ANNUAL REVIEW FUNDAMENTALS
LEON LAPIDUS
Applied Mathematics This review covers applied mathematical techniques outside as well as inside t h e chemical engineering literature t o help t h e chemical engineer in analyzing and solving problems
eginning with the current issue, this review will be
B broadened to emphasize and include many jour-
nals outside the area of chemical engineering. As such, the title of the review has been changed to correspond more closely to the material covered. These changes are motivated by the fact that the applied mathematical techniques outside as well as inside the-chemical engineering literature usually prove of specific interest in analyzing and solving problems. Obviously the area of applied mathematics is a most extensive one and care must be taken in separating the highly mathematical material from that which may have application in the next few years. At the same time, certain areas, such as optimal control theory, are covered in depth in other reviews in this series and are included in only a minimal sense here. The subject areas have been divided into suitable sections so that, along with the titles of each article, it is not necessary to develop tables covering specific articles. T h e articles and books listed cover roughly the period July 1967 to July 1968. General
Two new journals of interest to this review have recently been initiated. They are Theoretical Foundations of Chemical Engineering as a translation from the Russian by the Consultants Bureau and the Journal of Optimization Theory and Applications. I n addition, McGraw-Hill under its Schaum’s Outline Series has brought out two large-size paperbacked books of interest. One (75.4) deals with linear algebra and the other (78.4) with numerical analysis. These books are surprisingly advanced, although the depth of coverage is not excessive. Of primary interest is the extremely large number of detailed examples. A highly mathematically oriented book dealing with aspects of the Laplace transform has been published (ZA), and a n excellent text in the fluid mechanics area has come out (23.4). T h e latter book will be of interest 42
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to academicians for use in a n undergraduate transport course. A book on systems analysis, which includes such items as signal flow analysis, weighting patterns and filters, signal generators, frequency-domain representation, and a n analysis of zeroes and poles, has been developed (9A). While written from the electrical engineering point of view, the book contains much of interest to the chemical engineer. Knuth ( 7 7A) has developed a n outstanding book dealing with the mathematical foundations of algorithms and data structures. Of immediate interest to the chemical engineer is the recent book (70A) which summarizes a wide variety of computer experiences a t Michigan over the last five years. Details of the simulation and analysis of many physical systems related directly to chemical engineering make this book a must for those interested in these topics. Chemical Engineering Models
There have been a number of interesting papers in the last year dealing with the modeling or analysis of models developed for systems with chemical engineering relevance. These include structuring a number of systems in terms of signal flow graphs (4B, 5B) and the aspects of simulation of chemical reactors (723, 27B). The effect of mixing on the performance of tubular reactors has also been described (9B). A superb series of three papers analyzed very carefully all the theoretical and computational aspects of the design of reactor with back mixing (75B). The packed bed reactor has also received specific analysis (3B, 8B, 77B). I n particular, Crider and Foss ( 3 B ) were able to obtain a n analytical solution for this extremely complicated type reactor by assuming a specific form of the chemical rate equation. Other authors (7OB) analyzed the laminar flow of a reactant in a tubular reactor. Of concern was the numerical evaluation of a n integral kernel which arises in the analysis. Other works of interest in terms of model analysis were those dealing with gas absorption columns (73B), with
distillation towers (77B,20B), with flow in porous media (74B),and with chromatographic columns (72B). Residence-lime Analysis
Work has continued a t a fast pace in applying various aspects of residence-time and moment analysis to processes. This work involves, as a n example, the probing of a system model with specific inputs such as a pulse or step function. By analyzing the input and resulting output waves, it is possible to determine such items as the mean and variance of the wave form. These can then be related to the parameters in the model. Such a n approach or variations on same have been used for crystallizer behavior (4C, 74C), for systems with internal reflux (75C), for surface renewal models (5C), for porous systems with stagnant zones such as a liquid irrigated packed bed (7C), for fluid bed systems (2C, SC, 73C), for fixed beds and chromatographic columns (3C, SC), for a n absorption column (70C), for mass transfer across interfaces (IC), and to analyze various multistage or recycle reactors (5C, 7 IC,72C). Periodic Systems
Considerable activity has been evidenced recently in the nonsteady-state or periodic operation of many chemical engineering systems. These usually are systems forced into oscillation by periodic inputs in a n effort perhaps to improve the average yield of a chemical reactor over that associated with a steady-state type of operation. As a n illustration, certain authors ( 5 0 , 7 0 ) were interested in analyzing polymerization-type reactions. Others (30) discussed the behavior of countercurrent processes and more specifically the stage efficiency of a distillation column ( 6 0 ) . T h e periodic behavior of diffusion and transport systems has been analyzed ( 7 0 , 2 0 ) , and the velocity of a free particle suspended in a n oscillating fluid was also studied ( 4 0 ) . T h e paper by Wilhelm and coworkers ( 3 7 0 )is an example of the superb VOL 6 0
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work which can be done in this area from a n analysis and physical implementation point of view. Parameter Estimation
Since the advent of the digital computer there has been considerable interest in the parameter estimation and model discrimination problem. Here the question is how to fit (in a least square sense) parameters in a model or series of models to experimental data so as to find the “best ” model. I n this sense it does not make any significant difference if the model is linear or nonlinear. By contrast, one can change the problem to specify a linear model and seek the parameters in a transfer function when noise corrupts the experimental data. Because of the importance of this area of investigation, there have been a number of interesting papers in the past year. The well-known Efroyson regression procedure has been extended (4E), and certain aspects of the estimation method itself have been examined (74E). The vast majority of applications have been in kinetic systems where rate constants or exponential rate factors are estimated. Heinken et al. ( 7 E ) , in a superb paper, analyzed some enzymatic type reactions, while others (IOE, 75E, 76E, 79E, 27E) analyzed various aspects of homogeneous and heterogeneous reaction systems. Some experimental strategies which are useful in the estimation and modeling algorithm have also been presented (73E). Since parameter estimation of nonlinear systems actually proceeds by a successive series of linearizations, it would seem most feasible to use the algorithm of quasilinearization. This method is extremely versatile and computer oriented; it solves sets of linear equations in the to-be-determined parameters. Bellman et al. (ZE, 3 E ) and Lee (77E, 78E) have presented extensive details in this area. As an illustration, Lee determines the Peclet Number and the reaction rate constant for a tubular reactor system. Other interesting papers in the estimation area have been presented (5E, 8E, 273). The last two deal with the identification of purely linear systems. Finally we should point out that in the paper by Lapidus and Bard (76E) there are details not only of purely parameter estimation, but also of computer directed experiments to aid the estimation and model discrimination. This has importance in various aspects of on-line control systems. Analytical Solution of Equations
In this section we mention papers which have used analytical mathematics to solve the problem under investigation. This is taken to mean closed form or asymptotic solutions as distinct from numerical solutions. As expected, the papers cover the entire range from linear algebraic equations through ordinary differential equations and on through partial differential equations. The by-now classical linear algebra-type analysis of Prater and Wei on linear reaction systems has been extended (SF, 29F). I n the latter case, consideration is given to the handling of irreversible rate steps by de44
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composition into substeps while the former work deals with multicomponent problems. Other authors (27F, 33F) were concerned with solving the ordinary differential equations associated with catalyst effectiveness factors and the solutions relating to some form of chemical kinetic systems (ZF, 73F, ZOF). The differential equations associated with a spray column heat exchanger have also been analyzed (22F). I n the area of solving partial differential equations, there were a number of papers dealing with either fixed bed or chromatographic (ion exchange) multicomponent column behavior (5F, 76F, 27F, 3527). Others have considered adsorption kinetics in fixed beds (25F, 3ZF). Solutions of problems involving some form of heat transfer have been presented by many authors (3F, 9F, 77F, 36F). In the multicomponent transport and diffusion-type system, interesting solutions have been presented (74F, 78F), and a Stefan-type system with a moving boundary between phases has been analyzed (34F). Solutions in systems involving gas-solid operations, such as fluid beds and single and multiple particles in a secondary fluid, have been reported ( I F , 6F, 7F, 23F). Other works of interest are those (72F) dealing with the Hamilton- Jacobi equation of mechanics and optimal control and (7OF) dealing with propagation of flames. Optimization
The field of optimization continues to generate a vast flow of research papers which span the spectrum from the very theoretical to the very practical. Work in the linear programming area continues (6G, 30G, 44G), with the theme being to solve certain special problems. A recent book (78G) is of interest in this area as well as in the nonlinear programming area. Other special methods such as integer programming ( 7 7G), quadratic programming (35G), and the more recent geometric programming (24G, 32G, 33G) have also received further attention. The Fibonacci search (25G) and the golden block search (7G) methods have also been investigated. The gradient, Newton, and the Davidon approaches to optimization of multidimensional systems have been investigated by many authors. A detailed investigation of the first-order necessary conditions has been presented (79G), as well as a conjugate gradient with linear constraints approach (76G). The use of the penalty function approach for handling constraints has been investigated (3G, 73G), as well as a means of setting standards for measuring the efficiency of gradient methods (77G). Davidon (7G) has extended his original work, while another paper has been concerned with removing singular matrices which may have come up in the calculation (ZG). Other authors (5G, 46G) have detailed extensively all aspects of the Newton algorithm.
AUTHOR Leon Lapidus is Professor and Chairman of the Department of Chemical Engineering, Princeton University, Princeton, iV. J. T h i s is the author’sJrst year to compile the Fundamentals Annual Review on Applied Mathematics.
Finally, a recent book (42G) and two papers (40G, 47G) may become landmarks in covering all aspects of optimization. I n addition, three excellent books (27G, 37G, 38G) contain many aspects of optimization which are of direct interest to the chemical engineer. In the applications area there has been work on a twoderivative method for optimizing reactors (27G, 39G) and on a gradient method for temperature control in reactors (72G, 28G). Other papers deal with the decomposition method to design heat exchangers (36G), with heat exchanger systems with recycle (37G), with a polymerization reaction in a series of reactors (34G), with the use of adjoint variables for chemical reactors (8G, 22G), and to the design of a catalytic cracking system via the use of multilevel optimization (4G). Dynamic Programming
While the use of Bellman’s dynamic programming algorithm fits within the area of optimization, we have considered it here as a special topic. The use of this algorithm continues in almost every area of investigation from that of making managerial decisions (72H) to pattern recognition (3H, 4 H ) , to optimizing plant expansion ( 7 H ) , to the optimal design of evaporators (77H), and to minimizing the loss of an oxidation product (70H). Other papers of interest deal with branched allocation systems ( 7 H ) , with aspects of random replacement and maintenance policies (73H), with the optimal location and construction of nuclear fuel plants (23H), and with the optimal behavior of adiabatic bed reactors (771’3). A graphical representation of the dynamic programming algorithm in reactor problems ( 9 H ) has been presented, and a way to reconstruct the dynamic programming algorithm for parallel computers (8H) has been outlined. I n terms of the specific computational use of dynamic programming, a n excellent survey paper (74H) has been published. An extension of the normal algorithm by Larson has resulted in a book (75H) and a slight controversy on the specific advantages of extrapolation and interpolation in the use of the method (76H). An application of the method via a forward dynamic programming algorithm has been presented (20H),as well as some interesting results on decreasing the dimensionality in optimal control systems (6H). Stability
One of the more interesting mathematical problems which the chemical engineer encounters is the specification of the stability properties of a reaction or other system. T h e work of Amundson and coworkers, including Luss in particular, illustrates the superb mathematical and computational results which can be obtained. Necessary and sufficient conditions for stability have been determined for packed bed reactors (291) and for fluid beds (271). The use of the maximum principle for parabolic equations has been applied to tubular reactors with diffusion (201). Sufficiency conditions have been developed for specifying when unique steady states exist in a reaction system (221). Other
papers in the area of stability of catalyst particles (76Z) and of extractive tubular reactors (791) have been published. As an alternate approach, one may turn to the use of Liapunov’s method which hopefully bypasses almost all computation in establishing stability. A review of interest in this area (721) has been published, as well as a discussion of the different types of stability ( 7 7 1 ) and new theoretical results (241, 261, 331). Another review has covered the stability of distributed parameter systems (311). The use of Krasovskii’s theorem has been detailed (781), as well as the application of the Liapunov functional to distributed parameter systems of interest to chemical engineers (41). Other approaches to the stability of fluid dynamic systems (91, 70Z, 251, 28Z), to spherical catalyst particles (321), to distributed parameter systems via phase plane analysis (30Z), to fluid beds (21, 31), and to Marangoni films (771) have been presented. Optimal Control
Mathematical and computational progress continues a t a tremendous rate in the optimal control area. Here we mention only a limited number of these references and leave further discussion to another review in this series. At least five books of interest have been published. One of these (37J) is an outstanding introduction to the whole area of matrix manipulations, stability, and optimal control. Another ( 2 5 J ) contains both theory and computational results of direct interest to chemical engineers. A third ( 8 J ) deals with the control of distributed parameter systems. The final two books ( 6 J , 26J) are highly mathematically oriented. A recent review ( 7 9 4 of progress in the optimal control area is also of interest. Interesting papers in the optimal control area have been published dealing with a matrix minimum (maximum) principle ( 2 J ) and on the computational aspects of such methods as quasilinearization and the second variation (24J, 34J, 35J, 3 7 J ) . Consideration has been given to the so-called inverse control problem ( 7 4 J ) , to sensitivity considerations ( 9 J , 22J), and to optimum catalyst profiles in tubular reactors ( 3 8 J ) . The time optimal control of a batch distillation system (77J) has also been investigated. Extensions of the normal optimal control case to the variable time delay in the state and control have been made (5J, 78J, 3 6 J ) , as well as some most interesting work on distributed parameter systems (3J, 72J, 75J, 20J) 27J). Numerical
General. There have been at least seven excellent numerical analysis books in the past year. Included are a book on numerical integration ( 3 K ) which is absolutely first rate in detail and mathematical level, a book on the asymptotic solution of differential equations ( 2 K ) ,and an excellent book covering many up-to-date aspects of numerical analysis (77K),and two general introductory volumes (8K, 9K) (8K has 1130 pages and 9K has 750 VOL. 6 0
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pages) have been published which are equivalent to the Chemical Engineer’s Handbook but which deal with aspects of applied mathematics. A number of authors have published papers dealing with discretization methods (involving discrete-type approximations such as arise in many areas of numerical analysis). Convergence and stability of these methods were the theme of two papers (75K, 78K),while another paper was concerned with accelerating convergence by extrapolation to the limit (72K). Aspects of Romberg quadrature (the modern way to evaluate an integral numerically), such as error bounds (76K) and the effect of singularities ( 6 K ) , have been published. Also a monumental paper has been published on how linear programming can be used to solve ordinary and partial differential equations, matrix problems, and many others ( 7 4 K ) . This is an outstanding applied mathematics paper which cuts across the board area of analysis. Approximation. The work associated with approximating analytical functions or experimental data via some type of polynomial continues at a fast pace. I n the last year two areas have received attention, these being the use of spline functions and the use of Chebyshev polynomials. The spline function approximation, named after the use of a draftsman’s drawing tool, has developed quickly in the past five years as an alternative to normal polynomial approximation. This has culminated in a book (2L) by some of the pioneer researchers in the area. Other papers have applied the method to approximating thermodynamic data (7L),to the interpolation features (75L), to the convergence properties (3L),and to use, via least squares fitting, as a smoothing device (72L). A final paper (13L)has indicated the connection between the spline function and the Chebyshev approximation. In addition to the paper (73L) on the Chebyshev equioscillation polynomials above, there has been work associated with polynomial fitting (14L) and with the errors in such polynomial approximations ( 7 7L). In addition, and as part of the Handbook Series of Approximation in Numerical Mathematics, a broad-based article on Chebyshev methods has been presented (77L). Other material in the approximation area which do not fall within the two categories above are an excellent book (8L), some interesting work on Pade approximations ( 4 4 5L), and some features of smoothing and differentiating of experimental data via least squares fitting (6L)’ Linear algebra. In this area of numerical analysis dealing with the solution of linear algebraic equations, two new books were published (72M, 38M). The first, in particular, details the modern aspects of scaling and error monitoring for calculating such items as the inverse matrix. This short book should be read by all workers interested in numerical analysis. Papers of interest detail the solution of the simultaneous equations when the coefficient matrix has a special form such as a band matrix (34M, 35M) or is 46
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block symmetric ( 9 M ) . Others are concerned with solving an over-determined system via Chebyshev meththe use of mathematical programming to ods (3M), solve singular systems ( 2 9 M ) , the tridiagonalization of a symmetric matrix (ZOM), and the evaluation of catalyst effectively factors by means of the well-known Thomas algorithm (73M). Some authors have been concerned with the pseudoinverse which can be applied computationally to singular and rectangular matrices. This has extensive application in multivariable curve fitting and optimal control. Computational aspects have been presented ( 2 ? M ,32M, 33M, 37M),as well as all necessary proofs and relations (77M) and the use for generalized interpolation (75M). Along the same lines, one author has shown how to bound the solution of the Riccati matrix equation ( 2 M ) which may use the pseudo-inverse and another has detailed features of the monotonicity and stability of this special equation ( 8 M ) . Papers have also been published for evaluating eigenvalues and eigenvectors. In the Handbook Series on Linear Algebra in Numerical Mathematics, authors have detailed the evaluation of a symmetric eigenproblem ( 6 M ) , of the Jacobi method (TOM), and of features of the very promising QR-QL algorithms ( 6 M , 28M). The use of Graeffe’s method has been extended (27M), and features of calculating the maximum eigenvalue have been detailed (74M). Finally, the eigenvalues and -vectors have been calculated for the infamous Hilbert matrix of order 3 to 1 0 ( T I M ) . These last results should prove extremely useful to test computer matrix routines. Three authors have analyzed the normal equations or some variation of same arising in least squares analysis ( 5 M , 7 M , 76M). The first of these ( 5 M ) is concerned with special two-dimensional data and the third ( 7 6 M ) presents extensive computational details comparing various methods such as Gram-Schmidt. Ordinary differential equations. The solution of nonlinear ordinary differential equations continues to be of interest. I n general terms, one author has assessed the relative merits of most well-known methods ( 7 7 N ) , another has questioned the use of random numbers to simulate round-off error (28:V), and another has investigated parallel integration methods ( Z O N ) . Other authors have detailed the use of the backward stable Euler method (7:V), the solution of boundary-layer problems (27N), and the use of finite differences (22N). Finally, a detailed numerical comparison has been made using transient staged operation equations of most of the more sophisticated methods of integration ( 6 N ) . Runge-Kutta, or single-step, methods of integration have been analyzed in implicit form (2N-4:V) and when large Lipschitz constants (small step sizes) are involved (77N).One author has presented a new fifth-order Runge-Kutta equation ( 7 4 N ) , while another has presented a sixth-order equation with an extended range of stability (78N). Of particular interest to anyone in this field is the superb paper by Rosser (25N) showing what can really be done in the Runge-Kutta area. Finally,
79 first-order equations have been integrated by means of the classical Runge-Kutta equation (27N). Two authors have presented interesting analyses of predictor-corrector approaches. I n the first, the PECE (predict, evaluate, correct, evaluate) niethod (7SN) was analyzed in detail, whereas in the second only the PEC method (73N) was considered. We also note an excellent paper on stiff differential equations (widely separated time constants) ( 8 N ) and another which uses spline functions as the basis for integration in place of the usual polynomial (79N). These latter results are disappointing since the approach yields the familiar (trapezoidal, Milne) formulas. Partial differential equations. A revised and updated book by Richtmeyer (280) on the solution of partial differential equations has been published during the past year. This book will remain a classic in the field both from the mathematics and the analysis points of view. Papers dealing with the solution of partial differential equations arising from physical systems have been published. These include the parametric pumping system ( 3 7 0 ) ) the inviscid and viscous equations of a compressible gas (80, 2 9 0 ) , Poiseuille-type flow (200))flux components in potential flow (330)’ and the adiabatic fixed bed reactor ( 2 3 0 ) . I n terms of specific classes of partial differential equations, we first turn to work on parabolic equations. Rachford, in two superb papers, has considered the effect of rounding errors in parabolic equations with one space dimension (260) and in multiple space dimensions (250). These papers are pioneers in the field. Other authors have analyzed the biharmonic equation (760, 370))a Stefan-type problem ( 2 0 ) )the case of degenerate parabolic partial differential equations (30))the spherical symmetric diffusion equation ( 6 0 ) )and the influence of boundaries on Laplace’s equation (780, 3 8 0 ) . Further work dealing with orthogonal collocation ( 3 2 0 ) ) the Crank-Nicholson implicit method (700) and with convergence rates of finite difference representations (780) have appeared. An interesting paper has been published on estimating the truncation error in fourth-order elliptic equations (390) associated with a general shaped body. I n the area of hyperbolic partial differential equations, some extremely interesting work has been detailed. I n one case, the author combined nebencharacteristics with extrapolation methods to facilitate the numerical solution ( 3 6 0 ); in another, extrapolation methods by themselves were analyzed (730); and another has established explicit and implicit formulas for two-space variables and any boundary conditions (740,750). From an applications point of view, there are two papers (270, 3 4 0 ) dealing with integrations along the characteristic curves of packed bed and other-type reactors. Since most solutions of partial differential equations are iterative in nature, the SOR (successive over-relation) and AD1 (alternate direction implicit) iterative techniques have been carefully investigated. The use of Chebyshev polynomials to accelerate SOR has been de-
tailed (770, 770)’ while AD1 methods have been analyzed in general (220) and for hyperbolic equations (720). Finally, methods have been presented for the optimum SOR for Laplace’s equation (270) and for accelerating SOR with elliptic equations (240). Boundary-value problems. Boundary-value problems occur in an overwhelming number of physical systems such as those which include diffusion in almost any context. Both ordinary and partial differential equations may have the split type of boundary conditions, and most of the work on optimal control by means of maximum principle (or any other method) usually leads to split conditions on the state and adjoint variables. As a result, this is an area of continuing analytical and numerical investigation. The past year has seen the publication of an outstanding book in this area (6P). Not only is the mathematical level very high, but extensive discussion is given to most of the known methods for solving boundary value problems. This book is highly recommended. Papers in the area include one which extends Picard’s method (75P),one which is concerned with methods for converting the boundary-value problem to oriented linear flow graphs (4P) in any coordinate system, and another which is an extension of previous papers (74P). I n the latter case, work is described which is based on NewtonRaphson-Kantrovich methods which iterate on the unknown boundary conditions. As described, this last paper has certain features which are analogous to those used in invariant-imbedding as developed by Bellman. This particular technique has been described in terms of solving problems in unit operations (7P), and for solving boundary layer equations (77P)and tubular reactor systems (QP,70P)which involve split boundary conditions. Much of this work is summarized in the recent book (8P) which contains extensive numerical results on invariant-imbedding of direct interest to the chemical engineer. Data analysis. This section may be thought of as including many aspects of statistical analysis of data. Thus, work has been detailed on the analysis of size-age distributions (704)) on gravity decantation ( S Q ) , and on irrigated packed towers (24). However, while these are interesting, the exciting work in this area is the recently developed fast Fourier transform. Here we have a significant computational tool which facilitates signal analysis such as power spectrum analysis and filter simulation by digital computers. Mathematically, this fast transform is an efficient computational tool for developing the discrete Fourier transform of a series of data samples or time series. As such it should have application to many areas of chemical engineering research but, to date, it has evolved in the electrical engineering area. One recent paper (44) details the complete history of the development of the method, while others ( 3 Q , 94) stress the computation features. Another paper ( 7 4 ) deals with a specific application, of the Fourier transform, and finally there is a development suitable for a parallel processing mode of computer operation (8Q). VOL. 6 0
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Root location. Here we are concerned with methods for finding the roots of complex highly dimensional nonlinear algebraic equations. Alternately the material has application in optimization problems where the first necessary conditions are satisfied. Saaty (9R) has published a book in this area which provides excellent coverage of all aspects of nonlinear equations. This book is highly recommended for its mathematical depth of coverage. Papers have been published dealing with a random technique for locating on speeding up the convergence of the wellroots (4R), known polynomial Bairstow method (ZR), on an iterative method for locating roots based on bilinear forms which can handle complex and repeated roots (8R), and on a technique for terminating any iterative rootlocating method ( I R ) . Other work of interest directly to the chemical engineer is the use of a nested set of single variable equations to solve complex chemical equilibrium problems (TOR),
BIBLIOGRAPHY General a n d Miscellaneous (‘A) Balakrishnan, A . V., and Lions, J., L., “State Estimation for Infinite-Dimen. sional Systems,” J . Com.6. System Sa., 1,391 (1967). (2A) Berg, L., “Introduction to the Operational Calculus,” John Wiley and Sons, 1967. (3A) Bragg L. R . and Dettman J. W . “Related Partial Differential Equations and T h e & Applilations,” SZAM Appi.’Math., 16,459 (1968). (4A) Brick D. B. “ O n the Applicability of Weiner’s Canonical Expansions,” ZEEE T r i n r . Syrt&rSci. Cybern., 4, 29 (1968). (5A) Cox, R . G., and Brenner, H., “ T h e Lateral Migration of Solid Particles in Theory,” Chem. Eng. Sci.,23, 147 (1968). Poiseuille Flow-I. (6A) Fiore C. F and Rozwadowski, R . T., “ T h e Implementation of Process Models,”Manui: Sci., 14, B-360 (1968). (7A) ,Han, C. D., and Bixler, H. J., “Washing of the Liquid Retained by Granular Solids,” AZChE J . , 13, 1058 (1967). (8A) Howard, D. W., and Li htfoot, E. N., “Mass Transfer to Falling Films. I. Application of the Surface &retch Model to Uniform Wave Motion,” AZChE J., 14,458 (1968). (9A) H u gins, W H I and Entwisle, D. R., “Introductory Systems and Design,” Blaisde8,1968. ’ ‘ (10A) Katz, D. L., “Engineering Concepts and Perspectives,” John Wiley and Sons, 1968. (11A) Knuth, D. E . , ” T h e .4rt of Computer Programming,” Addison-Wesley, 1968. (12A) Koffman E. B “Learning Games Through Pattern Recognition,’’ ZEEE Tranr.Systemshi. Cy&., 4,12 (1968). (13.4) Kural, O., and Schoenhals, R . J., “Investigation of Feedback Controlled Diffusion Systems Using A High-speed Continuous Electrical Analog,” J . Basic Eng., 90,175 (1968). (14A) Liebelt, P . D., “ A n Introduction to Optimal Estimation,” Addison-Wesley, 1967. (15A) Lipschutz, S., “Linear Algebra,” Schaum’s Outline Series, McGraw-Hill, 1968. (16A) Loucks D. P ReVelle, C. S., and Lynn, W . R . , “Linear Programming Models for &ate, Pbllution Control,” M u n o g . Sci., 14,B-166 (1967). (17A) Piekaar, H. W., and Clarenburg L. .4.“Aerosol Filters-Pore Size Distribution in Fibrous Filters,” Chem. Eng. Sci.’, 22, l h 9 (1967). (18A) Scheid, F., “Numerical Analysis,” Schaum’s Outline Series, McGraw-Hill, 1968. (19A) Stakgqld, I., “Boundary Value Problems of Mathematical Physics. I and 11,” Macmillan, 1967 and 1968. (20A) Telles, A . S., and Dukler, A. E., “Similarity T pe Solutions of Turbulent Boundary Layers for Momentum and Energy,” A I C h J J . , 14,440 (1968). (21A) Vidal A . and Acrivos, A., “Effect of Nonlinear T e m p t u r e Profiles on Onset of 6on;ection Driven by Surface Tension Gradients, IND.END. CHEM., 7, 53 (1968). FUNDAY., (22A) W:aver, L. E., “Reactor Dynamics and Control: State Variable Techmques, American-Elsevier, 1968. (23A) Whitaker, S.,“Introduction to Fluid Mechanics,” Prentice-Hall, 1968.
>,
Chemical Engineering Models (1B) Boreskov G . K . and Slinko, M . G., “Simulation of Chemical Reactors,” Theor. Found: Chem. Eng., 1, (1967). (2B) Cloutier, L., and Cholette, A , , ‘‘Effect of Various Parameters on the Level of Mixing in Continuous Flow Systems, Can. J . Chem. Eng., 45,82 (1967). (3B) Crider, J. E., and Foss, A. S., “ A n Analytic Solution for the Dynamics of a Packed Adiabatic Chemical Reactor,” AZChE J , , 14,77 (1968). (4B) Greenfield G. G . and Ward T. J “Structural Analysis for Multivariable Process Contr&” IND:ENO.CHEM:,F U N ~ A M 6,564 ., (1967). (5B) Greenfield G . G and Ward, T. J., “Structural and Terminal Analysis in Multivariable‘Proces~ControI,’’ ibid., p 571. (6B) Hahn, G . J., and Shapiro, S. S., “Statistical Models in Engineering,” John Wiley and Sons, 1967. (7B) Harris, I. J., and Srivastava, R . D “ T h e Simulation of Single Phase Tubular Reactors with Incomplete Reactant d x i n g , ” Can. J . Chem. Eng., 46,66-69 (1968).
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tNDUSTRlAL A N D E N G I N E E R I N G CHEMISTRY
(8B) Hawthorn, R . D., Ackerman, G . H., and Nixon, A. C., “ A Mathematical Model of a Packed-Bed Heat-Exchanger Reactor for Dehydro enation of Methylcyclohexane: Comparison of Predictions and Experimental kesults,” AZChE J., 14, 69 (1968). (9B) Horn, F. J. M., ”,:d Parish, T. D . “ T h e Influence of Mixing on Tubular Reactor Performance, Chem. Eng. Sci., i2,1549 (1967). (10B) Kim, Chul-Hee, and Gilbert, R . E.,“Evaluation of a Kernel Associated with Laminar Flow Tubular Catalytic Reactors,” Math. Comp., 22,517 (1968). (11B) Kuchanov, S. I., and Pismen L. M. “ A Quasihomogeneous Model of a Packed Bed Reactor,” Theor. Found.’Chem. E&., 1, (1967). (12B) Lai, Cheng-Liang, and Roth, J. A . “Dvnamic Simulation of Gas Chromatographic Column,” Chem. Eng. Sci., 22,1599 (1967). (13B) Lees, F. P . I‘The Frequency Response of a Packed Gas Absorption Column,” rbzd., 23,97 (19k8). (14B) Litwiniszyn, J., “ O n Some Mathematical Models of the Suspension Flow in Porous Medium,” ibid., 22,1315 (1967). (15B) Mecklenburgh J. C. and Hartland, S., “Design of Reactors with BackmixApproxiIng-I. Exact l & o d s , ” “Design of Reactors with Backmixing-11. mate Methods Design of Reactors with Backmixing-111. Numerical Difference Betdeen Differential and Stagewise Models,” ibid., 23, 57, 67, 81 (1968). (16B) Meier, W . L. Jr. “Branch Corn ression and Absorption in Nonserial Multistage Systems,” J h A A , 21,426 (19687. (17B) Rafal M. D., and Stevens, W. F. “Discrete Dynamic Optimization Applied to On-Lineoptimal Control,” AIChE >., 14,85 (1968). (18B) Uppuluri, V. R . Rao, Feder, P. I., Shenton, L. R . “Random Difference Equations Occurring in One-Compartment Models,” Math. Biorci. 1, 143 (1967). (19B) Vaclavek, V. “Stochastic Linear Operator as a Characteristic of a Continuous Flow System,” Chkm. Eng. Sci., 22,1209 (1967). (20B) Wood, R . M . “ T h e Frequency Response of Ivfulticomponent Distillation Columns,” Trans. &st. Chem. Eng., 45,T190 (1967). (21B) Wright, B. S., “Computer Simulation of a Plug Flow Reactor for a Complex Reaction,” Brit. Chem. E n g . , 12,1750 (1967). Residence-Time Analysis (1C) Angelo, J. B., and Lightfoot, E. N., “Mass Transfer Across Mobil Interfaces,” AZChE J . , 14,531 (1968). (2C) Bhat, G. H . , and Srimathi, C. R . , “Evaluation of Residence Time for a GasSolids Interacting Moving Bed System,” Brit. Chem. Eng., 12,1597 (1967). (3C) Edwards, M. F., and Richardson, J. F., “Gas Dispersion in Packed Beds,” Chem. Eng. Sci., 23,109 (1968). (4C) H a n , C. D., and Shinnar, R . , “ T h e Steady-State Behavior of Crystallizers With Classified Product Removal,” AIChE J . , 14,612 (1968). (5C) Klinkenberg, A . , “Moments of Residence Time Distributions for Cascades of Stirred Vessels WithBackmixing,” Chem. E n g . Sci., 23,175 (1968). (6C) Kovisy, K.,“Different Types of Distribution Functions to Describe a Random Eddy Surface Renewal Model,” ibid., p 90. (7C) Levich, V. G., Markin, V. S., and Chismadzhev, Yu A . , “ O n Hvdrodvnamic Mixing in a Model of a Porous Medium With Stagnant Zones,” ibid., 22, 1357 (1967): (8‘2) ,Mireur, J. P., and Bischoff, K . B., “Mixing and Contacting Models for Fluidized Beds,” AIChE J.,13,839 (1967). (9C) Petho, A . , “ M e a n and Variance of Residence Time Distributions in Fixed Bed Multistage Exchange Processes of Linear Kinetics,” Chem. Eng. Sci., 22, 1793 (1967). (1OC) Prost, C., “Etude des Fluctuations d e la Texture du Liquide S’Ccoulant B Contre-courant ou Co-courant du Gaz Dans un Garnissage d e Colonne D’absorption,” ibid., p 1283. (11C) Rehikov;, M., and Novosad, Z . , “Residence Time Distribution and Fractional Conversion for a Multistage Reactor with Backmixing Between Real Stages,” ibid., 23, 139 (1968). (12C) Rippin, D . W. T., “Recycle Reactor as a Model of Incomplete Mixing,” I X D .ENC.CHEM.,FUNDAM,, 6,488 (1967). (13C). Sandblom, H . “ T h e Pulse Techni ue for Investigating Solids Mixing in a Fluidized Bed,” Brii. Chem. Eng., 13,677 ?1968). (14C) Sherwin M.B. Shinnar R., and Katz S. “ D namic Behavior of the WellMixed Isoth&nal C;ystallize;,” AIChE J., 1$,1!41 6967). (15C) Shinnar, R., and Naor, P., “Residence Time Distributions in Systems With InternalReflux,” Chem. Eng. Sci., 22, 1369 (1967). Periodic Systems (1D) H y r i s , H . G., and Goren, S. L., “Axial Diffusion in a Cylinder With Pulsed Flow. Chem. Ene.Sci.., 22.1571 11967). , . . (2D) Horn, F . J. M . and Kipp, K . L., Jr., “Mass Transport Under Oscillatory Fluid Flow Conditions,” ibid., p 1879. (3D) Horn, F. J. M., and May, R . A.,“Effect ofMixing on Periodic Countercurrent 7, 349 (1968). Processes,” IND. ENC.CHEM.,FUNDAM., (4D) Houghton G. “Particle Retardation in Vertically Oscillating Fluids,” Can. J . Chem. Eng., k 6 , ?9 (1968). (5D) Laurence, R . L., and Vasudevan, G., “Performance of a Polymerization DES. DEVELOP., 7, Reactor in Periodic Operation,” IND.EXO. CHEM.,PROCESS 427 (1968). May, R . A , , and Horn, F. J. M.,“Stage Efficiency of a Periodically Operated (6%tillation Column,” ibid., p 61. (7D) Ray, W . H.,“Periodic Operation of Polymerization Reactors,” ibid., p 422. (ED) Tomastik E. C. “Oscillation of a Nonlinear Second Order Differential Equation,” S i A M J . kpfippl. Math., 15, 1275 (1967). ij
Parameter Estimation (1E) Anderson R . L., and Crump, P. P., “Comparisons of Designs and Estimation Procedures fdr Estimating Parameters in a Two-Stage Nested Process,” Technomet., 9.499 -, - - (19671. (2E) Bellman,, R . , Kagiwada, H. H., and Kalaba, R . E. “Quasilinearization and the Estimation ofTime Lags,’’ Math. Bioscci., 1, 39 (1967): ):3( Bellman, R . E,, Kagiwada, H . H., Kalaba, R . E., and Vasudevan, R . , Quasilinearization and the Estimation of Differential Operators from Eigenvalues,” Comm. A C M , 11,255 (1968). (4E) Breaux, H. J., “ A Modification of Efroymson’s Technique for Stepwise Regression Analysis,” ibid., p 556. (5E) Carney T. M . , and Goldwyn, R . M.,“Numerical Experiments With Various Optimal Ekmators,” J . Opt. Theory Appppl., 1, 113 (1967). \
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(6E) Corrigan, T. E., Lander, H. R., Jr., Shaefer, R., and Dean, M . J., “ A TwoParameter Model for a Nonideal Flow Reactor,” AIChE J.,13,1029 (1967). (7E) Heinken, F. G., Isuchiya, H. M., and Aris, R., “On the Accuracy of Determining Rate Constants in Enzymatic Reactions,” Math. Biosci., l, 115 (1967). (BE) Heymann, M., McGuire, M. L., and Slipcevich, C. M.,“New Time-Domain Techni ue for Identification of Process Dynamics,” IND. END.CHEM.,FUNDAM., 6,555 (7967). (9E) Hill W. J Hunter, W . G., and Wichern D. W., “ A Joint Desi n Criterion for the’ Dual ‘)Problem of Model Discriminition and Parameter gstimation,” Technomet., 10,145 (1968). (10E) Himmelblau, D . M., Jones, C. R., and Bischoff, K . B., “Determination of 6, Rate Constants for Complex Kinetics Models,” IND. ENG. CHEM.,FUNDAM., 539 (1967). (1 1E) Hocking R R . and Leslie R . N.,“Selection of the Best Subset in Regression Analysis,” Tlchnome;., 9, 531 (1467). (12E) Vynter, W. G . , “Estimation of Unknown Constants from Multiresponse 6,461 (1967). Data, IND.ENC.CHEM.,FUNDAM., (13E) Hunter, W. G., Kittrell, J. R . and Mezaki R . “Experimental Strate ies for Mechanistic Modelling,” Trans.>nst. Chem. E&. \London), 45, T146 (19f7). (14E) Jennrich, R . I., and Sampson P F “ A p lication of Stepwise Regression to Non-Linear Estimation,” Technome;., Ib,23 (19f8). (15E) Kittrell, J . R . , and Erjavec, J.,“Response Surface Methods in Heterogeneous Kinetic Modeling,” IND.ENG.CHEM.,PROCESS DES.DEVELOP., 7,321 (1968). (16E) La idus, L., and Bard, Y . , “Kinetic Analysis by Digital Parameter Estimation,’’ &tal. Reo., 2,67 (1968). (17E), Lee, ;! S., “Invariant Imbedding, Nonlinear Filtering, and Parameter 7,164 (1968). Estimation, IND.END.CHEM.,FUNDAM., (18E) Lee, E. S., “ Quasilinearization and Estimation of Parameters in Differential Equations,” ibid., p 152. (19E) Mezaki, R., and Butt, J. B., “Estimation of Rate Constants from Multiresponse Kinetic Data,” ibid., p 120. (20E) Oza, K. G., and Jury, E. I., “System Identification and the Principle of Random Contraction Mapping,” SZAM J.Contr., 6,244 (1968). (21E) Sawyer, D. N., and Mezaki, R., “Catalytic Hydrogenation of Olefins,” AZChE J.,13,1221 (1967).
Analytical Solution of Equations (1F) Anderson T. B and Jackson R., “Fluid Mechanical Descri tion of Fluidized Beds. Equ:tions ‘,f Motion,” IN&.ENG.CHEM.,FUNDAM., 6,52?(1967). (2F) Bennett C. 0. “ A Dynamic Method for the Study of Heterogeneous Catalytic Kinetics,” h C h E >., 13,890 (1967). (3F) Cholewinski F. M and Haimo, D. T., “Classical Analysis and the Generalized Heat Eq;ation,’;’SZAM Reo., 10,67 (1968). (4F) Clements, W. C., Jr., “ A Simple Method for Derivation of Concentration Transfer Functions for Flow Systems,” AZChE J.,14,363 (1968). (5F) Collins C. G and Deans H . A. “Direct Chromatographic Equilibrium Studies in k h e m i c h y Reactivebas-Soljd Systems,” ibid., p 25. (6F) Cox R G . and Brenner H “ T h e Slow Motion of a S here Through a Viscou;Fl;id ‘I!owards a Plan6 &face 11,” Chem. Eng. Sci., 2 2 , 8 5 3 (1967). (7F) Doig I. D. and R o er G. H “Contribution of Continuous and Dispersed Phases t‘o Sus&nsion o?Siheres gy a Bounded Gas-Solids Stream,” IND. ENO. CHEM.,FUNDAM., 7,459 (1968). (SF) Faith, L. E., and Vermeulen, T., “Kinetics of Complex Is$hermal Reversible First-Order Reaction Systems Involving Three Components, AIChE J., 13, 936 119671. (9F) Fayon, A.M., Lehnsen, J. E., and Katz, S.,“HeatTransfer toFinite Cylinders With Variable Temperatures a t the Uninsulated Plane Surface-An Analytical Solution,” Chem. Eng. Sci., 22, 1875 (1967). (IOF) Fendell, F. E;; :‘Flame Structure in Initially Unmixed Reactants Under One-Step Kinetics, rbid., p 1829. (11F) Gavis, J., and Laurence, R . L., “Viscous Heating in Plane and Circular Flow Between Moving Surfaces,” IND.ENG.CHEM.,FUNDAM., 7,232 (1968). (12F) Ghaffari, A., “ O n Integration of Hamilton-Jacobi Partial Differential Equation,” Int. J.Eng.Sci., 5,747 (1967). (13F) Graham, R . R., Vidaurri, F. C., Jr., and Gully, A. J., “Catalytic Dehydrogenation of Cyclohexane: a Transport Controlled Model,” AZChE J., 14, 473 (1968). (14F) Gunn D. J “Diffusion and Chemical Reaction in Catalysis and Absorption,’’ Cheh. Eng.’)Sci., 22,1439 (1967). (15F) Hansen A G., and Na, T. Y., “Similarity Solutions of Laminar Incomressible BoGndary Layer Equations of Non-Newtonian Fluids,” J. Basickn!., 90, 71-74 (1968). (16F) Helfferich, F. G., “Multicomponent Ion Exchange in Fixed Beds. &Feralized Equilibrium Theory for Systems with Constant Separation Factors, IND. ENG.CHEM.,FUNDAM., 6,362 (1967). (17F) Howell, J. A., Sparrow, E. M., and Schmal, M . “Concentration Temperature and Reaction Surfaces in Laminar T u b e Flow h t h Radially Stgpwise Inlet Distribution,” Chem. Eng. Sci., 22,1383 (1967). (18F) Hudson J. L. “Transient Multicomponent Diffusion with Heterogeneous 13,961 (1967). Reaction,” hIChE (19F) Ishida, M., and Wen, C. Y.,“Comparison of Kinetic and Diffusional Models for Solid-Gas Reactions,” ibid., 14,311 (1968). (20F) Kilkson H “Generalization of Various Polycondensation Problems,” IND.ENC.CAEM.;)FUNDAM., 7,354 (1968). (21F) Klein, G., Tondeur D. and Vermeulen T.,“Multicomponent Ion Exchan e in Fixed Beds. Genera\ P;operties of Equilibrium Systems,” ibid., 6 , 339 (19677. (22F) Letan, R., and Kehat, E., “ T h e Mechanism of Heat Transfer in a Spray Column Heat Exchanger,” AIChE J., 14,398 (1968). (23F) McCarthy H. E., and Olson J. H. “Turbulent Flow of Gas-Solids Suspensions,” IND.’ENC.CHEM.,FUNDAM., 7 , 4 h (1968). (24F) Moran, M . J., and Gaggioli, R . A., “Reduction of the Number of Variables in Systems of Partial Differential Equations, with Auxiliary Conditions,” SZAM J.Appl. Math., 16,202 (1968). (25F) Morton E L and Murrill P. W “Analysis of Liquid Phase Adsorption Fractionation in Fiied Beds,” AZdhE J.,?3,965 (1967). (26F) Olmstead, W. E., “ A Class of Viscous Flows Derivable from Diffraction Theory,” SIAM J.Appl. Math., 16,586 (1968). (27F) Olson, J. H., “Rates of Poisoning in Fixed-Bed Reactors,” IND.ENG.CHEM., FUNDAMEN., 7,185 (1968). (28F) Ozawa, Y.,and Bischoff, K. B., “Coke Formation Kinetics on Silica-Alumina Catalyst. Basic Experimental Data,” IND.ENO. CHEM.,PROCESS DES.DEVELOP., 7,67 (1 968). ~I
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(29F) Prater, C. A., Silvestri, A. J., and Wei, J., “ O n the Structure and Analysis of Complex Systems of First-Order Chemical Reactions Containing Irreversible Steps,” Chem. Eng.Sci., 22, 1587 (1967). (30F) Ratkowsky, D. A., and Epstein, N . “ F m i n a r Flow in Regular Polygonal Shaped Ducts with Circular Centered Cores, Cun. J . Chem. Eng., 46,22 (1968). (31F) Re1 ea D. L. and Permutter D. D “Stirred Reactor with Porous Catalyst Wall,” AD:ENG. &HEM., PROCESS ~ E S . D & E L O P7,261 ., (1968). (32F) Rim el, A. E., Jr., Camp, D. T., Kostecki, J. A,, and Canjar, L. N “Kinetics of ghysical Adsorption of Propane from Helium on Fixed Beds of Acgvated Alumina,” AZChE J., 14,19 (1968). (33F) Sagara, M . Masamune and Smith, J. M “Effect of Nonisothermal Operation on Cdtalyst Foulin;, ‘;bid., 13, 1226 (1967):) (34F) Solomon, A., “ A Steady State Phase Change Problem,” Math. Comp., 21, 355 (1967). (35F) Tondeur D and Klein G . “Multicomponent Ion Exchan e in Fixed Beds Constant-SeGaraYion-Factor’Eq;ilibrium,” IND. ENG. CHEM., 6, 3 5 i (1967). (36F) Yu, W. S., and Chen, J. C., “Slug Flow Heat Transfer with Mass Injection and Linearly Varying WallTemperature,” AZChE J., 13,1127 (1967).
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Optimization (1G) ,Avriel, M . , and Wilde, D. J., “Golden Block Search for the Maximum of Unimodal Functions,” Manag. Sci., 14,307 (1968). (2G) Bard, Y . , “ O n A Numerical Instability of Davidon-Like Methods,” Math. Comp., 22,665 (1968). (3G) Beltrami, E. J., “ O n Infinite-Dimensional Convex Programs,” J. Comp. SystemSci., 1, 323 (1967). (4G) Brosilow, C., and Nunez, E., “Multi-Level 0 timiration Applied to a Catalytic Cracking Plant,” Can. J. Chem. Eng., 46,205 8968). (5G) Broyden, C. G. “ uasi-Newton Methods and Their Application to Function Minimisation,” MLth.%omp‘. 21, 368 (1967). (6‘2) Dantzig G. B. and Van Slyke R . M., “Generalized Upper Bounding Techniques,” J.bomp. LiystemSci., I, 213 )(1967). (7G) Davidon, W. C., “Variance Algorithm for Minimization,” Comp. J., 10, 406 (1968). (8G) Dyson, D. C., and Horn, F. J. M . “ O p t i m u m Distributed Feed Reactors for Exothermic Reversible Reactions,” J. bpt. Theory Appl., 1,40 (1967). (9G) Dyson, D . C., Horn, F. J. M Jackson R., and Schlesinger C. B., “Reactor Optimization Problems for Re&sible Elothermic Reactions,”’ Con. J. Chem. Eng., 45, 310 (1967). (10G) Eben, C. D., and Ferron, J. R . , “ A qpnjugate Inequality for General Means with Applications to Extremum Problems, AZChE J., 14, 32 (1968). (11G) Echols, R . E., and Cooper L “ T h e Solution of Integer Linear Programming ProblemsofDirect Search,” J.k d k , 15,75 (1968). van elista J. J. and Katz S. “Best Temperature Schedules in Batch (1E!3zOt‘S,’kIND.k N C . CLEM., 60 (3): 24’(1968). (13G) Fiacco, A. V., “Second Order Sufficient Conditions For Weak and Strict Constrained Minima,” S I A M J. Appl. Math., 16,105 (1968). (14G) Forsythe, G . E., “On the Asymptotic Directions of the s-Dimensional Optimum Gradient Method,” Numer. Math., 11, 57 (1968). (15G) Gatehouse, J. S., “Optimization in the Presence of Constraints,” Control, 10,355 (1966). (16G) Goldfarb, D., and Lapidus, L., “Conjugate Gradient Method for Nonlineai Pro ramming Problems with Linear Constraints,” I N D .ENC. CHEM.,FUNDAM.. 7, 1% (1968). (17G) Greenstadt, J., “ O n the Relative Efficiencies of Gradient Methods,” Math, Comp., 21, 360 (1967). (18G) Gue, R . L., and Thomas, M . E.,“Mathematical Methods in Operations Research,’’ Macmillan, (1968). (1 9G) Guinn, T., “ First-Order Necessar Conditions for Generalized Optimization Problems,” J.Comp. System Sci., 1,235 6967). (20G) Guinn, T., “Solutions of Generalized Optimization Problems,” ibid., p 227. (21G) Himmelblau, D. M., and Bischoff, K. B., “Process Analysis and Simulation,” John Wiley and Sons, 1967. (22G) Horn, F . J. M . and Tsai, M . J., “ T h e Use of the Adjoint Variables in the Develo ment of Improvement Criteria for Chemical Reactors,” J. Opt. Theory Appl., 131 (1967). (23G) Hu, T. C., “ A Decomposition Algorithm for Shortest Paths in a Network,” Operations Res. 16, 91 (1968). (24G) Kermode, R. I “Geometric Programming: A Simple, Efficient Optimization Technique,” Ciim. Eng., 74, 97 (1967). (25G) Krolak, Patrick D., “Further Extensions of Fibonaccian Search to Nonlinear Programming Problems,” SZAM J. Contr., 6 , 258 (1968). (26G) Mehndiratta, S. L., “ Self-Duality in Mathematical-Programming,” SZAM J.Appl. Math., 15,1156 (1967). (27G) Methot, J. C., and Cholette, A,, “Adaptation of the Two-Derivative Method ofoptimization in Cases ofNon-Convergence,” Can. J . Chem. Eng., 45,319 (1967). (28G) Millman, M . C., and Katz, S., “Linear Temperature Control in Batch Reactors,” IND.END.CHEM.,PROCESS DES.DEVELOP.,6,447 (1967). (29G) Morrison, D. D., “Optimization by Least Squares,” S I A M J. Nume,. Anal., 5 , 8 3 (1968); (30G) Orden, A,, and Nalbandian, V., “ A Bidirectional Simplex Algorithm,” J.A C M , 15,221 (1968). (31G) Pacey, W. C., and Rustin, A,, “Optimization of a Reaction and Heat Exchange System with Recycle,” Can. J. Chem. Eng., 45, 305 (1967). (32G) Passy, U., and Wilde, D. J., “Generalized Polynomial Optimization,” SZAM J. Appl. Math., 15, 1344 (1967). (33G) Passy, U., and Wilde, D. J., “ A Geometric Programming Algorithm for Solving Chemical Equilibrium Problems,” SZAM J.Appl. Math., 16, 363 (1968). (34G) Ray,, W. H., “Modelling Polymerization Reactors with Applications to Optimal Design,” Can. J. Chem. Eng., 45, 356 (1967). (35G) Ritter, K., “ A Decomposition Method for Structural Quadratic ProgramSci... 1., 241 (1967). mine Problems.“ J . Comb. Svstem , . (36G) Rudd, D. F., “ T h e Synthesis of System Design: I . Elementary Decomposition Theory,” AZChE J., 14, 343 (1968). (37G) R u d d , D. F., and Watson, C. C., “Strategy of Process Engineering,” John Wilev and Sons. 1968. (38G) Schechter, R. S., “ T h e Variational Method in Engineering,” McGraw-Hill, 1967. (39G) Smith, W. R., and Missen, R . W., “On the Two-Derivative Method of Optimization,” Can. J.Ckem. Eng., 45,346 (1967).
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(40G) Temes, G . C., and Calahan, D. A., “Computer-Aided Network Optimization-The State-of-the-Art,” Proc. I E E E , 55, 1832 (1967).
(41G) Warren, A. D., Lasdon, L. S., and Suchman, D . F., “Optimization in Engineering Design,” ibid., p 1885. (42G) Wilde, D. J., and Beightler, C. S., “Foundations of Optimization,” PrenticeHall, 1967. (43G) Zangwill, W. I., “ A n Algorithm for the Chebyshev Problem-With an ApDlication to Concave Proerammine.” Manae. Sci.. , 14.,58 119671. . (44G) Zangwill, W. I . , “ T h e Convex Simplex Method,” ibrd., p 221. (45G) Zangwill, W. I., “Minimizing a Function Without Calculating Derivatives,” Comp. J . , 10,293 (1967). (46G) Zeleznik, F. J., I ‘ Quasi-Newton Methods for Nonlinear Equations,” J . A C M , 15,265 (1968). Y
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Dynamic Programming (1H) Beightler, C. S., and Meier W . L. Jr “Design of an Optimal Branched Allocation System,” I N D . END.CLEM., 66, 44’11968). (2H) Brogan, W. L., “Dynamic Programming and a Distributed Parameter Maximum Principle,” J . Basic Eng., 90, 152 (1968). (3H) Cardillo, G. P. and Fu, K. S., “ A Dynamic Programming Procedure for Sequential Pattern Classification and Feature Selection,” Math. Biosci,, 1, 463 (1967). (4H) Chien, Y. T., and Fu, K . S . , “ A n Optimal Pattern Classification System Using Dynamic Programming,” ibid., p 439. (5H) Davis, R . C., “Stochastic Final-Value Control Systems with a Fuel Constraint,” J M A A , 21, 62 (1968). (6H) Detchmendy, D. M., and Kalaba, R . E., “Reduction in Decision Dimensionality in Dynamic Programming,’’ I E E E Auto. Conlrol, 13,296 (1968). (7H) G::eroso, E., Jr., and Hitchcock, L. B., “Optimizing Plant Expansion-Two Cases, IND.END.CHEY.,60, 12 (1968). (8H) Gilmor, P . A.,“Structuring ofparallel Algorithms,” J . A C M , 15, 176 (1968). (9H) Hellinckx, L., and Rijckaert, M., “ A Graphical Representation of the DynamicProgramming Algorithm,” Chem. Eng. Sci., 22,1149 (1967). (10H) Ishimoto, S., Sasano, T . , and Kavamura, K . , “ Liquid-Phase Oxidation of DES.DEVELOP., 7, 469 (1968). Cyclohexanol,” IXD.END.CHEM.,PROCESS (11H) Itahara, S., and Stiel, L. I., “Optimal Design of Multiple-Effect Evaporators with Vapor Bleed Streams,” ibid., p 6. (12H) Kaufmann, A,, and Cruon, R . , “Dynamic Programming,” Academic Press, 1967. (13H) King, R . P., “Optimal Replacement and Maintenance Policies,” IND.ENG. CHEM.,60,29 (1968). (14H) LaSson, R . E., “ A Survey of D namic Programming Computational Procedures, I E E E Auto. Control, 12, 767 (7967). ’ (l5H) Larson, R . E., “State Increment Dynamic Programming,” American-Elsevier, 1968. (16H) Larson, R . E., Keckler, W. G., and Bard, Y., “Comments on Interpolation and Extrapolation Schemes in Dynamic Programming,” ZEEE Auto. Control., 13, 294 (1968). (17H) Malenge, J.-P., and Willermaux, J., “Optimal Adiabatic Bed Reactor with Cold Shot Cooling,” IND.END.CHEW,PROCESS DES.DEVELOP., 6 , 535 (1967). (18H) Neu, R., and Nixdorf, K., “Nesting of Subprograms in Dynamic Programming,” Elektron. Datenuernrb. 5, 219 (1967). (19H) Pruzan, P . M., and Munch-Andersen, B., “ O n the Application of Dynamic Programming-Type Algorithms to Antenna Design,” S I A M J . Appl. Math., 15, 1113 (1967). (20H) Seinfeld J. H. and Lapidus L “Aspects of Forward Dynamic Programming Algorithm,” f N D . ENO.CHEhf., PkOZESS DES.DEVELOP., 7, 475 (1968). (21H) Seinfeld, J. H., and Lapidus, L., “Discrete Dynamic Programming Algorithms,” ibid., p 479. (22H) Shapiro, J. F., “Dynamic Programming Algorithms for the Integer Programming Problem-I: T h e Integer Programming Problem Viewed as a Knapsack Type Problem,” Operationr Res., 16, 103 (1968). (23H) Thiriet, L., and Deledicq, A,, “Applications of Suhoptimality in Dynamic Programming,” IND.END.CHEY.,60, 23 (1968). Stability (11) Albertoni, S., 4pllina A and Szego G. P. “ O n the Stability of Homogeneous Nuclear Reactors, J . F&k: Inst., 284, i 7 9 (lj67). (21) Anderson, T. B., and Jackson, R . , “Fluid Mechanical Description of Fluidized Beds. Stability of State of Uniform Fluidization,” IND.END.CHEM.,FUNDAY., 7, 12 (1968). (31) Anderson, T. B., and Jackson, R., “Hydrodynamic Stability of a Fluidized Bed, ibid., 6, 487 (1967). (41) Berger,,A. J., and Lapidus, L.,“An Introduction to the Stability ofDistributed Systems Via a Liapunov Functional,” AZChE J . , 14, 558 (1968). (51) Blodgett R . E. and King R . E., “Absolute Stability of a Class of Nonlinear Systems Cohtaini& Distributid Elements,” J . Franklin Inst., 284, 153 (1967). (61) Busse F . H. “ T h e Stability of Finite Amplitude Cellular Convection and Its R e l a t i o i to an kxtremum Principle,” J.Fluid Mech., 30, 625 (1967). (71) Chen, T. S., and Sparrow, E. 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50
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(181) Luecke, R. H., and McGuire, hl. L., “Stability Analysis by Liapunov’s Direct Methods. Investigation and Extension of Krasovskii’s Theorem,” IND. E N G .CHEM.,FUNDAM., 6,432 (1967). (191) Luss, D., and Amundson, N.R., “Performance and Stability of Countercurrent Two-Phase Extractive Tubular Reactors,” ibid., p 436. (201) Luss, D., and Amundson, N. R. “Some General Observations on Tubular Reactor Stability,” Can. J . Chem. Eng.: 45, 341 (1967). (211) Luss, D., and Amundson, N . R.,“Stability ofBatch Catalytic Fluidized Beds,” AIChE J . 14,211 (1968). (221) Luss, D., and Amundson, N . R . , “Uniqueness of Steady State for an Isothermal Porous Catalyst,” IND.END.CHEM.,FUKnAM., 6,457 (1967). (231) Maurer, C. J., and Garlid, K . L., “Stability of Katurally Bounded Nonlinear Systems,’’ AIChE J . , 14, 3 (1968). (241) Newman, A . K., “ N e w Liapunov Function for Nonlinear Time-varying Systems,” J . Basic Eng., 90, 208 (1968). 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(335) Porter, William A., “On Function Space Pursuit-Evasion Games,” SZAM J . Contr., 5,555 (1967). (345) Rothenberger, B. F., and Lapidus, L., “ T h e Co?frol of Nonlinear Systems: Part IV. Quasilinearization as a Numerical Method, AZChE J.,13, 973 (1967). (35J) Rothenberger B F and La idus, L., “ T h e Control of Nonlinear Systems: Part V. Quasili&a;iz;tion a n a State-Constrained Systems,” AZChE J . , 13, 982 (1967). (365) Sebesta H R and Clark, L. G., “On the 0 timal-Control Problem for Dynamical k-o&ss;s with Variable Delays,” J.Basic Eng., 90,181 (1968). (375) Tapley B D and Lewallen J M “ C o m arison of Several Numerical Optimizati6n Metzods,” J . Opt. Tiear; Ap$l., 1, 1 8967). (38J) Thomas W. J. and Wood R . M . “Use of the Maximum Principle to Calculate Optirktm Cktalyst Corni,osition’Profiles for Bifunctional Catalyst Systems Contained in Tubular Reactors, Chem. Eng. Sci.,22,1607 (1967). (395) Traca, G. S., “ A Selected Bibliography on the Application o f o p t i m a l Control Theory to Economic and Business Systems, Management Science, and Operations Research,’’ Operations Res. 16, 174 (1968). (40J) Warga, J., “Restricted Minima of Functions of Controls,” SZAM J . Contr., 5 , 642 (1967). (415) Wen, C. Y . , and Chan T. M., “ 0 timal Design of Systems Involving Parameter Uncertainty,” IND.%NG.CHEM.,~ R O C E S SDES.DEVELOP.;7,49 (1968). (42J) Wilde, D. J., “Automatic Partial Inventory Control with Interaction,” I N D .END.CHEM.,60 (2), 60 (1968). Numerical-General (1K) Ballester, C., and Pereyra, V., “ O n the Construction of Discrete Approximations to Linear Differential Expressions,” Math. Camp., 21, 297 (1967). (2K) Cole, J. D., “Perturbation Methods in Applied Mathematics,” Blaisdell, 1968. (3K) Davis, P. J., and Rabinowitz, P.,“Numerical Integration,” ibid., 1967. 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(2M) Aoki M . “ A Note on Aggregation and Bounds for the Solution of the Matrix Riccati iquahons,” J . Math. Anal. Appl., 21, 377 (1 968).
(3M) Bartels, R . H., and Golub, G. H.,“Stable Numerical Methods for Obtaining the Chebyshev Solution to a n Overdetermined System of Equations,” Comm. A C M , 11,401 (1968). (4M) Bjorck, A., “Iterative Refinement of Linear Least Squares Solutions I,” B I T , 7,257 (1967). (5M) Bjarck A “Iterative Refinement of Linear Least Squares Solutions 11,” ibid., 8 , 8 (i968j. (6M) Bowdler, H., Martin, R . S., Reinsch, C., and Wilkinson, J. H., “ T h e Q R and Q L Algorithmsfor Symmetric Matrices,” Num. Math., 11, 293 (1968). (7M) Bucharen, J. E., and Thomas, D. H., “On Least-Squares Fitting of TwoDimensional Data with a Special Structure,” SIAM J.Num. Anol., 5 , 252 (1968). (8M) Bucy, R . S., “Global Theory of the Riccati Equation,” J . Camp. System Sci., 1, 349 (1967). (9M) Charmonman, S., “ A n Efficient Algorithm of Inverting a Block-Symmetric Matrix,” Math. Camp., 21, 714 (1967). (10M) Eberlein P. J and Boothroyd J. “Solution to the Eigenproblem by a Norm Reducing JacAbi T’;pe Method,” N;m.’Math., 11, 1 (1968). (11M) Fettis, H. E., and C a s h , J. C “Eigenvalues and Eigenvectors of Hilbert MatricesofOrder 3 through 10,” M a k Comp., 21,431 (1967). (12M) Forsythe, G. E., and Moler, C. B., “Computer Solution of Linear Algebraic Systems,” Prentice-Hall, 1967. (13M) Foster, R . N., and Butt, J. B. “Some Surface Transport Effects on Activity 6, in Diffusionally Limited Cata1yti)c Systems,” IND. ENG. CHEM., FUNDAM., 481 (1967). (14M) Hall, C . A., and Porsching, T. A.,“Com uting the Maximal Eigenvalue and Eigenvectorofa Positive Matrix,” SIAM J . A m . Anal., 5,269 (1968). (15M) Herring, G. P., “ A Note on Generalized Interpolation and the Pseudoinverse,” ibid., 4, 548 (1967). (16M) Jordan, T. L., “Experiments on Error Growth Associated with Some Linear Least-Squares Procedures,” Math. Camp., 22, 579 (1968). (17M) Langenhop, C. E., “On Generalized Inverses of Matrices,” SIAM J. Appl. Math., 15, 1239 (1967). (18M) Loizou, G., “ A n Empirical Estimate of the Relative Error of the Computed SolutioniofAx = 6,”Cump. J . , 1 1 , 9 1 (1968). (19M) Martin, R . S., Reinsch, C., and Wiikinson, J. H., “Householder’s Tridiagonalization of a Symmetric Matrix,” Num. Math., l l , 181 (1968). @OM) Martin, R. S., and Wilkinson, J. H., “Reduction of the Symmetric Eigenproblem Ax = XBx and Related Problems to Standard Form,” ibid., p 99. (21M) Morris, G . L., and Odell, P. L., “ A Characterization for Generalized Inverses of Matrices,” SZAM Rev., 10,208 (1968). (22M) Morris, G. L., and Odell, P. L.,“Common Solutions for n Matrix Equations with Applications,” J . ACM, 15,272 (1968). (23M) Nathan A. and Even R. K “ T h e Inversion of Sparse Matrices by a Strategy Der:vedlfrom Their Graphs;” Comp. J., 10, 190 (1967). (24M) Pease, M . C., “Matrix Inversion Using Parallel Processing,” J . A C M , 14, 757 (1967). (25M) Porsching, T. A., “ O n the Spectrum of a Matrix Arising from a Problem in Reactor Kinetics,” SZAM J . Appl. Math., 16 301 (1968). (26M) Price, H. S . , “ M o ~ o t o n eand Oscillation Matrices Applied to Finite Difference Approximations, Moth. Camp., 22, 489 (1968). (27M) Polya, G., “Graeffe’s Method for Eigenvalues,” Num. Math., 11, 315 (1968). (28M) Reinsch, C., and Bauer, F. L., “Rational Q R Transformation with Newton Shift for Symmetric Tridiagonal Matrices,” ibid., p 264. (29M), Re logle, J., Holcomb, B. D., and Burrus, W. R., “ T h e Use of Mathematical %rogrammin for Solvin Singular and Poorly Conditioned Systems of Equations,” J M A A , $0, 310 (196%. (30M) Ruhe, A,, “On the Quadratic Convergence of the Jacobi Method for Normal Matrices,” B I T , 7, 305 (1967). (31M) Schwartz, C., “Numerical Techniques in Matrix Mechanics,” J . Camput Phys., 2 , 90 (1967). (32M) Teyarson, R . P., “ A Computational Method for Evaluating Generalized Inverse, Comp. J., 10,411 (1968). (33M) Tewarson, R . P., “ A Direct Method for Generalized Matrix Inversion,” SZAM J . Num. Anal., 4,499 (1967). (34M) T e w a m m , R . P., “Solution of Linear Equations with Coefficient Matrix in Band Form, B I T , 8 , 53 (1968). (35M) Tewarson, R . P., “Solution of a System of Simultaneous Linear Equations with a Sparse Coefficient Matrix by Elimination Methods,” ibid., 7,226 (1967). (36M) Tuel, W. G “Computer Algorithm for Spectral Factorization of Rationa Matrices,” ZBM 2 Res. Duel., 12, 163 (1968). (37M) Urquhart, N. S., “Computation of Generalized Inverse Matriccs Which Satisfy Specified Conditions,” SIAM Rev., 10,216 (1968). (38M) Westlake, J. R . , “ A Handbook of Numerical Matrix Inversion and Solution of Linear Equations,” John Wiley and Sons (1968).
Numerical-Ordinary
Differential Equations
(1N) Brush, D . G., Kohfeld, J. J., and Thompson, G. T., “Solution of Ordinary Differential Equations Using Two‘Off-Step’ Points,” J . A C M , 14, 769 (1967). (2N) Cooper, G . J “ A Class of Single-Step Methods for Systems of Nonlinear Differential Equatio&“ Math. Comp., 21, 597 (1967). (3N) Cooper G. J., “Interpolation and Quadrature Methods for Ordinary Differential E&ations,” ibid., 22, 69 (1968). (4N) Cooper G J., and Gal E “Single Step Methods for Linear Differential Equations,:’ N i m . Math., 10, jOf’(l967). (5N) Davison, E. J “ T h e Numerical Solution of Large Systems of Linear Differential Equations,” 2ZChE J . , 14,’46 (1968). (6N) Distefano, G. P., “Mathematical Modeling and Numerical Integration of Multicomponent BatchDistillation Equations,” ibid., p 190. (7N) El-Sherif H. H “Implicit I m lementation of the Weighted Backwards Euler Formula,” I b M J.’hes.Develop., 18, 335 (1968). (8N) Fowler, M . E., and Warten, R . M . , “ A Numerical Integration Technique for Ordinary Differential Equations with Widely Separated Eigenvalues,” ibid., 11, 537 (1967). (9N) Goldwyn, R . M., and Sloan, L. De. L., “ A Geometrical Method of Solving Second Order Nonlinear Differential Equations,” SIAM J . Appl. Math., 16, 146 (1968). (10N) Hirt, C . W and Harlow, F. H., “ A General Corrective Procedure for the Numerical Solut?on of Initial-Value Problems,” J . Cumput. Phys., 2, 114 (1967). (11N) Hull T. E. “ A Search for Optimum Methods for the Numerical Integration of Ordiniry Differential Equations,” SIAM Rev., 9 , 647 (1967).
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.(12N),Jackson, R., “An Approach to the Numerical Solution of Time-Dependent Optimization Problems in Two-Phase Contacting Devices,” Trans. I n s t . Chem. Engrs., 45, T160 (1967). (13N) Klopfenstein R. W., and Millman R . S. “Numerical Stability of a OneEvaluation PrediAtor-Corrector Algoritdm for )Numerical Solution of Ordinary Differential Equations,” Math. Camp., 22, 557 (1968). (14N) Konen, H. P. and Luther, H. A., “Some Sin ular Explicit Fifth Order Runge-Kutta Solutions,” S I A M N u m . Anal., 4,607 (19f7). (15N) Krogh, F. T. “ A Note on the Effect of Conditionally Stable Correctors,” Math. Comp., 21, 7 i 7 (1967). (16N) Lambert, R . J., “ A n Analysis of the Numerical Stability of Predictor-Corrector Solutions of Nonlinear Ordinary Differential Equations,” SZAM J . Nun. Anal., 4,597 (1967). (17N) Lawson, J . D., “Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants,” ibid., p 372. (18N) Lawson, J. D., ”An Order Six Runge-Kutta Process with Extended Region ofstability,’, rbid., p 620. (19N) Loscalzo, F. R., and Talbot, T. D., “Spline Function Approximations for Solutions of Ordinary Differential Equations,” ibid., p 433. (20N) Miranker, W . L., and Liniger, W., “Parallel Methods for the Numerical Integration of Ordinary Differential Equations,” Math. Camp., 21, 303 (1967). (21N) Murphy, W. D., “Numerical Analysis of Boundary-Layer Problems in Ordinary Differential Equations,” ibia‘., p 583. (22N) Newman J. “Numerical Solution of Coupled, Ordinary Differential Equations,” INO.E~o.’CHEM., FUNDAM., 7,514 (1968). (23N) Osborne, M . R., and Watson, G. A , , “Note on T w o Methods of Solving Ordinary Linear Differential Equations,” Comp. J., 10, 383 (1968). (24N) Riekert, L., and Wei, I., “Kinetics of Coupled First-Order Reactions with Time-Dependent R a t e Coefficients in Ternary Systems,” IND.ENC. CHEM., FUNDAM., 7, 125 (1968). (25N) Rosser, J. B., “ A Runge-Kutta for All Seasons,” SIAM Rev., 9, 417 (1967). (26N) Sankar-Rao, M . “ A Method of Solution for a System of Two Second-Order Differential Equations Arising in the Theory of the Mean Atmospheric Waves,” J . Comput. Phys., 2, 228 (1968). (27N) Thorogood R . M., “ T h e Dynamic Response of an Air Rectification Column to Interruptions’of the Feed Flow,” Chem. Eng. Sci., 22,1457 (1967). (ZEN) Urabe, M . “Roundoff Error Distribution in Fixed-point Multiplication,” SIAM J . N u n . Anal., 5,202 (1968). Numerical-Partial
Differential Equations
( 1 0 ) Birkhoff, G., Schultz, M . H., and Varga, R . S., “Piecewise Hermite Interpolation in O n e and Two Variables with Applications to Partial Differential Equations,” Num. Math., 11,232 (1968). ( 2 0 ) Cannon J. R and Dou las J. Jr. “ T h e Cauchy Problem for the Heat Equation,” h A M 3. Num. Ana7., 4: 315 (1667). ( 3 0 ) ,Cannon, J. R., and Hill, C. D., “ A Finite-Difference Method for Degenerate Elliptic-Parabolic Equations,” ibid., 5, 211 (1968). ( 4 0 ) Cheng, K . C.,“Dirichlet Problems for Laminar Forced Convection with Heat Sources and Viscous Dissipation in Regular Polygonal Ducts,” AZChE J . , 13, 1175 (1967). ( 5 0 ) Demin J. L., T,ao, L. C., and Weber, J. H., “Numerical Solutions of Disk Source Proglems,” ibid., p 1214. ( 6 0 ) Eisen, D., “On the Numerical Analysis of a Fourth Order Wave Equation,” S I A M J . iVum. Anal., 4,457 (1967). ( 7 0 ) Eisen, D . , “ O n the Numerical Solution ofu, = u , ~ Z/r(u,),” N u m . Math., 10, 397 (1967). ( 8 0 ) Emery, A. F., “ A n Evaluation of Several Differencing Methods for Inviscid Fluid Flow Problems,” J . Comput. Phys. 2, 306 (1968). ( 9 0 ) Fagela-Alabastro, E. B., and Hellums, J. D., “Laminar Gas Jet Impinging on an Infinite Liquid Surface. Numerical Finite Difference Solution Involving Boundary and Free Streamline Determinations,” IND. ENG. CHEM.,FUNDAM., 6 , 580 (1967). (100) Gavalas, G. R., Reamer, H . H., and Sage, B. H., “Homogeneous Phases at Elevated Pressures,” ibid., 7, 306 (1968). (110) Gourla A. R . “ T h e Acceleration of the Peaceman-Rachford Method by Chebyshev l!~lynomials,” Comp. J . , 10, 378 (1968). ( 1 2 0 ) Gourlay, A. R., and Mitchell, A. R . “Split Operator Methods for Hyperbolic Systems inp-Space Variables,” Malh.’Comp., 21, 351 (1967). ( 1 3 0 ) Gourlay, A. R . , and Morris, J . LI.,‘‘Deferred Approach to the Limit in NonLinear Hyperbolic Systems,” Comp. J., l 1 , 4 5 (1968). (140) Gourlay, A. R., and Morris, J. LI.,“Finite-Difference Methods for Nonlinear Hyperbolic Systems,” Math. Comp., 22, 28 (1968). (150) Gourlay, A. R., and Morris, J. LI., “Finite-Difference Methodsf or Non. linear Hyperbolic Systems, II.,” ibid., p 549. ( 1 6 0 ) Greenspan, D., “ A Numerical Approach to Biharmonic Problems,” Camp. J., 10, 198 (1967). (170) Hagem:n L. A and Kellogg R . B., “Estimating Optimum Overrelaxation Parameters, h a t h . ;om$., 22, 60 (i968). (180) Hedstrom, G . W. “ T h e Rate of Convergence of Some Finite Difference Schemes,” SIAM J . N u x . Anal., 5, 363 (1968). ( 1 9 0 ) Jarratt, P., and Mack, C . , “ A %ast Squares Method for Laplace’s Equation with Dirichlet Boundary Conditions, Comp. J . , 11, 83 (1968). ( 2 0 0 ) Kettlehorough, C. F., “Poiseuille Flow with Variable Fluid Properties,” J.BaszcEng., 89,666 (1967). (210) Lehr, C. G., Yorchak, S., and Kabel, R . L.,“Response of a Tubular Heterogeneous Catalytic Reactor to a Step Increase in Flow Rate,” AZChE J . , 14, 627 (1968). ( 2 2 0 ) Lynch, R . E., and Rice J . R “Convergence Rates of AD1 Methods with Smooth Initial Error,’’ Math.’Comp,:’22, 311 (1768). ( 2 3 0 ) Olson, K . E. Luss D., and Amundson, N. R . , “Regeneration of Adiabatic . PROCESS DES.DEVELOP., 7,96 (1968). Fixed Beds,” I N D~. N C &EM., ( 2 4 0 ) Poussin, F. D., “An Accelerated Relaxation Algorithm for Iterative Solution of Elliptic Equations,” SIAM J . Num. Anal., 5,340 (1968). ( 2 5 0 ) Rachford, H. H., Jr., “Rounding Errors in Parabolic Problems. I : T h e One-Space Variable Case,” ibid., p 156. ( 2 6 0 ) Rachford, H. H., Jr. “Rounding Errors in Alternating Direction Methods for Parabolic Problems,” &AM J . N u m . Anal., 5, 407 (1968). ( 2 7 0 ) Randall, T. J . , “ A N o t e on the Estimation of the Optimum Successive Overrelaxation Parameter for Laplace’s Equation,’’ Con$. J., 10,400 (1968). ( 2 8 0 ) Richtmeyer, R . D., and Morton, K . W., “Difference Methods for InitialValue Problems,” Interscience, 1967. (290) Rubin, E. L., “Difference Methods for the Inviscid and Viscous Equations o f a Compressible Gas,” J . Comput. Phyr., 2,178 (1967).
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(300) Sigillito, V. G “ O n a Continuous Method ofApproximating Solutions of the H e a t Equation,” J:’ACM, 14,732 (1 967). (310) Smith, J . , “ T h e Coupled Equation Approach to the Numerical Solution of the Biharmonic Equation by Finite Differences,” S I A M J. Arum. Anal., 5 , 323 (1968). (320) Villadsen, J. V., and Stewart, W. E.,“Solution of Boundary-Value Problems by Orthogonal Collocation,” Chem. Eng. Sci., 22, 1483 (1967). (330) Von Rosenberg D . U. “Numerical Solution for Flux Components in PotentialFlow,” Math. bomomp., Zi,620 (1967). (340) Weekman, V. W., Jr., “Mod,$ of Catalytic Cracking Conversion in Fixed, Moving, and Fluid-Bed Reactors, I h D . END. CHEM.,PROCESS DES. DEVELOP., 7.90 (1968). ( 3 5 0 ) Wendroff, B., “ Well-posed Problems and Stable Difference Operators,” SIAM J.Num.Anal..5.71 . , 11968). , (360) Werner, W., “Numerical Solution of Systems of Quasi-linear Hyperbolic Differential Equations by Means of the Method of Nebencharacteristics in Combination with Extrapolation Methods,” Arum. Math., 11, 151 (1968). (370) Wilhelm, R . H., Rice, A. W., Rolke, R . W., and Sweed, N. H., “Parametric Pumping,” I N D END. . CHEM.,FUNDAM., 7,337 (1968). (380) Wood, W. L . , “ O n an Explicit Numerical Method for Solving the Biharmonic Equation,” Num. Math., 11,413 (1968). ( 3 9 0 ) Zlamal, M., “Discretization and Error Estimates for Elliptic Boundary Value Problems of the Fourth Order,” SIAM J . Nun. Anal., 4,626 (1967). ~
Numerical-Boundary Value Problems (1P) Bucy, R . S., “Two-Point Boundary Value Problems of Linear Hamiltonian Systems,” SZAM J . Appl. Math., 15, 1385 (1967). (ZP), Ciarlet, P. G., Schultz, M . H., and Varga, R . S., “Numerical Methods of High-Order Accuracd): ,for ,?linear Boundary Value Problems. 11. Nonlinear Boundary Con itions, Num. Math., 11, 331 (1968). (3P) Denman, E. D., and Kelley, J. P., “ Invariant Imbedding and the Band Theory ofSolids,” JMAA, 21, 53 (1968). (4P) Feeser, L. J., and Feng, C. C.,“Flow Graphs and Boundary Value Problems,” J . Fronklzn Inst., 284, 251 (1967). (5P) Hagin, F. G., “Some Asymptotic Behaviour Results for Initial Value Problems: An Application of Invariant Imbedding,” JMAA, 20, 540 (1967). (6P) Keller, H. B.,“Numerical Methods for Two-Point Boundary-Value Problems,” Blaisdell, 1968. (7P) Koenig, D. M., “Invariant Imbedding: New Design Method in Unit Operations,” Chem. Eng., 74, 181 (1967). (8P) Lee, E. S., “Quasilinearization and Invariant Imbedding,” Academic, 1968. (9P) Lee, E. S., “Quasilinearization, Difference Approximation, and Nonlinear Boundary Value Problems,” AZChEJ., 14,490 (1968). (1OP) Lee, E. S., “Quasilinearization in Optimization: A Numerical Study,” tbzd., 13, 1043 (1967). (11P) Lee E. S and Fan L. T., “Quasilinearization Technique for Solution of BoundaIy Lay& Equatio&” Can. J . Chem. Eng., 46,200 (1968). (12P) Meyer, G . H., “On a General Theory of Characteristics and the Method of Invariant Imbedding,” SIAM J . Appl. Moth., 16,488 (1968). (13P) Na, T. Y . ,“Further Extension on Transforming from Boundary Value to Initial Value Problems,” SIAM Rev., 10, 85 (1968). (14P)Roberts, S. M., and Shipman, J. S., “Jus;!fication for the Continuation Method in T w o Point Boundary Value Problems, J M A A , 21, 23 (1968). (15P) Shampine, L. F., “Boundary Value Problems for Ordinary Differential Equations,” SIAM J . Num. Anal., 5,219 (1968). (16P) Srinivasan, S. K., and Koteswara-Rao N . U ”Invariant Imbedding Technique and Age-Dependent Birth and Death broces&,’’ J M A A , 21,43 (1968). Numerical-Data
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(1Q) Bergland G . D., “ A Fast Fourier Transformation Algorithm Using Base 8 Iterations,” Math. Camp., 22, 275 (1968). ( Z Q ) Buchanan, J. E., “Holdup in Irrigated Ring-Packed Towers Below the Loading Point,” I N D .END.CHEY.,FLWDAM., 6,400 (1967). ( 3 4 ) Cochran, W . T . , and others, “ W h a t Is the Fast Fourier Transform?,” Proc. IEEE,55,1664 (1967). (4 ) Cooley J. W., Lewis, P. A . W., and Welch, P . D., “Historical Notes on the g a s t Fouridr Transform,” ibid., 55, 1675 (1767). (5 ) Duffin, R . J., “Extrapolating Time Series by Discounted Least Squares,” Y M A A , 20, 325 (1967). (6Q) Manchanda, K. D., and Woods, D. R . , “Significance Design Variables in Continuous Gravity Decantation,” IXD. EKO. CHEM.,PROCESS DES. DEVELOP., 7,182 (1968). ( 7 4 ) Meister B “ A n Application of the Cooky-Tukey Algorithm to Equalization,” Z E M j . :as. Develop., 12, 331 (1968). (8Q) P e a x , M . C. “ A n Adaptation of the Fast Fourier Transform for Parallel Processing,” J . A b M , 15, 252 (1968). (9Q) Singleton, R. C., “ O n Computing the Fast Fourier Transform,” Camm. ACM, 10.647 (1967). (lOQ) Valentas, K. J., and Amundson, N . R., “Influence Droplet Size-Age Distribution on Rate Processes in Dispersed-Phase Systems, IND.ENO. CHEM., FUNDAM., 7, 66 (1968).
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Numerical-Root Location (1R) Adams D . A. “ A Stopping Criterion for Polynomial Root Finding,” Camrn. ACM, 10, i 5 5 (19i7). (2R) Birtwistle, G . M . , and Evans, D. J., “ O n the Generalization of Bairstow’s Method,” B I T , 7,175 (1967). (3R) Bohte, Z. “Numerical Solution of the Inverse .4lgebraic Eigenvalue Problem,” Camp.J., lo,;, 385 (1968). (4R) Booth, R . S.,“Random Search for Zeroes,” JMAA, 20,239 (1967). (5R) Dennis, J. E,, J r . , “ O n Newton-Like Methods,” iVum. Moth., 11, 324 (1968). (6R) Epstein, M . P., “ T h e Use of Resultants to Locate Extreme Values of Polynomials,” SZAMJ. Appl. Moth., 16, 62 (1968). (7R) Fletcher, R . , “Generalized Inverse Methods for the Best Least Squares Solution of Systems ofNon-linear Equations,” Comp. J., 10,392 (1968). (8R) Garside, G. R., Jarratt P. and hlack, C . , “ A New Method for Solving Polynomial Equations,” ibid.,
