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Author: David Y. Gao
Publisher: CRC Press
ISBN: 1420011731
Category : Mathematics
Languages : en
Pages : 270
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Book Description
Among the theoretical methods for solving many problems of applied mathematics, physics, and technology, asymptotic methods often provide results that lead to obtaining more effective algorithms of numerical evaluation. Presenting the mathematical methods of perturbation theory, Introduction to Asymptotic Methods reviews the most important m
Author: David Y. Gao
Publisher: CRC Press
ISBN: 1420011731
Category : Mathematics
Languages : en
Pages : 270
Get Book Here
Book Description
Among the theoretical methods for solving many problems of applied mathematics, physics, and technology, asymptotic methods often provide results that lead to obtaining more effective algorithms of numerical evaluation. Presenting the mathematical methods of perturbation theory, Introduction to Asymptotic Methods reviews the most important m
Author: S.H. Patil
Publisher: Springer Science & Business Media
ISBN: 3642573177
Category : Science
Languages : en
Pages : 178
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Book Description
Quantum mechanics and the Schrodinger equation are the basis for the de scription of the properties of atoms, molecules, and nuclei. The development of reliable, meaningful solutions for the energy eigenfunctions of these many is a formidable problem. The usual approach for obtaining particle systems the eigenfunctions is based on their variational extremum property of the expectation values of the energy. However the complexity of these variational solutions does not allow a transparent, compact description of the physical structure. There are some properties of the wave functions in some specific, spatial domains, which depend on the general structure of the Schrodinger equation and the electromagnetic potential. These properties provide very useful guidelines in developing simple and accurate solutions for the wave functions of these systems, and provide significant insight into their physical structure. This point, though of considerable importance, has not received adequate attention. Here we present a description of the local properties of the wave functions of a collection of particles, in particular the asymptotic properties when one of the particles is far away from the others. The asymptotic behaviour of this wave function depends primarily on the separation energy of the outmost particle. The universal significance of the asymptotic behaviour of the wave functions should be appreciated at both research and pedagogic levels. This is the main aim of our presentation here.
Author: Yuri A. Mitropolsky
Publisher: Springer Science & Business Media
ISBN: 9401588473
Category : Technology & Engineering
Languages : en
Pages : 352
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Book Description
Many dynamical systems are described by differential equations that can be separated into one part, containing linear terms with constant coefficients, and a second part, relatively small compared with the first, containing nonlinear terms. Such a system is said to be weakly nonlinear. The small terms rendering the system nonlinear are referred to as perturbations. A weakly nonlinear system is called quasi-linear and is governed by quasi-linear differential equations. We will be interested in systems that reduce to harmonic oscillators in the absence of perturbations. This book is devoted primarily to applied asymptotic methods in nonlinear oscillations which are associated with the names of N. M. Krylov, N. N. Bogoli ubov and Yu. A. Mitropolskii. The advantages of the present methods are their simplicity, especially for computing higher approximations, and their applicability to a large class of quasi-linear problems. In this book, we confine ourselves basi cally to the scheme proposed by Krylov, Bogoliubov as stated in the monographs [6,211. We use these methods, and also develop and improve them for solving new problems and new classes of nonlinear differential equations. Although these methods have many applications in Mechanics, Physics and Technique, we will illustrate them only with examples which clearly show their strength and which are themselves of great interest. A certain amount of more advanced material has also been included, making the book suitable for a senior elective or a beginning graduate course on nonlinear oscillations.
Author: D.M. Klimov
Publisher: CRC Press
ISBN: 1482265222
Category : Mathematics
Languages : en
Pages : 239
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Book Description
Group analysis of differential equations has applications to various problems in nonlinear mechanics and physics. Group-Theoretic Methods in Mechanics and Applied Mathematics systematizes the group analysis of the main postulates of classical and relativistic mechanics. Exact solutions are given for the following equations: dynamics of rigid body, heat transfer, wave, hydrodynamics, Thomas-Fermi, and more. The author pays particular attention to the application of group analysis to developing asymptotic methods for problems with small parameters. This book is designed for a broad audience of scientists, engineers, and students in the fields of applied mathematics, mechanics and physics.
Author: Jean Cousteix
Publisher: Springer Science & Business Media
ISBN: 3540464891
Category : Science
Languages : en
Pages : 437
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Book Description
This book presents a new method of asymptotic analysis of boundary-layer problems, the Successive Complementary Expansion Method (SCEM). The first part is devoted to a general presentation of the tools of asymptotic analysis. It gives the keys to understand a boundary-layer problem and explains the methods to construct an approximation. The second part is devoted to SCEM and its applications in fluid mechanics, including external and internal flows.
Author: Carl M. Bender
Publisher: Springer Science & Business Media
ISBN: 1475730691
Category : Mathematics
Languages : en
Pages : 605
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Book Description
A clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.
Author: V.I. Arnol'd
Publisher: Springer Science & Business Media
ISBN: 1475720637
Category : Mathematics
Languages : en
Pages : 530
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Book Description
This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.
Author: Wolfgang Wasow
Publisher: Courier Dover Publications
ISBN: 0486824586
Category : Mathematics
Languages : en
Pages : 385
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Book Description
This outstanding text concentrates on the mathematical ideas underlying various asymptotic methods for ordinary differential equations that lead to full, infinite expansions. "A book of great value." — Mathematical Reviews. 1976 revised edition.
Author: Milton Van Dyke
Publisher: Parabolic Press, Incorporated
ISBN:
Category : Science
Languages : en
Pages : 296
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Book Description
Author: George C. Sih
Publisher: Springer Science & Business Media
ISBN: 9789001798604
Category : Science
Languages : en
Pages : 578
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Book Description
It is weH known that the traditional failure criteria cannot adequately explain failures which occur at a nominal stress level considerably lower than the ultimate strength of the material. The current procedure for predicting the safe loads or safe useful life of a structural member has been evolved around the discipline oflinear fracture mechanics. This approach introduces the concept of a crack extension force which can be used to rank materials in some order of fracture resistance. The idea is to determine the largest crack that a material will tolerate without failure. Laboratory methods for characterizing the fracture toughness of many engineering materials are now available. While these test data are useful for providing some rough guidance in the choice of materials, it is not clear how they could be used in the design of a structure. The understanding of the relationship between laboratory tests and fracture design of structures is, to say the least, deficient. Fracture mechanics is presently at astandstill until the basic problems of scaling from laboratory models to fuH size structures and mixed mode crack propagation are resolved. The answers to these questions require some basic understanding ofthe theory and will not be found by testing more specimens. The current theory of fracture is inadequate for many reasons. First of aH it can only treat idealized problems where the applied load must be directed normal to the crack plane.
