads naturally to methods of orders three and four The novel algorithms are tested on a heat equation with constant coefficients in which a discontinuity between the initial values and boundary values exists
TL;DR: In this article, a Crank-Nicolson-Galerkin approximation with extrapolated coefficients is presented along with a conjugate gradient iterative procedure which can be used efficiently to solve the different linear systems of algebraic equations arising at each step from the Galerkin method.
Abstract: Three cases for the nonlinear Sobolev equation c(x,u)(Ou/Ot)-V.(a(x,u)Vu+ b(x, u, Vu)V(Ou/Ot))=f(x, t, u, Vu) are studied. In case I, the coefficients a and b have uniform positive lower bounds in a neighborhood of the solution; in case II, b b(x, u) is allowed to take zero values and possibly cause the Sobolev equation to degenerate to a parabolic equation; in case III we only require a bound of the form la(x, u)l
TL;DR: In this paper, the authors trace out the broad characteristics of a class of higher order finite difference schemes which are applicable to the solution of parabolic partial differential equations associated with viscous fluid flow problems.
TL;DR: In this paper, the application of Partial Differential Equation (PDE) models for restoration of noisy images is considered and performance bounds based on PDE model theory are calculated and implementation tradeoffs of different algorithms are discussed.
Abstract: Application of Partial Differential Equation (PDE) models for restoration of noisy images is considered. The hyperbolic, parabolic, and elliptic classes of PDE's yield recursive, semirecursive, and nonrecursive filtering algorithms. The two-dimensional recursive filter is equivalent to solving two sets of filtering equations, one along the horizontal direction and other along the vertical direction. The semirecursive filter can be implemented by first transforming the image data along one of its dimensions, say Column, and then recursive filtering along each row independently. The nonrecursive filter leads to Fourier domain Wiener filtering type transform domain algorithm. Comparisons of the different PDE model filters are made by implementing them on actual image data. Performances of these filters are also compared with Fourier Wiener filtering and spatial averaging methods. Performance bounds based on PDE model theory are calculated and implementation tradeoffs of different algorithms are discussed.
TL;DR: Using these techniques, the governing nonlinear partial differential equation (PDE) can be replaced locally by a vector field on a low dimensional manifold and the bifurcations thus detected yield a useful description of the qualitative dynamics of the original PDE.
TL;DR: In this article, the authors give a functional-analytical treatment of discretization methods such as quadrature formula method for nonlinear integral equations, difference method for nonsmooth boundary value problems, etc.
Abstract: THIS survey paper gives a functional-analytical treatment of discretization methods such as quadrature formula method for nonlinear integral equations, difference method for nonlinear boundary value problems, etc. Two approaches to the convergence problem have been developed. The first of them (Section 3) is applicable to an equation with differentiable operator and rests on a remark that such an operator is locally almost linear. The second, less traditional approach (Section 4) is based on a topological concept, namely the invariance of the fixed point index under suitable approximations of an operator. As regards the approximation concepts, the paper is built on a relatively novel principle of regular convergence of operators (Section 2). In our fixed opinion, this concept is rather appropriate to applications, and we hope that the reader agrees with us familiarizing himself with the proof ideology of Sections 5-7. Another methodological prop of the paper is the concept of discrete convergence (Section 1). In Sections 5-7 the abstract results of Sections l-4 have been applied to the quadrature formula method for nonlinear integral equations and to the collocation, subregion, Galerkin and difference methods for nonlinear boundary value problems. Only ordinary differential equations are considered. For partial differential equations our approaches are still weakly developed : first works (e.g. [l-3]) concern linear equations. Sections l-3 contain more material than is urgently needed for applications, our significant goal. By stars are labelled the sections, propositions etc. that can be omitted if one wishes to get to applications more quickly. The main text contains only few references. For the reference notes, see the end of the paper.
Journal Article•
TL;DR: In this article, Variational methods are used to obtain new existence theorems for critical points of a real valued function on a Banach space, and applications are made to semilinear elliptic boundary value problems.
Abstract: : Variational methods are used to obtain some new existence theorems for critical points of a real valued function on a Banach space. Applications are made to semilinear elliptic boundary value problems. (Author)
01 Jan 1978
TL;DR: In this article, the authors highlight the applicability of the abstract considerations to several concrete problems, explain what useful information semigroup theory provides and what it does not, and demonstrate how in certain cases, various partial differential equations techniques can be used to extend the theoretical understanding.
Abstract: Publisher Summary This chapter highlights the facts about nonlinear semigroups in arbitrary Banach spaces and an exposition of some applications and extensions of this general theory to certain nonlinear partial differential equations of parabolic type. The chapter focuses on the applicability of the abstract considerations to several concrete problems, to explain what useful information semigroup theory provides and what it does not, and to demonstrate how in certain cases, various partial differential equations techniques can be used to extend the theoretical understanding. The chapter presents the basic working information about nonlinear semigroups, including Crandall–Liggett generation theorem and various regularity and perturbation results. It also presents the nonlinear Chernoff theorem and several related topics, with a view toward the applications. These are the porous medium and related equations, certain variational and quasi-variational inequalities of evolution, and Bellman's equation.
15 Mar 1978
TL;DR: The dual-shaped reflector synthesis problem was first solved by Galindo and Kinber in the early 1950's for the circularly symmetric shaped reflectors as mentioned in this paper, and it was shown that the surfaces required to transform this feed pattern by geometrical optics into any specified phase and amplitude pattern in the specified output aperture are found by the integration of two simultaneous nonlinear ordinary differential equations.
Abstract: The dual-shaped reflector synthesis problem was first solved by Galindo and Kinber in the early 1950's for the circularly symmetric-shaped reflectors. Given an arbitrary feed pattern, it was shown that the surfaces required to transform this feed pattern by geometrical optics into any specified phase and amplitude pattern in the specified output aperture are found by the integration of two simultaneous nonlinear ordinary differential equations. For the offset noncoaxial geometry, however, it is shown that the equations found by this method are partial differential equations which, in general, do not form a total differential. Hence the exact solution to this problem is generally not possible. It is also shown, however, that for many important problems the partial differential equations form a nearly total differential. It thus becomes possible to generate a smooth subreflector by integration of the differential equations and then synthesize a main reflector which gives an exact solution for the specified aperture phase distribution. The resultant energy (or amplitude) distribution in the output aperture as well as the output aperture periphery are then approximately the specified values. A representative group of important solutions are presented which illustrate the very good quality that frequently results by this synthesis method. This includes high gain, low sidelobe, near-field Cassegrain, and different ( f/D ) ratio reflector systems.
TL;DR: In this paper, the authors present an asymptotic behalrior of the scattering phase for exterior domains, based on the partial differential equation (PDE) of the exterior domain.
Abstract: (1978). Asymptotic behalrior of the scattering phase for exterior domains. Communications in Partial Differential Equations: Vol. 3, No. 12, pp. 1165-1195.
TL;DR: It is explained how complete connectivity conditions can be used to replace boundary integral equations in many situations, and it is shown that completeness is satisfied not only in steady-state problems such as potential, reduced wave equation and static and quasi-static elasticity, but also in time-dependent problemssuch as heat and wave equations and dynamical elasticity.
Abstract: In previous papers Herrera developed a theory of connectivity that is applicable to the problem of connecting solutions defined in different regions, which occurs when solving partial differential equations and many problems of mechanics. In this paper we explain how complete connectivity conditions can be used to replace boundary integral equations in many situations. We show that completeness is satisfied not only in steady-state problems such as potential, reduced wave equation and static and quasi-static elasticity, but also in time-dependent problems such as heat and wave equations and dynamical elasticity. A method to obtain bases of connectivity conditions, which are independent of the regions considered, is also presented.
01 Jan 1978
TL;DR: Attention is given to the types of boundary-fitted coordinate systems, coordinate system control, operation of the coordinate codes, solution of partial differential equations, application to free-surface flow, and other applications of interest.
Abstract: The technique of boundary-fitted coordinate systems is based on a method of automatic numerical generation of a general curvilinear coordinate system having a coordinate line coincident with each boundary of a general multi-connected region containing any number of arbitrarily shaped bodies. Once the curvilinear coordinate system is generated, any partial differential system of interest can be solved on this coordinate system by transforming the equations and solving the resulting system in finite difference approximation on the rectangular transformed plane. This method of automatic body-fitted curvilinear coordinate generation is used to construct finite-difference solutions of the full, time dependent Navier-Stokes equations for the unsteady viscous flow about arbitrary two-dimensional airfoils, or any other two-dimensional bodies. Finally, initial results for three-dimensional applications are also presented.
TL;DR: In this article, numerical solutions for the problem of laminar natural convection in a square enclosure using the penalty function, finite-element method are presented, and the simplicity and general applicability of the method are shown especially in the context of extensions to threedimensional geometries and irregular computational grids.
Abstract: Numerical solutions for the problem of laminar natural convection in a square enclosure using the penalty function, finite-element method are presented. Solutions are obtained for values of the Rayleigh number up to 107 using primitive fluid variables, and the efficacy of the method is demonstrated through a qualitative and quantitative evaluation of the results. The simplicity and general applicability of the method are shown especially in the context of extensions to three-dimensional geometries and irregular computational grids.
15 Sep 1978
TL;DR: In this article, the finite element method is applied to collinear reactive scattering problems, where no basis set expansion of the wave function is required and a direct solution of the two-dimensional partial differential equation is achieved.
Abstract: The finite element method is applied to collinear reactive scattering problems. In this way no basis set expansion of the wave function is required and a direct solution of the two-dimensional partial differential equation is achieved. It is shown how to generally formulate this approach and achieve fast and accurate results. As a test calculation the method was applied to H + H2, yielding excellent agreement with close coupling results. Since no basis sets are used in the finite element calculation, no question of basis set convergence or closed channel behavior arises. Some discussion on applications to higher dimensions is also included.