John L. Casti

Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations☆

Abstract The numerical solution of nonlinear partial differential equations plays a prominent role in numerical weather forecasting, optimal control theory, radiative transfer, and many other areas of physics, engineering, and biology. In many cases all that is desired is a moderately accurate solution at a few points which can be calculated rapidly. In this paper we wish to present a simple direct technique which can be applied in a large number of cases to circumvent the difficulties of programming complex algorithms for the computer, as well as excessive use of storage and computer time. We illustrate this technique with the solution of some partial differential equations arising in various simplified models of fluid flow and turbulence.

Metaphors for manufacturing: What could it be like to be a manufacturing system?

Abstract In the world of modern manufacturing, considerable emphasis is placed upon properties of manufacturing systems described by the terms “flexibility,” “complexity,” “reliability,” “self-repairing,” and so forth. To understand and deal with such properties, one needs a theoretical framework allowing one to pose and analyze various questions surrounding the meaning and interrelations of these terms. This paper addresses the questions of what sort of system-theoretic frameworks would be likely candidates to form the basis for such a theory of manufacturing . Our approach to the modeling problem is to examine several different paradigms that have proven useful in other fields—engineering, biology, linguistics, computer science, chemistry—and to explore the degree to which these “metaphors” can be used to characterize manufacturing systems. The paper concludes with an assessment of the strengths and weaknesses of each metaphor and a suggestion for a program devoted to further development of the most promising approaches.

Numerical experiments in linear control theory using generalized X - Y equations

Numerical investigations of the relative efficiency of Riccati versus non-Riccati based approaches to the determination of optimal feedback gains for linear dynamics-quadratic cost control processes over a finite interval are presented. The non-Riccati algorithms used are the so-called generalized X- Y functions [1] or Chandrasekhar-type [2] algorithms. The results of the experiments show that the generalized X- Y approach has significant computational advantages over the usual Riccati equation and, in many cases, the computational gain exceeds rough estimates based solely upon a count of the number of equations to be integrated.

Catastrophe Theory and Urban Processes

Phenomena exhibiting discontinuous change, divergent processes, and hysteresis can be modelled with catastrophe theory, a recent development in differential topology. Exposition of the theory is illustrated by qualitative interpretations of the appearance of functions in central place systems, and of price cycles for urban housing.

LINEAR METABOLISM-REPAIR SYSTEMS

Metabolism-repair systems represent a formal mathematical framework for representing characteristic properties of living systems such as repair, replication, adaptation and so forth. In this paper, the concrete realization of such structures is developed in the case when the system “metabolism” is linear. Explicit results are given to show when system repair operations can counteract environmental and metabolic fluctuations. Additional results pertaining to the replication operation and the possibility for ‘Lamarckian” inheritance are also given, together with a formal demonstration of the increase in complexity as we proceed from the processes of metabolism to repair to replication. The paper concludes with a discussion of several application areas, together with a consideration of several conceptual and mathematical questions requiring attention for further development of this non-Newtonian systems paradigm

Manufacturing as a System-Determined Science

This paper addresses the general issue of manufacturing as a system-determined science. To adequately assess the degree to which systems thinking enters into considerations of modern manufacturing, several interconnected schemes are developed which, taken together, provide a taxonomy of manufacturing problems. It is shown that each manufacturing problem exists at a certain hierarchical level, the lowest being Raw Materials, the highest Values. Further, each problem can be labeled as a Design, Production, or Distribution problem, provided these terms are taken in a general sense. Finally, it is shown that each problem has associated with it one or more foundational system concepts (flexibility, complexity, adaptation, etc.), lending the problem its characteristic system flavor. Putting the hierarchical, Design-Production-Distribution and system concepts labels together supplies the basis for a classification scheme which, at the same time, enables us to unequivocally answer the question as to whether there is a significant systems component to most problems of modern manufacturing. There is!

Forest monitoring and harvesting policies

This paper considers the problem of what information must be measured in a forest management model in order to generate optimal feedback harvesting policies. Together with this question, the paper also addresses the issue of how LP-based models can be embedded within a dynamic programming framework, so that feedback rather than open-loop decisions can be determined. The appendix to the paper presents a more general system-theoretic framework within which forest management may be studied. Issues of surprise, response to unknown disturbances, and robustness of policies are examined and a program for systematically investigating such management questions is outlined.

A rapprochement of the theories of radiative transfer and linear stochastic estimation

Parallels between linear stochastic estimation theory and radioactive transfer in the atmosphere are investigated. It is shown that the two theories are virtually equivalent and that basic functions in one may be meaningfully interpreted in the context of the other.

New equations for the time-dependent regulator problem

This correspondence presents a derivation of an alternate set of equations for determination of the optimal feedback gain matrix for the time-dependent linear regulator problem. In contrast to the usual Riccati equation approach, the new system is composed of only nm equations, where m is the number of systems input terminals. As payment for this reduction in number of equations, however, the new system is not entirely general, being restricted to constant control matrices G , with arbitrary dynamics and cost matrices F and Q , respectively.

Invariant imbedding and the solution of Fredholm integral equations with displacement Kernels— comparative numerical experiments

This paper compares the relative efficiencies of the invariant imbedding method with the traditional solution techniques of successive approximations (Picard method), linear algebraic equations, and Sokolovs method of averaging functional corrections in solving numerically two representatives of a class of Fredholm integral equations. The criterion of efficiency is the amount of computing time necessary to obtain the solution to a specified degree of accuracy. The results of this computational investigation indicate that invariant imbedding has definite numerical advantages; more information was obtained in the same length of time as with the other methods, or even in less time. The conclusion emphasized is that a routine application of invariant imbedding may be expected to be computationally competitive with, if not superior to, a routine application of other methods for the solution of some classes of Fredholm integral equations.