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Languages : en
Pages :
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Book Description
Problems of technological interest can very often be described by partial differential equations (PDEs) with one dependent and two independent variables (call them c, z, and t, respectively). Many such PDEs are invariant to one-parameter families of one-parameter affine groups. Similarity solutions are solutions of the PDE that are invariant to one group of the family. The great utility of similarity solutions is that they may be calculated by solving an ODE rather than a PDE and are thus much more easily accessible than other solutions. The form of the principal ODE depends, of course, on the form of the PDE, but it can be proved quite generally that the principal ODE is itself invarient to the one-parameter affine group or associated group. because of the invariance of the principal ODE to the associated group, the dependence on the boundary and initial conditions of certain special values of the function y(x), e.g., y(O), y(infinity), y(O), ets., may be predicted a priori without solving the principal ODE. The nonlinear PDE of heat transport in superfluid He-II, is used as an illustration of these ideas in this review.
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Languages : en
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Book Description
Problems of technological interest can very often be described by partial differential equations (PDEs) with one dependent and two independent variables (call them c, z, and t, respectively). Many such PDEs are invariant to one-parameter families of one-parameter affine groups. Similarity solutions are solutions of the PDE that are invariant to one group of the family. The great utility of similarity solutions is that they may be calculated by solving an ODE rather than a PDE and are thus much more easily accessible than other solutions. The form of the principal ODE depends, of course, on the form of the PDE, but it can be proved quite generally that the principal ODE is itself invarient to the one-parameter affine group or associated group. because of the invariance of the principal ODE to the associated group, the dependence on the boundary and initial conditions of certain special values of the function y(x), e.g., y(O), y(infinity), y(O), ets., may be predicted a priori without solving the principal ODE. The nonlinear PDE of heat transport in superfluid He-II, is used as an illustration of these ideas in this review.
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Category : Aeronautics
Languages : en
Pages : 968
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Book Description
Author: Dominic G. B. Edelen
Publisher: World Scientific
ISBN: 9789810209339
Category : Mathematics
Languages : en
Pages : 348
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Book Description
The purpose of the book is to provide research workers in applied mathematics, physics, and engineering with practical geometric methods for solving systems of nonlinear partial differential equations. The first two chapters provide an introduction to the more or less classical results of Lie dealing with symmetries and similarity solutions. The results, however, are presented in the context of contact manifolds rather than the usual jet bundle formulation and provide a number of new conclusions. The remaining three chapters present essentially new methods of solution that are based on recent publications of the authors'. The text contains numerous fully worked examples so that the reader can fully appreciate the power and scope of the new methods. In effect, the problem of solving systems of nonlinear partial differential equations is reduced to the problem of solving families of autonomous ordinary differential equations. This allows the graphs of solutions of the system of partial differential equations to be realized as certain leaves of a foliation of an appropriately defined contact manifold. In fact, it is often possible to obtain families of solutions whose graphs foliate an open subset of the contact manifold. These ideas are extended in the final chapter by developing the theory of transformations that map a foliation of a contact manifold onto a foliation. This analysis gives rise to results of surprising depth and practical significance. In particular, an extended Hamilton-Jacobi method for solving systems of partial differential equations is obtained.
Author: P.L. Sachdev
Publisher: CRC Press
ISBN: 1000611418
Category : Mathematics
Languages : en
Pages : 172
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Book Description
Nonlinearity plays a major role in the understanding of most physical, chemical, biological, and engineering sciences. Nonlinear problems fascinate scientists and engineers, but often elude exact treatment. However elusive they may be, the solutions do exist-if only one perseveres in seeking them out. Self-Similarity and Beyond presents
Author: Mi-Ho Giga
Publisher: Springer Science & Business Media
ISBN: 0817646515
Category : Mathematics
Languages : en
Pages : 307
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Book Description
This work will serve as an excellent first course in modern analysis. The main focus is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. This textbook will be an excellent resource for self-study or classroom use.
Author: Gerry E. Hendershot
Publisher:
ISBN: 9780840602220
Category : Birth control
Languages : en
Pages : 982
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Book Description
The 1982 statistics on the use of family planning and infertility services presented in this report are preliminary results from Cycle III of the National Survey of Family Growth (NSFG), conducted by the National Center for Health Statistics. Data were collected through personal interviews with a multistage area probability sample of 7969 women aged 15-44. A detailed series of questions was asked to obtain relatively complete estimates of the extent and type of family planning services received. Statistics on family planning services are limited to women who were able to conceive 3 years before the interview date. Overall, 79% of currently mrried nonsterile women reported using some type of family planning service during the previous 3 years. There were no statistically significant differences between white (79%), black (75%) or Hispanic (77%) wives, or between the 2 income groups. The 1982 survey questions were more comprehensive than those of earlier cycles of the survey. The annual rate of visits for family planning services in 1982 was 1077 visits /1000 women. Teenagers had the highest annual visit rate (1581/1000) of any age group for all sources of family planning services combined. Visit rates declined sharply with age from 1447 at ages 15-24 to 479 at ages 35-44. Similar declines with age also were found in the visit rates for white and black women separately. Nevertheless, the annual visit rate for black women (1334/1000) was significantly higher than that for white women (1033). The highest overall visit rate was for black women 15-19 years of age (1867/1000). Nearly 2/3 of all family planning visits were to private medical sources. Teenagers of all races had higher family planning service visit rates to clinics than to private medical sources, as did black women age 15-24. White women age 20 and older had higher visit rates to private medical services than to clinics. Never married women had higher visit rates to clinics than currently or formerly married women. Data were also collected in 1982 on use of medical services for infertility by women who had difficulty in conceiving or carrying a pregnancy to term. About 1 million ever married women had 1 or more infertility visits in the 12 months before the interview. During the 3 years before interview, about 1.9 million women had infertility visits. For all ever married women, as well as for white and black women separately, infertility services were more likely to be secured from private medical sources than from clinics. The survey design, reliability of the estimates and the terms used are explained in the technical notes.
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Category : Power resources
Languages : en
Pages : 648
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Book Description
Author: E.E. Rosinger
Publisher: Elsevier
ISBN: 0080872573
Category : Science
Languages : en
Pages : 429
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Book Description
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concerning existence, uniqueness regularity, etc., of generalized solutions for nonlinear partial differential equations can be reduced to elementary calculus in Euclidean spaces, combined with elementary algebra in quotient rings of families of smooth functions on Euclidean spaces, all of that joined by certain asymptotic interpretations. In this way, one avoids the complexities and difficulties of the customary functional analytic methods which would involve sophisticated topologies on various function spaces. The result is a rather elementary yet powerful and far-reaching method which can, among others, give generalized solutions to linear and nonlinear partial differential equations previously unsolved or even unsolvable within distributions or hyperfunctions.Part 1 of the volume discusses the basic limitations of the linear theory of distributions when dealing with linear or nonlinear partial differential equations, particularly the impossibility and degeneracy results. Part 2 examines the way Colombeau constructs a nonlinear theory of generalized functions and then succeeds in proving quite impressive existence, uniqueness, regularity, etc., results concerning generalized solutions of large classes of linear and nonlinear partial differential equations. Finally, Part 3 is a short presentation of the nonlinear theory of Rosinger, showing its connections with Colombeau's theory, which it contains as a particular case.
Author: W. F. Ames
Publisher: Academic Press
ISBN: 1483221504
Category : Mathematics
Languages : en
Pages : 335
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Book Description
Nonlinear Partial Differential Equations: A Symposium on Methods of Solution is a collection of papers presented at the seminar on methods of solution for nonlinear partial differential equations, held at the University of Delaware, Newark, Delaware on December 27-29, 1965. The sessions are divided into four Symposia: Analytic Methods, Approximate Methods, Numerical Methods, and Applications. Separating 19 lectures into chapters, this book starts with a presentation of the methods of similarity analysis, particularly considering the merits, advantages and disadvantages of the methods. The subsequent chapters describe the fundamental ideas behind the methods for the solution of partial differential equation derived from the theory of dynamic programming and from finite systems of ordinary differential equations. These topics are followed by reviews of the principles to the lubrication approximation and compressible boundary-layer flow computation. The discussion then shifts to several applications of nonlinear partial differential equations, including in electrical problems, two-phase flow, hydrodynamics, and heat transfer. The remaining chapters cover other solution methods for partial differential equations, such as the synergetic approach. This book will prove useful to applied mathematicians, physicists, and engineers.
Author: Victor A Galaktionov
Publisher: CRC Press
ISBN: 9780367389970
Category :
Languages : en
Pages : 528
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Book Description
Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book focuses on the existence of new exact solutions on linear invariant subspaces for nonlinear operators and their crucial new properties. This practical reference deals with various partial differential equations (PDEs) and models that exhibit some common nonlinear invariant features. It begins with classical as well as more recent examples of solutions on invariant subspaces. In the remainder of the book, the authors develop several techniques for constructing exact solutions of various nonlinear PDEs, including reaction-diffusion and gas dynamics models, thin-film and Kuramoto-Sivashinsky equations, nonlinear dispersion (compacton) equations, KdV-type and Harry Dym models, quasilinear magma equations, and Green-Naghdi equations. Using exact solutions, they describe the evolution properties of blow-up or extinction phenomena, finite interface propagation, and the oscillatory, changing sign behavior of weak solutions near interfaces for nonlinear PDEs of various types and orders. The techniques surveyed in Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics serve as a preliminary introduction to the general theory of nonlinear evolution PDEs of different orders and types.
