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Perturbation methods
Book Reviews classical way (based on the frequency response concept) but from the point of view of the time domain. The work also deals with such problems as self-adjusting and hierarchic systems, not found in most introductory textbooks, but still being researched intensively. Part I (Ch. l-5) summarizes the basic principles of cybernetics, while the second part (Ch. 6-15) describes signal transmission and state behaviour in linear continuous-time systems. The introductory concepts and the relations between cybernetics and engineering are dealt with first, followed by a conceptual summary of motion, state and behaviour of cybernetical systems. The third chapter discusses model building and the characteristic functions of the steady-state and dynamic behaviour. The fourth chapter contains the requirement-s and global structures as well as the system characteristic of automatic control. The fifth chapter deals with the problems of information and information processing. The sixth chapter treats the basic problems of analysis, identification and synthesis of control systems. The principles of the state space method are discussed next. Chapters 8-10 describe signal transmission by decomposing the signals into sine waves and exponential impulses, signals. The application of the convolution integrals, the Fourier and Laplace transforms are also dealt with. Next, the basic types of transfer elements are classified. The last two chapters discuss the computation of signal transmission and the establishing of st,ate space equations for transfer elements in elementary connections. Though the book has an introductory character it is necessary to know linear differential equations, vector and matrix the calculus with complex analysis, numbers, the set theory and the Laplace transformation. The material is clearly written. However, beyond a doubt, the inclusion of a detailed bibliography, being absent entirely, could have increased the value of the book immeasurably. PETERMAGYAR Polytechnical University Budapest, Hungary
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PERTURBATION METHODS, by Ali Hasan Nayfeh. 445 pages, diagrams, 6 x 9 in. John Wiley, New York, 1973. Price, $17.95 (approx. $7.45). This is the most extensive book available on singular perturbation methods. It covers the method of strained coordinates, the method of matched expansions, the method of averaging, the method of multiple scales and the various W.K.B. methods. There are no theorems or proofs. The author’s approach is to solve problems and to give references to more solved problems. The author begins by discussing what is meant by asymptotic and some elementary operations on asymptotic series. He then shows the various kinds of singularities and non-uniformities which can arise in solving differential equations by perturbation expansions. The next chapter deals with strained coordinates. The author, using examples, gives the Lindstedt-Poincare method, Whittaker’s method, the RayleighSchroedinger method, Lighthill’s method (P-L-K), Temple’s method and Pritulo’s method, plus examples of where the methods fail. He does this by working out 20 examples, a couple of them by different methods, and gives references to 85 papers where these methods are used. The next chapter is devoted to matched expansions. The author does an elementary boundary-layer example and turns to Prandtl’s idea of matching, van Dyke’s idea and Kaplun’s idea, plus composite solutions. He then gives the method of composite expansions of Visik and Lyusternik, as well as Latta’s method. These are done by working out 10 examples, with 41 references to various papers. The fifth chapter covers the method of variation of parameters, van der Pol’s technique, the Krylov-Bogoliubov technique, the generalized method of averaging, Struble’s technique, the KrylovBogoliubov-Mitripolski technique, avervon aging using canonical variables, Zeipel’s procedure, Lie series, Whitham’s technique and Kamel’s algorithm. This is done by using 21 examples, many of which are of the same problem done different ways, with 70 references to examples in the literature.
Journalof The Franklin Institute
Book Reviews The next chapter covers the area in which the author has done most of his own research-that of multiple scales. He covers three versions of this idea: the derivative expansion or many variable technique; the two variable technique; and the non-linear scales technique. These are applied to 23 examples, many of which are treated in earlier chapters. There are nearly 200 references to the literature. The last chapter covers linear differential equations with slowly varying coefficients, the kinds of problems lumped under the name of W.K.B. problems, and/ or turning-point problems. The author covers the W.K.B. theory, Lange& approach, expansions near caustics and the Born approximations. Each chapter includes about 20-30 problems. There are no solutions. The book’s strengths and weaknesses are suggested by the above. The author has adopted, with some exception, the role of reporter. He does that in an outstanding fashion. The computations are clear and it is very easy to see how to use each method and where the methods have been used. However, with one exception-his own specialty: the derivative expansion technique ; in this he rarely adopts the role of critic or teacher. A great many problems are solved several ways, and he reports that the solutions are the same. Sometimes two methods give the same answer (or the exact answer is known) and a third gives a different answer, so he reports that the third method fails here. The author rarely analyzes the example to try to conjecture what property of the problem causes the failure. He does report that various other authors say certain methods will fail on certain problems, but he rarely tries to teach why that is so, or why, given several methods, one is preferable to another. That, plus the feeling that he reports too much, is the book’s glaring weakness as a text. However, the tremendous wealth of solved problems in a large number of fields is the book’s towering strength. CRAIG COMSTOCK Naval Postgraduate School Monterey, California
Vol.300,No.3,September 1975
INTEGERPROGRAMMING, byHaroldGreenberg. 196 pages, diagrams, 6 x 9 in. Academic Press, New York, 1971. Price $12.50 (approx. $5.10).
This book is ideal for students and practitioners who wish to develop a basic knowledge of integer programming into a deeper insight and understanding. The theory is logically developed from a modest introductory chapter which whets the appetite with such classics as the travelling salesman, pilot scheduling and the knapsack problem; and the reader is carefully led from theorem to algorithm to example in a way that encourages interest. Ample references at the end of each chapter give a lead into further study. Particularly interesting is the inclusion of a chapter on pure number theory which neatly explains some of the underlying mathematical concepts. It is a disappointment that branch and bound techniques are not discussed at greater length since, in Europe at least, b and b has emerged as the most practical method for obtaining optimal or close-tooptimal integer solutions, and real problems, Ref. (1) for example, are being successfully solved for the first time using this method. Also, some mention of the real world of mathematical programming systems would have been welcome since a knowledge of matrix-generators, reportwriters and post-optimal analysis has become as important as the underlying theory (Ref. IL)-but then maybe Dr. Greenberg has plans for a second volume. R.N. ADAMS Queen Mary College University of London London, England References (1) R. N. Adams and M. A. Laughton, “Optimal planning of power networks using mixed integer programming-l. Static and timephased network synthesis”, Proc. IEE, Vol. 121, No. 2, Feb. 1974. 12) F. Beglari and M. A. Laughton, “Model building with particular reference to power system planning”, J. Energy Policy-Special
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236
PERTURBATION METHODS, by Ali Hasan Nayfeh. 445 pages, diagrams, 6 x 9 in. John Wiley, New York, 1973. Price, $17.95 (approx. $7.45). This is the most extensive book available on singular perturbation methods. It covers the method of strained coordinates, the method of matched expansions, the method of averaging, the method of multiple scales and the various W.K.B. methods. There are no theorems or proofs. The author’s approach is to solve problems and to give references to more solved problems. The author begins by discussing what is meant by asymptotic and some elementary operations on asymptotic series. He then shows the various kinds of singularities and non-uniformities which can arise in solving differential equations by perturbation expansions. The next chapter deals with strained coordinates. The author, using examples, gives the Lindstedt-Poincare method, Whittaker’s method, the RayleighSchroedinger method, Lighthill’s method (P-L-K), Temple’s method and Pritulo’s method, plus examples of where the methods fail. He does this by working out 20 examples, a couple of them by different methods, and gives references to 85 papers where these methods are used. The next chapter is devoted to matched expansions. The author does an elementary boundary-layer example and turns to Prandtl’s idea of matching, van Dyke’s idea and Kaplun’s idea, plus composite solutions. He then gives the method of composite expansions of Visik and Lyusternik, as well as Latta’s method. These are done by working out 10 examples, with 41 references to various papers. The fifth chapter covers the method of variation of parameters, van der Pol’s technique, the Krylov-Bogoliubov technique, the generalized method of averaging, Struble’s technique, the KrylovBogoliubov-Mitripolski technique, avervon aging using canonical variables, Zeipel’s procedure, Lie series, Whitham’s technique and Kamel’s algorithm. This is done by using 21 examples, many of which are of the same problem done different ways, with 70 references to examples in the literature.
Journalof The Franklin Institute
Book Reviews The next chapter covers the area in which the author has done most of his own research-that of multiple scales. He covers three versions of this idea: the derivative expansion or many variable technique; the two variable technique; and the non-linear scales technique. These are applied to 23 examples, many of which are treated in earlier chapters. There are nearly 200 references to the literature. The last chapter covers linear differential equations with slowly varying coefficients, the kinds of problems lumped under the name of W.K.B. problems, and/ or turning-point problems. The author covers the W.K.B. theory, Lange& approach, expansions near caustics and the Born approximations. Each chapter includes about 20-30 problems. There are no solutions. The book’s strengths and weaknesses are suggested by the above. The author has adopted, with some exception, the role of reporter. He does that in an outstanding fashion. The computations are clear and it is very easy to see how to use each method and where the methods have been used. However, with one exception-his own specialty: the derivative expansion technique ; in this he rarely adopts the role of critic or teacher. A great many problems are solved several ways, and he reports that the solutions are the same. Sometimes two methods give the same answer (or the exact answer is known) and a third gives a different answer, so he reports that the third method fails here. The author rarely analyzes the example to try to conjecture what property of the problem causes the failure. He does report that various other authors say certain methods will fail on certain problems, but he rarely tries to teach why that is so, or why, given several methods, one is preferable to another. That, plus the feeling that he reports too much, is the book’s glaring weakness as a text. However, the tremendous wealth of solved problems in a large number of fields is the book’s towering strength. CRAIG COMSTOCK Naval Postgraduate School Monterey, California
Vol.300,No.3,September 1975
INTEGERPROGRAMMING, byHaroldGreenberg. 196 pages, diagrams, 6 x 9 in. Academic Press, New York, 1971. Price $12.50 (approx. $5.10).
This book is ideal for students and practitioners who wish to develop a basic knowledge of integer programming into a deeper insight and understanding. The theory is logically developed from a modest introductory chapter which whets the appetite with such classics as the travelling salesman, pilot scheduling and the knapsack problem; and the reader is carefully led from theorem to algorithm to example in a way that encourages interest. Ample references at the end of each chapter give a lead into further study. Particularly interesting is the inclusion of a chapter on pure number theory which neatly explains some of the underlying mathematical concepts. It is a disappointment that branch and bound techniques are not discussed at greater length since, in Europe at least, b and b has emerged as the most practical method for obtaining optimal or close-tooptimal integer solutions, and real problems, Ref. (1) for example, are being successfully solved for the first time using this method. Also, some mention of the real world of mathematical programming systems would have been welcome since a knowledge of matrix-generators, reportwriters and post-optimal analysis has become as important as the underlying theory (Ref. IL)-but then maybe Dr. Greenberg has plans for a second volume. R.N. ADAMS Queen Mary College University of London London, England References (1) R. N. Adams and M. A. Laughton, “Optimal planning of power networks using mixed integer programming-l. Static and timephased network synthesis”, Proc. IEE, Vol. 121, No. 2, Feb. 1974. 12) F. Beglari and M. A. Laughton, “Model building with particular reference to power system planning”, J. Energy Policy-Special
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