Richard Bellman

Differential-Difference Equations

Publisher Summary A systematic development of the theory of differential–difference equations was not begun until E. Schimdt published an important paper about fifty years ago. The subsequent gradual growth of the field has been replaced, in the last decade or so, by a rapid expansion due to the stimulus of various applications. This chapter introduces the study of differential–difference equations and discusses some of the main features of the theory. The role of differential–difference equations is vital in some areas, such as engineering problem and fluid mechanics. In engineering problem, the problem of controlling the temperature in a reaction tank is addressed using differential difference equations. The temperature variation is reported because of random disturbances, inherent effects due to u being non-zero, and the operation of the control device. The chapter discusses the asymptotic behavior of solutions and the problem of stability.

Quasilinearization and nonlinear boundary-value problems

Quasilinearization and nonlinear boundary-value problems , Quasilinearization and nonlinear boundary-value problems , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

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.

On Structural Identifiability

In this article we introduce a new concept, structural identifiability, which plays a central role in identification problems. The concept is useful when answering questions such as: To what extent is it possible to get insight into the internal structure of a system from input-output measurements? What experiments are necessary in order to determine the internal couplings uniquely? The definition of the concept of an identifiable structure is given. Criteria as well as certain identifiable structures are discussed. Particular emphasis is given to compartmental models.

Numerical inversion of the Laplace transform

Abstract : Usual analytic methods of inverting the Laplace transformation are mostly impractical for numerical work. A method applicable to the numerical analysis of the inverse Laplace transform is discussed. Numerical examples are given to illustrate this method.

On the analytic formalism of the theory of fuzzy sets

The extension of standard concepts of set theory (like union, intersection, etc.) to the theory of fuzzy sets is not obvious. Several policies for answering questions of this type have been discussed, but it appears that the operations of max and min play a central role in the arithmetic of fuzzy sets. Letting μA(x) denote our willingness to accept x as a member in the fuzzy set A, Zadeh defines intersection A ∩ B and union A ∪ B of the two fuzzy sets A and B by A ∩ B = {(x; min {μA(x), μB(x)})} and A ∪ B = {(x; max {μA(x), μb(x)})}. The object of this paper is to show that these definitions are not only natural, but under quite reasonable assumptions the only ones possible.

Local and fuzzy logics

Fuzzy logic differs from conventional logical systems in that it aims at providing a model for approximate rather than precise reasoning.

Invariant Imbedding and Mathematical Physics. I. Particle Processes

With the use of invariance principles in a systematic fashion, we shall derive not only new analytic formulations of the classical particle processes, those of transport theory, radiative transfer, random walk, multiple scattering, and diffusion theory, but, in addition, new computational algorithms which seem well fitted to the capabilities of digital computers. Whereas the usual methods reduce problems to the solution of systems of linear equations, we shall try to reduce problems to the iteration of nonlinear transformations.Although we have analogous formulations of wave processes, we shall reserve for a second paper in this series a detailed and extensive treatment of this part of mathematical physics.

Mathematical Optimization Techniques

The papers collected in this volume were presented at the Symposium on Mathematical Optimization Techniques held in the Santa Monica Civic Auditorium, Santa Monica, California, on October 18-20, 1960. The objective of the symposium was to bring together, for the purpose of mutual education, mathematicians, scientists, and engineers interested in modern optimization techniques. Some 250 persons attended. The techniques discussed included recent developments in linear, integer, convex, and dynamic programming as well as the variational processes surrounding optimal guidance, flight trajectories, statistical decisions, structural configurations, and adaptive control systems. The symposium was sponsored jointly by the University of California, with assistance from the National Science Foundation, the Office of Naval Research, the National Aeronautics and Space Administration, and The RAND Corporation, through Air Force Project RAND.

Curve Fitting by Segmented Straight Lines

Abstract In many situations approximation of a set of data by a polygonal curve is more advantageous than approximation by a polynomial. If the join points of the polygonal curve are known, the problem is quite simple. If, however, they are to be chosen in some expeditious fashion, considerable numerical difficulties can arise if the curve-fitting problem is approached directly. In this paper it is shown that dynamic programming offers a simple direct approach to the determination of an optimal fit.