Applied Mathematics This year’s review lists new journal, general books, and articles pointing out special ones of interest to chemical engineers
his review follows the format introduced last year in which pubT lications outside of chemical engineering are included. T o aid the reader, the subject areas have been divided into sections; however, it should be emphasized that the distribution of publications within the different sections is not unique. The articles and books listed cover roughly the period July 1968 to July 1969. A new journal, Cornfluting Surveys, was introduced this year and a t least one paper (27) in the first two issues is of direct interest since it covers the development of the digital computer as a viable computational tool. T h e December 1968 issue of J . Franklin Znst. also deserves special attention for its coverage of “Modern Aspects of Large Scale System Science.” In particular, the highly intriguing “tearing” method of Kron which decomposes large complex systems into smaller ones is detailed (79,27). It is important to point out the Algorithm section of the Comm. ACM which presents worked out and verified ALGOL algorithms for almost all areas of numerical calculations. As an illustration, Algorithm 351 details a program for a modified Romberg quadrature, which is one of the best and most efficient ways to evaluate a n integral. A number of new general books were published during the past year. These include a superb volume on all aspects of chemical reactor analysis ( 2 ) [see (3)also], a book on system analysis (6) and on computer process control (ZO), a book on the use of moment methods (72),another on the solution of difference equations (23), and a very readable one on numerical analysis (75). A number of articles have been concerned with various aspects of on-line automation of laboratories (7, 77), including a whole issue of a journal ( 2 9 ) . Topological and network analysis have been of interest to a number of writers (5,7, 70, 77, 24, 28), as well as the connection between economic and biological systems (4). Broad articles on the equations of transfer processes (8), the use of multivector calculus and funtions ( 7 3 , 74),random number generation techniques (78,25), Monte Carlo methods (30),and aspects of parallel-type computations (26) have been published. I n ad-
dition, a number of authors (9, 76) have been concerned with aspects of allowing the computer to derive its own differential equations to represent physical systems. Models and Analysis
As expected, a large number of papers in the past year have been devoted to the modeling or simulation of various systems. There have been publications devoted to the use of input-output data to develop mathematical models (33A, 45A) and to various forms of state (not transform) domain models ( Z I A , 46A). One paper was concerned with model simplification ( Q A ) , and another was a pioneer effort in the reaction system area (47A). Simulation details for tubular reactors ( 6 A ) , batch distillation (76A), rubber plants (79A), refining networks (34A), styrene production (35A), ammonia synthesis (36A), heat exchangers (&A), production plants (47A, 42A), and reservoir behavior ( 4 8 A ) have been presented. The special cases of staged systems have also been analyzed ( 7 A , 25A, 37A) and, in particular, for diffusion cascadcs (ZZA), reverse osmosis (23A), and plate distillation columns ( 3 9 A ) . Hybrid simulation methods have been analyzed ( 5 4 7 3 A ) and thr cell model approach to systems has been detailed (7A, 8A, 20A). T h e analysis of the models or mathematical representations of reaction systems or chemical reactors has been detailed ( 1 7 4 49A). Such items as effectiveness factors (30A, 4 3 A ) , the influence of axial diffusivity (504 57A), and mixing phenomena ( 7 4 A ) in tubular reactors have been presented. T h e pioneering analysis of the Amundson-Ark-Luss group on various aspects of reaction and biochemical systems has continued in the past year (2A-4A, 72A, 26A-29A, 3 8 A ) . These papers should be required reading for all those interested in analysis of detailed and complex systems. Residence-lime and Probability
Publications continue in the area of residence-time or moment analysis as a vehicle for model building and in those areas which VOL. 6 1
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involve stochastic phenomena. As an illustration, residence-time analysis of turbulent bed contactors (5B), mixing models ( 7 B ) , coupled and multicomponent systems (98),two-phase flow in packed beds (7413),fluid beds (23B, 27B), spray towers (25B), packed beds (77B, 35B), adsorption in packed beds (34B), and selectivity systems (32%)have been presented, Further work on specifically reducing the model of a system by analyzing its moment distribution (IOB), using imperfect pu1sr.s in fluid beds (Z8R), and for nonlinear systems where superposition does not hold ( 7 B ) have also been reported. This approach is also of interest in dispersion phenomena in different systems (4B,6B, 73B). Stochastic effects have been noted in chemical reactor problems (ZSB), in distributions in catalysts (798, 37B), and in particulate processing (ZOB). The last paper is highly recommended reading for material in this area. Further work on unfolding a distribution into its parts ( Z B ) ,on analyzing a system with random outputs derived from a deterministic tracer input ( Z Z B ) , and on the concept of fuzzy events (36B)has been detailed.
Periodic System
Periodic systems have been of interest to many workers in the past year. Such systems can evolve from a variation in the parameters of the system (8C, 9C), from the structure of the system itself (ZC, 7C, 77C-73C, 77C),including a limit cycle system (3C, 6 C ) or a vibrating string (75C), and from a periodic forcing function. I n the latter case, work has been reported on feed concentration perturbations to tubular reactors (4C) and to a diffusion flame (5C), heat pulses for conduction studies in fluids (79C), and sinusoidal inputs to general systems (74C). An analysis of cyclic distillation control (22C)has also been reported.
Analytical Techniques
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 largest area of investigation deals with the solution of partial differential equations, PDE. Thus, we note the solution of wave flow (40), moving boundary diffusion problems (?OD), liquid distribution in packed beds ( 7 ID),concentration fields near electrodes (730),photoreaction systems (750, 3 6 0 ) , nonisothermal flow between rotating disks (760),liquid drop profiles (ZOD), adsorption in fixed beds (ZZD), Stokes flow equations ( 2 7 0 ) , diffusion and chemical reaction near an interface (370), gas-solid reactions ( 3 2 0 ) , ordinary and forced diffusion ( 3 5 0 ) , laminar convection ( 3 7 0 ) , multicomponent mass transfer (do), and oxygen adsorption by blood (431)). Other papers have dealt with an asymptotic representation for parabolic PDE’s ( 5 D ) , the question of Laplace’s equation and elliptic problems in general ( 6 0 , 2 5 0 ) , and of the chromatographic column ( 2 7 0 ) . This last paper has a superb analysis of the previous literature in the field. Other papers in this area have dealt with integral transforms and methods for parabolic equations ( 2 3 0 ) , shock studies (ZO), and reaction in porous catalysts (780), with a closure method for converting an infinite dimensional problem into an equivalent finite dimensional problem ( 3 0 ) , a n imbedding procedure for connecting local and global properties ( 2 8 0 ) , and maximum properties of O.D.E.’s ( 4 2 0 ) . Finally, there has been some interesting work using Laplace transforms for drying problems ( 7 0 ) , on generalized inverse transforms ( g D ) , Fourier transforms (770), on the application of generalized finite integral transforms ( 3 3 0 ) ,and the use of transform techniques for heat flow systems ( 2 4 0 ) .
Qplimization
The field of optimization continues to generate a vast flow of research papers which span the spectrum from the very theoretical to the practical. Work in the linear programming area continues (4E, 29E, 47E, 49E, 55E, 58E, 64E, 70E), with the main themes being extensions or reviews of the standard approach. Three books in the nonlinear programming area have appeared (70E, 27E, 69E). Other special methods, such as separable programming (ZOE), integer programming (24E, 67E, 67E, 69E), convex 44
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programming (40E), quadratic programming (45E), stochastic programming (76E),and geometric programming (%E,3523) have also rcceived attention. Papers dealing with programming with increasing constraints (S‘E), rnultiple or vector objective functions (74E), the usc of the Kuhn-Tucker conditions (28E), and with many aspects of game theory (37E, 43E, 44E, 53E) have also appeared. The Davidon method of optimization has been investigated in Hilbert space (37E) and for regression problcms (G3E), while an improved version of Rosenbrock’s method has been prrscnted and used for optimization of an existing reactor system (59E). Thc two-level method of optimization has received further attention (5E, 8E), as well as a means for determining a relative minimum without calculating derivatives (7E). The question of optimization of design calculations has also been analyzed in two most interesting and readable papers (75E, 4ZE). I n the applications area, some of the theoretical approaches have found extensive use. Thus, various authors have been concerned with production scheduling (79E, 4723)and with optimization of complete chemical processes ( I E , 38E, 39E), communication networks (48E), plant location (60E), separation-blending problems (54E), allocation of resources (6223) and natural gas transmission networks (66E). Further papers have considered convex allocation ( 3 E ) ,the optimization of large interconnected systems (77E), the optimization of chemical reactors with a delayed addition of reactant (33E), and the adiabatic reacior sequence with cold shot cooling (SE, 78E, 30E). Dynamic Programming
Dynamic programming continues to interest many research workers in the allocation, optimization, and control areas. I n particular, a number of authors have been concerned with the reduction of computer memory requirements (7F, 7F, 74F), with extensions and further developments of the theory (#F, 70F), and with stochastic problems (ZF, 8 F ) . Definitive applications to the Kronigsberg bridge problem ( 3 F ) , the linear-quadratic control problem (SF),water resource and capital allocations (6F, 9 F ) , hospital patient care ( 7 7 F ) , the shortest route problem (TZF), and some control systems (73F)have been detailed in the past year. Invarianf Imbedding
Invariant imbedding i? a variation of dynamic programming which has certain advantages over the more well-known Bellman technique. As such, it has found application for analyzing the asymptotic behavior of systems (1G),scattering processes (2G), calculating eigenvalues ( 3 G ) , and evaluating integral equations (5G). Further interest in its use in biological systems (4G),countercurrent dialyzers (GG), and optimization problems (76, 8 G ) have been reported. Stability
The publications dealing with stability of systems continues at a fast pace. Pioneer work comparing the adiabatic CSTK (an ODE system) with the catalyst particle (a PDE system) in all guises ( Z H ) has been presented, as well as work on stability of distributed systems which does not require steady-state information (5ZH). Other papers dealing with stability of distributed systems have been presented (4ZH, 43H, 64H, 74H, 75H). I n addition, an excellent new book (G6N) in this area has recently become available. Further papers of interest have analyzed stability in autorefrigerated reactors (57H), Marangoni phenomena in unsteady diffusion (32H), coupled reaction and diffusion in opcn systems (59H), solid-liquid interfaces (77H), an enclosed laminar flame (ZZH), stochastic equations (48H), the Kiccati equation (ZOH, 45H), and finite-difference equations (46H). Stability of various fluid devices has been analyzed (73H, 28H) and applied to viscosity stratification (74H, 47H), Poiseuille flow in pipes (78H, 37H), wave propagation in channels (30H),flow down an inclined plane (34H),fluid in a rotating pipe (60H),and a liquid jet (79H). Popov’s test for stability has been further analyzed ( 3 H ) and ap-
Leon Lapidus is Professor and Chairman of the Department of Chemical Engineering, Princeton University, Princeton, N . J . T h i s is the author’s second year to compile the Fundamentals Annual Reuiew on Applied Mathematics. AUTHOR
plied to an adiabatic packed bed reactor (37H). Zubov’s method also has been examined and extended in terms of a Lie series for global stability ( I H ) , while a stochastic form of Lurie’s approach has been developed (53H,55H, 56H). Stability via Liapunov’s method has also received attention during the past year. A review paper is available (68H),as well as one detailing methods of construction of Liapunov functions (35H). Szego’s method of construction has been analyzed (38H) and. the use of higher derivatives examined (8H). One paper has examined the Liapunov approach to population genetics (27H),while another has applied Krasovskii’s version to high dimensional reaction systems (4H). The application of a Liapunov function as a control algorithm has also been detailed (9H, 77H, 50H, 62H). Optimal Control
The field of optimal control is rapidly reaching the point where definitive textbooks are available. Thus, in the past year, a number of excellent texts (71, 731, 281, 321, 501, 621, 951) have been published. Three others are highly recommended for all workers in this area (91, 471, 691). Papers have been published dealing with time-optimal control (81),various types of delays (31, 141, 241, 271, 371, 651, 661, 981), singular control (41, 741,851),suboptimal control (21,171,541,581, 681, 921), and stochastic control (51, 461, 521, 771, 891, 971). Other papers have dealt with the synthesis of control (IOZ), the computational aspects of control (761, 271, 341, 371), control in Banach space (221),the connection of game theory to control (231), a survey of incorrect problems in control (361),a general theory of extremals (561),the inclusion of constraints (471, 571, 901), the use of singular perturbations (701), the application of the sweep method (721), the pursuit problem (801),the connection to programming techniques (201, 971), and the discrete maximum principle (291). There has been an excellent review article (191),consideration of a vector-type index (601), extensions to the linear-quadratic problem (251, 331, 862, 941), consideration of modal control which adjusts the system eigenvalues (871),work on control of growth processes (551), consideration of hospital in-patient admission (61), excellent discussions of the application of control theory to economic problems (151, 492), and application to batch polymerization reactors (351), reactors with catalyst decay (591),and tubular reactors (671). The sensitivity analysis of control systems has also been actively pursued during this past year. This work includes two superb summaries of work to date (451, 881),an adaptive sensitivity analysis (38Z), performance index sensitivity analysis (391, 441), plus others of interest in this area (261,641, 771, 751, 761, 821, 931). While most of the above references deal with the control of lumped-parameters systems, extensive work was also carried out on distributed-parameter systems. An excellent survey was presented ( 7 71), computational considerations were discussed (731), and integral constraints were analyzed (791). Other papers in this area covered such topics as the control of heated slabs ( 7 1 , 481), tubular reactors (121), and general systems (991-1071). Identification
This area of investigation spans the parameter-estimation problem in which a system model is partially specified and includes the black-box problem in which no a priori information about the model is known. I n all cases, experimental data are required to complete the specification of the problem. I n terms of parameter estimation, a number of papers have been published dealing with estimation of chemical rate constants (7.25, 33J), parameters in packed beds ( 2 7 J ) ,for plants (38J, 39J), recirculating systems (20.7, 27J), and to designing experiments to aid in the estimation (25J, 29J). Not only have ODE systems been examined in this sense, but also PDE systems (3J,5J,47J, 48J). General papers on identification have been reported (6J, 245), as well as a paper using Hermite funtions (55J), one involving residuals ( Q J ) one , using only normal operating inputs and outputs ( I O J ) , one discussing white noise corrupted output measurements ( 4 5 4 , and one deriving a feedback control law as a linear function of past observations (75). The question of observability of systems has been analyzed ( 7 5 , 74J, 30J, 37J, 44J), as well as the use of observers when state variables are only partially measurable (4J,35J, 41J). A specific learning algorithm for identification has been proposed ( 5 7 J )and an outstanding review of pattern classification algorithms pub-
lished (225). The latter paper is of decided interest since pattern recognition is, in many aspects, really identification. Optimal filtering as due to Kalman has also received considerable attention (25, 77J, 265, 32J, 435, 5.25, 53J, 60J). Here, we seek to estimate the states of a system in the presence of noise on the measurements. The connection to sensitivity analysis ( 7 8 J ) and to Monte Carlo techniques (795)has been presented, as well as an on-line filtered investigation of a tubular reactor with catalyst decay (16J). Numerical-Approximation
The past year has seen a number of interesting papers dealing with polynomial or alternate approximation of many functions. One book (77K) has presented actual computer programs and codes for evaluating Chebyshev series, continued fractions, and asymptotic series, while two papers ( I K , 3 K ) have been concerned with spline functions. Orthogonal polynomials have been analyzed (2K, 7 5 K ) and, in particular, Chebyshev approximations in all forms (8K-70K, 17K, 33K). An excellent book dealing with Chebyshev polynomials has also been published (12K). Chebyshev quadrature ( d K ) , as well as Gauss (7K, 7dK, 25K) and Clenshaw-Curtis quadratures (24K), has been investigated. The numerical evaluation of the convolution integral (32K),further remarks on Romberg-type integration (79K),and straight-line ( 2 6 K ) and exponential (31K) approximate algorithms have been detailed. I n addition, the Laguerre polynomial for smoothing and extrapolation (22K),and approximation for pure time delay (27K),and interval-interpolation for mass transfer problems (ZOK), and the evaluation of Volterra integral equations ( 6 K , 76K, 78K, 27K, 28K) have all been the subject of publications. Numerical-linear
Algebra
I n this area of investigation, a number of papers dealt either with the direct solution of a set of linear algebraic equations or with the calculation of an inverse coefficient matrix. I n particular, optimal scaling of the matrix elements (5L, 3015) and the influence of the pivot size ( 6 L ) were detailed. Further papers dealt in an equivalent sense with special tridiagonal and block matrices ( Z Z L ) , the use of conjugate gradient methods (34L), partitioning (38L), the conversion to Hessenberg form (37L), and the LR (32L), QL (33L), and QR (37L) algorithms. Another paper analyzed the special class of tridiagonal matrices which result from a finitedifference conversion of ODE (40L). The calculation of eigenvalues and eigenvectors was also of interest in the past year. Papers dealing with the calculation of these items for tridiagonal matrices (70L) and for general matrices (78L) have appeared, as well as papers dealing with mean eigenvalues (3L), maximum eigenvalues (73L, 74L), a separation theorem for eigenvalues (75L), the condition of a matrix to assure all negative real eigenvalues (QL),the use of iteration (42L) and inverse iteration (50L) to calculate eigenvalues, and an analysis of round-off error in such calculations (7L). Three papers have also proposed certain test matrices which can be used to check any computer routine which calculates eigenvalues (72L, 76L, 36L). The Riccati matrix equation has been the subject of a number of papers (ZOL,24L, 57L), as well as the Penrose generalized or pseudoinverse matrix (27L, 35L, 45L). I n addition, serial and parallel addition involving the pseudoinverse ( 2 L ) and a n analog method of calculation of this inverse matrix (2QL)have been presented. Other papers in the linear algebra area have dealt with round-off and stability considerations in evaluating a determinant (25L), evaluating the determinant of a pentadiagonal matrix (47L), the appearance and use of tridiagonal matrices in chemical engineering (23L)and in network synthesis (53L),the algebraic properties of certain special matrices ( I L ) ,commenting on tensor notation (77L, 79L, 47L), and calculating the eigenvalues of Fredholm integral equations (27L). Finally, we mention the use of projection methods for solving linear equations (4QL)and to generate artificial straight-line reaction paths in a Prater-Wei-type analysis
(44L). Numerical-Ordinary
Different Equations
Interest in the numerical solution of ODE’Scontinued during the past year. Excellent review articles were published which, as an illustration, compared the single- and multiple-step methods on VOL. 6 1
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different systems in terms of accuracy, stability, and computation times (ZM, 5 M , 7 7 M ) . These papers are outstanding in delineating the various features of the numerical methods. Stability features have been analyzed. Thus, one author described the use of Pade approximations to yield A-stable algorithms ( 7 M ) , another author developed a theorem for stability of predictor-corrector methods ( Q M ) another , developed a sufficient and almost-necessary condition for A-stable methods ( 7 3 M ) , and A-stable high-order methods were suggested ( 7 6 M ) . I t has also been shown that a suitable combination of the predicted and corrected values can be used to extend the region of stability ( 2 3 M )in predictor-corrector methods. Other papers have included the use of spline functions rather than polynomials ( 3 M ) , the use of multiple-step methods to solve Volterra integral-differential equations ( 74M), a n analysis of stiff equations ( 7 5 M ) , new details on implicit Runge-Kutta methods ( 2 4 M ) . and a means to accelerate converEence of single-step methods (25M). A hvbrid version of Nordsieck’s method (7OM) and a hybrid method for the case of a second-order O D E with no first derivative (6.U) have been proposed. These hybrid forms use special off-step points in the integration space to achieve higher accuracy. In addition, two further papers have proposed algorithms for the above special second-order ODE (72M,2 2 M ) . T h e integration of ODE’S in chemical reaction systems ( 7 M ,27iM) and in sequences of CSTR’s ( 4 M )have also been reported. I
Numerical-Parlial
Differential Equalions
T h e numerical solution of PDE’s has been the subject of a large number of papers in the past year. Considering purely parabolic PDE’s work dealing with the method of moments (IN), the explicit Saulyev stable algorithms ( 7 N , 4 5 N ) , various versions of the successive over-relaxation method (46N, 6 7’V, 7 8 N ) , generalized alternating direction methods (23’VV, 26!V, 27N, 37N, 3 9 N ) , various boundary conditions (5IAr, 751V), round-off error analysis ( 5 Q N ) , high order and stable procedures ( 2 N , 28N), and periodic systems ( 5 2 N )have all been reported. I n terms of elliptic PDE’s, invariant imbedding ( 5 N ) and dynamic programming ( 6 N ) have been proposed to solve this essentially boundary value problem. Further, convergence of difference schemes (9.\’),special boundary conditions ( 3 8 N ) ,successive line over-relaxation ( 7 9 N ) ,and accelerating a relaxation algorithm (441V) have been detailed. A conversion to boundary-value form for hyperbolic PDE’s has also been presented (3ON),as well as the use of multistep methods in two-space variables ( Z 8 N ) . I n more general terms, three new books in this area have been published (Z5N, 29N, 7 Q N ) ,the use of weighted residuals has been discussed (24iV), and analog and hybrid techniques have been outlined (77N, 48N, 4 Q N ) . Other work of interest has involved a preconditioning of matrices to achieve convergence in iterative methods ( Z l N ) ,finite diffcrencing for high-order derivatives ( S Z N ) , convergence of general matrices in a Jacobi iteration (65.V), construction and comparison of different difference schemes (7UN, 72N), use of parallel solution methods (841X’), convergence and stability of difference schemes (341V, 47X), and iterative solution of large problems (68N, SZiV). I n an applications sense, PDE’s involving cold front motion (3N), transport of turbulence ( 4 N ) , supersonic two-dimensional gas flow (SlV), the Navier-Stokes equation ( 7 I N , 72N), ion exchange bed problems ( 7 4 N ) , a global ocean model ( 7 5 N ) , neutron transport (76iV), diffusion in pulsating flow ( Z Z N ) , almost compressible flow ( 3 2 N ) , transient flow in porous media (36N), stochastic control systems ( 4 3 N ) ,Stokes flow (SON),laminar flow over spheres ( 5 7 N ) , two-dimensional elastic flow ( 5 6 N ) , parametric pumping ( 5 7 N , 64iXr, 7 3 N ) , diffusion-convection problems (58h‘),thin plate heat transfer ( 6 7 N ) , dissolution of solid cylinders by a laminar flowing fluid ( 6 9 N ) ,freezing of a saturated solution ( 7 4 N ) ,countercurrent heat exchange (7G1V, SON), inviscid compressible flow (77N), transient shock waves (87iV), and reverse osmosis ( 8 3 N ) have all been investigated.
Numerical-Boundary
Value Problems
Two-point boundary value problems occur in many fields of scientific and engineering interest. A new book covers both the theory and appIications in a superb fashion (740),while a second book presents many interesting cxamples (50). A series of papers
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on nonlinear boundary value problems has been presented which emphasize Galerkin’s method (60-80), and an equivalent method has been applied to convection (700) and reactor design problems (720). Other papers have proposed a globally convergent shooting method ( I O ) , analyzed the use of contraction mapping (ZOO), detailed the use of invariant imbedding ( 2 0 )and quasilinearization ( l Q O ) , and suggested the use of cubic spline functions (30). Further work has analyzed the integration of inherently unstable boundary-value problems (QO), the use of Banach space (730),asked questions on convergence ( 7 7 0 , 780) and instability of discretization algorithms in such problems (250), and discussed the problems in reaction systems (260) and boundary layer equations ( 7 7 0 ) . An initial adjoint boundary value search method in optimal control problems has been described ( 7 5 0 ) , as well as in general adjoint systems ( 2 7 0 ) . Numerical-Data
Analysis
Data and statistical analysis continues to interest many workers. New books on evolutionary operation ( 4 P ) and spectral analysis ( 73P)are recommended. Least-squares and regression procedures have been applied in many situations. These include a special data form ( 6 P ) , a comparison with the method of averaging ( Q P ) , a two-dimensional analysis ( 7 8 P ) ,and a superb paper on the whole question of least-squares (ZOP). Applications for the Schrodinger equation (ZP), a crystallization process (74P), and chemical reaction orders (7QP) have been reported. T h e stepwise regression method ( 5 P ) , design of experiments for regression ( 7 5 P ) , pitfalls of regression (23P), and the use of the Davidon algorithm (24P) are also of interest. Other papers have indicaied methods of transforming experimental data (3P, 7OP),of fitting non-Newtonian viscosity data ( S P ) , and they have presented case histories of the statistical analysis of real complex systems (12P, 76P). Numerical-Root
Location
A number of papers have discussed various versions of Newton’s method for solving nonlinear algebraic equations ( 4 4 , 74,SQ). Another papcr showed how to obtain an optimal initial approximation for Newton’s mpthod of evaluating a square root ( 5 Q ) , and another square root algorithm was proposed ( Q Q ) . Other topics in this area of investigation included a general method for solving nonlinear algebraic equations ( 3 Q ) , another which operated in Banach space ( I Q ) , and a detailed investigation of parallel-type algorithms (GQ).
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(22) Mastascusa E. J., “A Fast Algorithm for Computing the Response of a Linear System’to a Large Number of Inputs,” Proc. I E E E , 57, 338 (1969). (23) Miller, K . S., “Linear Difference Equations,” W. A. Benjamin, Inc., 1968. (24) Morse, S. P., “Concepts of Computer Use in Contour M a p Processing,” Comm. ACM, 12, 147 (1969). (25) Payne, W. H., Rabung, J.,? and Bogyo T P “Coding the Lehmer PseudoRandom Number Generator, &mm. ACM,’12; 8; (1969). (26) Reiter, R., “Scheduling Parallel Computations,” J A C M , 15, 590 (1968). (27) Rosen, S., “Electronic Computers: A Historical Survey,” Comp. Rev., 1 , 7 (1969). (28) Sakai T. and Nagao M., “Simulation of Traffic Flows in a Network,” Comm. AbM,’12, 311 (19d9). (29) Swalen J. D., “Laboratory Automation Papers,” I B M J . Res. Develop., 13 (Jan. lb691. (30) Zaremba S K . “The Mathematical Basis of Monte Carlo and Quasi-Monte Carlo MethAds;” SiAM Rev., 10, 303 (1968). (31) Zemanian A H “The Postulational Foundations of Linear Systems,” J M A A , 24,4b9 ii968’j.
MODELS AND ANALYSIS (1A) AhI?walia, M . S., and Levenspiel, D., “Reaction Between Two Immiscible Fluids in Countercurrent Flow in a Series of Stirred Tanks,’’ Can. J. Chem. Eng., 46, 443 (1968). (2A) Amundson, N. R., and Luss, D., “Qualitative and Quantitative Observations on the Tubular Reactor,” ibid., p 424. (3A) Aris, R., “A Note on Mechanism and Memory in the Kinetics of Biochemical Reactions,” Math. Biosci., 3, 421 (1968). (4A) Aris, R., “Sufficient Conditions for the Uniqueness of the Steady State,” Chem. Eng. Sci., 23, 1501 (1968). (5A) Bekey, George, “Hybrid Computation,” Wiley, 1968. (6A) Binns, D. T., Kantyka, T. A., and Welland, R . C., “Design of a Tubular Reactor with Optimum Temperature Profile,” Trans. Inst. Chem. Eng., 47, T53 (1 969). (7A) Buffham, B. A., and Gibilaro, L. G., “The Analytical Solution of t$e DeansLevich Model for Dispersion in Porous Media,” Chem. Eng. Sci.,23, 1399 (1968). (8A) Buffam, B. A., Gibilaro, L. G., and Kropholler, H. W., “Network Combing of Complex Flow-Mixing Models,” ibid., 24, 7 (1969). (9A) , C h y C F and Shieh L. S “A Novel Approach to Linear Model Simplification, h.’J.’bontrol, 8, 5 i l (1469). (10A) Coggan G . C. and Bourne J. R “The Design of Gas Absorbers with Heat General P;ograi for Adiabatic Plate Absorbers,” Trans. Eflects: Pakt I, I n s t . Chem. Eng., 47, T96 (1969). (11A) Crum,p K. S., and Mode, C. J., “A General Age-Dependent Branching Process. I, J M A A , 24, 494 (1968). (12A) Drott, D. W., and Aris, R:, “Communications on the Theory of Diffusion ’ and Reaction. I,” Chem. Eng. Sci., 24, 541 (1969). (13A) Eteson, D. C., and Zwiebel, I. “Hybrid Computer Solution of the Simple Fixed Bed Absorption Model,” A.I.6h.E. J., 15, 124 (1969). (14A) Felder, R. M., and Hill F. B., “Mixing Effects in Chemical Reactors. I,” Chem. Eng. Sct., 24, 385 (19695. (15A) Fromm, J. E “A Method for Reducing Dispersion in Convective Difference Schemes,” J . Corn&. Phys., 3, 176 (1968). (16A) Goldman, M . R., and Robinson E. R “The Corn uter Simulation of Batch Distillation Processes,” Brit. Chem. Ekg., 13,’ 1713 (19687. (17A) Hlavacek V Marek M and Kubicek, M., “Modelling of Chemical Reactors. x,’’Chkm.’hng. 23;’1083 (1968). (18A) Jarvelainen, Martti, and Norden, H. V., “A Theoretical Study of Filter Cake Washing,” B I T , 8, 295 (1968). (19A) Johnson A. I Aizawa M and Petryschuk W. F., “Simulation of a Synthetic Rubber Piknt,” Bri;. Chi&. Eng., 13, 1432 (i968). (20A) Kafarov, ;Y. V “Mathematical Analysis of Cell Model with Backmixing between Cells, Theb;.. Found. Chem. Eng., 2, 59 (1968). (21A) Kesavan, H . K ; , Sarma I. G and Prasad U. R., “Sensitivity-State Models for Linear Systems, Int. J. bontroi,’9, 291 (1964). (22A) ,Khqlpanov, L. P., Zhavoronkov, N. M., and Malyusov, V. A., “Calculation ofDiffusion Cascades,” Theor. Found. Chem. Eng., 2,471 (1968). (23A) Kimura, S., Sourirajan S and Ohya H., “Stagewise Reverse Osmosis ~ E S ~ CDEVELOP., N 8, 79 (1969). Process Design,” IND.ENG. HE;. PROCESS (24A) Kuchanov S. I Pis’men L. M and Levich V. G “Thermal Conductivity of a Granular Bkd in’steady donditi&” Theor. kound. b c m . Eng., 2, 485 (1968). (25A) Lebedev, Y . N., “Combined Hydrodynamic Models ofFractionating Column Trays Under Crossflow Conditions,” ibid., p 155. (26A) Luss D . “Sufficient Conditions for Uni ueness of the Stead State Solution in Distributkd Garameter Systems,” Chem. Eng.%ci., 23, 1249 (19687. (27A) Luss, D., “Bounds on the Effectiveness Factors for Exothermic Catalytic Reaction,” A.I.Ch.E. J.,14, 966 (1968). (28A) Luss D “ O n the Uniqueness of a Large Distributed Parameter System with ChemicaiRd)action and Heat and Mass Diffusion,” Chem. Eng. Sci., 24,879 (1969). (29A) Luss D and Amundson N R “Maximum Temperature Rise in GasSolid Re&&,” A.I.Ch.E. J.,’15,‘194‘ (1969). (30A) McGreavy, C., and Cresswell, D. L., “Non-Isothermal Effectiveness Factors,” Chem. Eng. Sci., 24, 608 (1969). (31A) Mecklenburg J. C., and Hartland S “Two Phase Countercurrent Extraction with High Bafkmixing, zbzd., 23, 1621’?1968). (32A) Murakami, Y., Kobayashi, T., Hattori, T., and Masuda, M., “Effect of Intra article Diffusion on Catalyst Fouling,” IND.END. CHEM.,FUNDAMENTALS, 7, 59f (1968). (33A) Murrill, P. W., Pike R W., and Smith, C. L., “Dynamic Mathematical Models” Chem Eni. 117’(Se‘ 1968). 177 (Oct 1968). 165 (Nov. 1968)‘ 103 (Dec. 1668); i67 t i a n . 19697; 105 (Feb. 1969); 111 (Mar. 1969); 151 IApr. 1969); and 195 (May 1969). (34A) Petryschuk W. F and Johnson A. I “The Mathematical R e resentation of a Light Hydrbcarbog Refining Netkork,;: Can. J . Chem. Eng., 46, 383 (1968). (35A) Pliskin L. G and Rzaev T. G “Mathematical Model and Algorithm for Optimal CAntrol gf Styrene P;oduct?on,” Auto. Remot: Control, 11, 1864 (1968). (36A) Porobszky, I., Simonyi, E., and Ladanyi, G., “An Investigation of Mathematical Models Dcscribing Ammonia Synthesis Reactors,” Brit. Cham. Eng., 14, 495 (1969). (37A), Raskin, A. Y., “Mathematical Model and Calculation Algorithm for Radial Adiabatic Reactor.3,” Theor. Found. Chem. Eng., 2, 186 (1968). (38A) Rester, S., and Aris R “Communications on the Theory of Diffusion and Reaction. 11,” Chem. En;. Si:., 24, 793 (1969).
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(39A) Sargent, R. W. H and Murtagh B. A. “The Design of Plate Distillation Columns for MulticomGonent Mixture:,” Trans. Inst. Chem. Eng., 47, T85 (1969). (40A) Schmidt J. R and Clark D. R “Analog Simulation Techniques for Modeling Parallel-Fiow Heat Exchanger:,” Simulation, 12, 15 (1969). (41A) Shigin, E. K “Classification of Dynamic Models of Controlled Objects in Chemical Enginediing Processes. 11,” Auto. Remole Control, 7, 1152 (1968). (42A) Shigin E. K “Classification of Dynamic Models of Chemical Production Plants. 111:” ibid.: 10, 1703 (1968). (43A) Stewart, W. E., and Villadsen, J. V., “Graphical Calculation of Multiple Steady States and Effectiveness Factors for Porous Catalysts,” A2.Ch.E. J., 15, 28 (1969). (44A) Stewart, R. R., and Bruley, D. F., “Process Dynamics of Distributed Parameter Counterflow Svstems in Laminar Flow.” tbid.. D 220. (45A) Todes, 0. M., “Several Ways to Determine Mixing Coefficients from Response Curves,” Theor. Found. Chem. Eng., 2, 89 (1968). (46A) Wang, P. K . C., “A Method for Approximating Dynamical Processes by Finite-State Systems,” Int. J . Control, 8, 285 (1968). (47A) Wei, J., and Kuo, J. C. W., “A Lumping Analysis in Monomolerular Reaction Systems,” INn. ENG.CHEM.,FUNDAMENTALS, 8, 114, 124 (1969). (48A) Weinaug C F., “Mathematical Modeling of Reservoir Behavior,” Simulation, 12, 179 (i969). (49A) Wen, C. Y “Noncatal tic Heterogeneous Solid Fluid Reaction Models,” IND. ENC.CHFM::ti0 (9), 34 6968). (50A) Wissler, E. H., “On the Applicability of the Taylor-Aris Axial Diffusion Model to Tubular Reactor Calculations,” Chem. Eng. Sci., 24, 527 (1969). (51A) Wissler, E. H., “On the Asymptotic Behavior of a Tubular Reactor in t h e Limit of Small Axid Diffusivity,’ ibid., p 829. I
RESIDENCE-TIME A N D PROBABILITY (1B) Akita, S., Shibata, S., Nishimura, Y., and Matsubara M “Modellin a Non, 433 (7969). linear Process with Asymmetrical Dynamics,” Chem. En;. Si‘;.24, (2B) Baker, G. A., Jr., “A Certain Unfolding Problem,” J . Comput. Phys., 3, 486 (1969). (3B) Bayens, C. A., and Laurence, R . L., “A Model for Mass Transfer in a Coalescing Dispersion,” IND. END.CHEM. FUNDAMENTALS, 8, 71 (1 969). (4B) Bischoff, K. B “Accuracy of the Axial Dispersion Model for Chemical Reactors,” A.I.Ch.2. J.,14, 820 (1968). (5B) Chen B. H and Douglas W. J. M “Axial Mixing of Liquid in a TurbulentBed Coitacto;,” Can. J . Chek. Eng., 4?,113 (1969). (6B) Dayan, J and Levenspiel 0 “Longitudinal Dispersion in Packed Beds of Porous Adsogbing Solids,” C h h . 2.g. Sci., 23, 1327 (1968). (7B). Fan, L. T., Chen, M . S. K Ahn Y K and Wen C Y “Mixing Models with Varying Stage Size,” Can.”J. Chkm.’Eni., 47, 141 t19i9):’ (8B) Fuller, A. T., “Analysis of Nonlinear Stochastic Systems by Means of the Fokker-Planck Equation,” Int. J . Control, 9, 603 (1969). (9B) Gal-Or, B., and Padmanabhan, L., “Coupled Energy and Multicomponent Mass Transfer in Dispersions and Suspensions with Residence Time and Size Distributions,” A.I.Ch.E. J., 14, 709 (1968). (10B) Gibilaro, L. G., and Lees, F . P., “The Reduction of Complex Transfer Function Models to Simple Models Using the Method of Moments,” Chem. Eng. Sci., 24, 85 (1969). (1lB) Grethlein, H., “Exact Weight Fraction Distribution in Linear Condensation 8, 206 (1969). Polymerization,” INn. ENG.CHEM.FUNDAMENTALS, (12B) Gwyn, J. E., “JZjgital Resolution of Residence Time Distributions from Pulse Response Data, A.I.Ch.E. J., 15, 126 (1969). (13B) Hassinger, R . C., and Von Rosenberg, D. D., “A Mathematical and Experimental Examination of Transverse Dispersion Coefficients,” Soc. Petrol. Eng. J., 8, 195 (1968). (14B) Hochman, J. M., and Effron, E., “Two-Phase Cocurrent Downflow i n Packed Beds,” IND. ENCI.CHEM.FUNDAMENTALS, 8, 63 (1969). (15B) Horn G., and Atherton, A., “Statistical Techniques for Evaluatin Experimental Performance Results on a Batch of Heat Exchanger Elements,” &am. Inst. Chem.Eng.,47,T43 (1969). (16B) Hulburt H. M., and Akiyama, T., “Liouviile Equations for Agglomeration and Dispersion Processes,” INn. ENG.CHEM. FUNDAMENTALS, 8,319 (1969). (17B) Jefferson, C. P., A Further Note on “Dynamics of Packed Beds with Interphase Heat or Mass Transfer,” Chem. Eng. Sci., 24, 613 (1969). (18B) Kafarov, V. V., “Statistical Estimates of the Parameters in Mathematical Models Describing the Hydrodynamic Structure of Flows,” Theor. Found. Chem. Eng., 2, 228 (1968). (19B) Kapur, P. C., and Fuerstenau, D. W., “A Coalescence Model for Granulation,’’ IND. ENO.CHEM.PROCESS DESIGN DEVELOP., 8, 56 (1969). (20B) Katz, S., and Shinnar, R “Particulate Methods in Probability Systems,” IND. END. CHEM.,61 (4) 60 ((969). (21B) Kazakov, I. E., and Slovtsov, N. I., “Application of the Method of Moments to a n Investigation of the Accuracy of Dynamic Systems that are Subjected to Random Perturbations in the Form of Polynomials with Random Parameters,” Auto. Remote Control, 11, 1748 (1968). (22B) King, R . P “Continuous Flow Systems with Stochastic Transfer Functions,” Chem. Eng. Sci.,’i3, 1035 (1968). (23B) Kunii, Daizo, “Fluidization Engineering,” Wiley, 1968. (24B) Larson, V. H., “Some Differential Equations of Probability Theory i n Stochastic Processes and Control Systems,” Int. J. Control, 9, 709 (1969). (25B) Letan, R., and Kehat, E., “Residence Time Distribution of the Dispersed Phase in a Spray Column,” A.I.Ch.E. J.,15, 4 (1969). (26B) Morse, P. M., and Elston, C., “A Probabilistic Model for Obsolescence,” Operations Res., 17, 36 (1969). (27B) Nauman E B and Collin e, C. N., “The Theory of Contact Time Distrihutions in Gas Pluihizkd Beds,” Cfem. Eng. Sci., 23, 1309 (1968). (28B) Ostergaard, K., and Michelsen, M . L., “On the Use of the Imperfect Tracer Pulse Method for Determination of Hold-Up and Axial Mixing,” Can. J. Chem. Eng., 47, 107 (1969). (29B) Pell, T. M., and Aris, R., “Some Problems in Chemical Reactor Analysis with Stochastic Features,” IND.ENC.CHEM.FUNDAMENTALS, 8,339 (1969). (30B) Randolph A. D “Effect of Crystal Breakage on Crystal Size Distribution in a Mixed Suspdnsion brystallizer,” ibid., p 58. (31B) Rony, P. R., “Supported Liquid-Phase Catalysts,” Chem. Eng. Sci., 23, 1021 (1968). (32B) Rosas, C. B. “Axial Mixing and Selectivity,” IND. ENG. CHEM.FUNDAMENTALS, 8, 361 (i969). (33B) Sampson, R. E., and Springer, G. S., “Condensation on and Evaporation from Droplets by a Moment Method,” J. Fluid Mech. 36, 577 (1969).
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{34B) Schneider P. and Smith J. hl “Adsorption Rate Constants from Chromatography,” )A.I.’Ch.E. J., 14,’762 (ib68). (35B) van den Bleek, C. M., van der Wiele, K and van den Berg, P. J., “The Effect of Dilution on the Degree of Conversionk Fixed Bed Catalytic Reactors,” Chem. Eng. Sci., 24, 681 (1969). (36B) Zadeh, L. A., “Probability Measures of Fuzzy Events,” J M A A , 23, 421 (1968). PERIODIC SYSTEMS (1C) Anderson, C. H., “Asymptotic Oscillation Results for Solutions to FirstOrder Nonlinear Differential-Difference Equations of Advanced Type,” J M A A , 24, 430 (1968). (2C) Baile P. B and Shampine L F “Concerning Periodic Solutions of Y” 4F(T, Y , = 6,” ibid., 23, 558 ili68j: (3C) Bypokes, B. E., and Johnson, R. A., “Limit Cycles in Time Delay Relay Systems, Znt. J . Control, 9, 387 (1969). (4C) Chany; K S and Bankoff S. G “Oscillatory Operation of Jacketed Tubular ~ I N D & N ~ A L S , 7, 633 (1968). Reactors, I N ’ D . ’ ~ NCHEW. G. (5C) Clarke, J;,F. and Stegen G R “Some Unsteady Motions of a Diffusion Flame Sheet, J.’FIuid Mech., 54,’343”(1969). Y 0,” J M A A , (6C) Comstock, C., “ O n the Limit Cycles of Y” - U Sin Y’ 26, 128 (1969). (7C) Dasarathy, B. V., and Srinivasan, P., “ O n the Synthesis of Self-Oscillatory Systems,” Znt. J . Control, 8, 97 (1968). (8C) Dasarathy, B. V., and Srinivasan, P., “ O n a Class of Time-Dependent Systems with Periodic Solutions,” ibid., p 205. (9C) Dasarathy B. V and Srinivasan, P., “A New Approach to the Study of Systems with Perjodicaiiy Varying Parameters,” ibid., p 265. (IOC) Datko R . “A Fixed-point Theorem for a Class of Integral Operators,” J M A A , 23: 704 (1968). (11C) Diliberto, S. P., “Scale Factors for Periodic Surfaces,” SZAM J . Appl. Math., 16, 1119 (1968). (12C) Farlow, S. J., “An Existence Theorem for Periodic Solutions of a Parabolic Boundary Value Problem of the Second Kind,” ibid., p 1223. (13C) Grimmer R C “As mptotically Almost Periodic Solutions of Differential Equations,” idid.,’17;’109 4969). (14C) James, D. J. G., ‘LStabilityAnalysis of a Model Reference Adaptive Control System with Sinusoidal Inputs,” Znt. J . Control, 9, 311 (1969). (15C) Johnson H. L. “The Existence o r a Periodic Solution o f a Vibrating Hanging String,” S I A h J . &Ld. Moth., 16, 1048 (1968). (16C) Kartsatos, A. G. “ O n Oscillations of Nonlinear Equations of Second Order,” J M A A , 24, 665 (1nst. Ma& Appl., 4,276 (1968). (17H) Datko, R “An Extension of a Theorem of A. M . Lyapunov to Semi-Groups ofOperators,”‘>MAA, 24,290 (1968). (18H) Dave A and Drazin P. G., “The Stability of Poiseuille Flow in a Pipe,” J . Fluid dch.,”36, 209 (196b). (19H) Davison, E. J., “Some Sufficient Conditions for the Stability of a Linear Time-Varying System,” Int. J . Control, 8, 377 (1968). (20H) Davison, E. J., and Man, F. T. “The Numerical Solution of A‘Q QA -C,” ZEEE Auto. Control, 13, 448 (1668). (21H) Davidson, E. J., and Cowan, K. C., “A Computational Method for Determining the Stability Region of a Second-Order Nonlinear Autonomous System,” Znt. J . Control, 9, 349 (1969). (22H) Decker, B. E. L., and Scrinivas, R. A., “An Investigation into the Stability of Enclosed Laminar Diffusion Flames,” Can. J . Chem. Eng., 46, 412 (1968). (23H) Desoer, C. A., and Wu M. Y.,“Stability of Multiple-Loop Feedback Linear Time-Invariant Systems,” jMAA, 23, 121 (1968). (24H) Dmitriev, Y.A. “Absolute Stability of Sampled-Data Control Systems with a Single Nonlinearit;,” Auto. Remote Control, 8, 1242 (1968). (25H) Duffin, R. J., “Stability of Systems with Nonlinear Damping,” J M A A , 23, 428 (1968). (26H) Eskin, V. I., “Local Parametric Synthesis of Optimal Control i n Complex Systems,’’Aulo. Remole Control, 8, 1211 (1968).
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(27H) Falk, H . , and Falk, C. T., “Stability of Solutions to Certain Nonlinear Difference Equations of Population Genetics,” BiometricJ, 25, 27 (1969). (28H) Frinberg, M . R.. and Schowalter, W.R., “Sufficient Conditions for Stability of Fluid Motions Constitutively Described by the Infinitesimal Theory of Viscoelasticity,” IND. ENO.CHEM. FUNDAMENTALS, 8, 332 (1969). (29H) Fink 4. bf and Seifert, G., “Liapunov Functions and Almost Periodic Solutionskdr Alm’Ast Periodic Systems,” J . D@. Equations, 5 , 307 (1969). (30H) Gradowczyk, M . H “Wave Propagation and Boundary Instability in Erodible-Bed Channels,” 3. Fluid Mech., 33, 93 (1968). (31H) Grosch, C. E., and Salwen. H.? “The Stability of Steady and Time-Dependent Plane Poiseuille Flow,” ibid., 34, 177 (1968). (32H) Gross, B., and Hixson, A. iX. “Marangoni Instability with Unsteady Diffusion in the Undisturbed State,” I ~ DEND. . CHEM. FUNDAMENTALS, 8,288 (1969). (33H) Gulchuk G . G “Frequency Conditions for Absolute Stability of Automatic Control Systeks for”Certain Classes of Nonlinearities,” Auto. Remote Control, 9, 1529 (1968). (34H) Gupta, A. S., and Rai, L., “Note on the Stability of a Viscoelastic Liquid Film Flowing Down an Inclined Plane,” J.Fluid Mech., 33, 87 (1969). (35H) Gurel, O.,,and,,Lapidus L “A Guide to Methods for the Generation of Liapunov Functions, IND.E&. ?HEM., 61 (3), 30 (1969). (36H) Hale, J. L., “Dynamic Systems and Stability,” J M A A , 26, 39 (1969). (37H) Ha,, C. D., “Stability, Analysis of Adiabatic Packed-Bed Reactors Via Popov s Frequency Method, INn. END.CHEM.FUNDAMENTALS, 8, 16 (1969). (38H) Hewit, J. R., and Storey, C., “Numerical Application of Szego’s Method for Constructing Liapunov Functions,” I E E E Trans. Auto. Control, 14, 106 (1969). (39H) Hill,y, J., and Keenan, R . K., “Stability of Finite-Width Sampled-Data Systems, Int. J. Control., 8, 1 (1968). (40H) Hirt C. W. “Heuristic Stability Theory for Finitr-Difference Equations,” J. Compu;. Phys.,’2, 339 (1968). “Role of Viscosity Stratification in the Stability of Two-Layer (41H) Kao, T. W., Flow Down an Incline,” J . Fluid Mech., 33, 561 (1968). (42H) Kastenberg W.E. “ O n the Stability of Nonlinear Space-Dependent Reactor Kinetics,” k u c l . &;. Eng., 31, 67 (1968). (43H) Kastenberg W.E and Chambre, P. L., “ O n the Asymptotic Stability of a Particular Class’ of Di’sturbed Parameter Feedback Control Systems,” Int. J . Control, 8, 339 (1968). (44H) Knowles, C. P., and Gebhart, B. “The Stability of the Laminar Natural Convection Boundary Layer,” J . Fluid‘ Mech., 34, 657 (1969). (45H) Kreindler, E., “ O n the Linear Optimal Servo Problem,” I n l . J . Control, 9, 465 (1969). (46H) Kreiss H . “Stability Theor for Difference Approximations of Mixed Initial Bouhdar; Value Problems. !,” MaZh. Cornp., 22, 703 (1969). (47H) Kuntsevich V. M., “Investigating the Stability of a Multivariable Control System with Pulbe-Frequency hlodulation Using the Direct Lyapunov Method,” Auto. Remote Control, 9, 1424 (1968). (48H) Kushner, H. J., “ O n the Stability of Processes Defined by Stochastic Difference-DifferentialEquations,” J . Di’. Equations, 4, 424 (1968). (49H) Lakshmikantham V. Leela, S., and Tsokos, C. P., “Stability of Controlled hlotion,” J M A A , 26, i 9 6 ’(1969). (SOH) Loo, S. G . “Stability of Linear Stationary Systems with Time Delay,” Int. J . Control, 9,’103 (1969). (51H) L u ) - p n W L “Effect of Imperfect Mixing on Autorefrigerated Reactor Stability, A:I.Ci.E:’J., 14, 880 (1968). (52H) Luss D and Lee. J. C M. “ O n Global Stability in Distributed Parameter Systems,”C/&. Eng. Sci., 23,’ l 2 j 7 (1968). (53H), Mahalanabis. A. K., and Purkayastha, S . “ O n the Stability of Stochastic Lurie Type Systems,” Int. J.Control, 8, 365 (19Gk). (54H) Michel, A. N,, and Wu, S. H., “Stability of Discrete Systems Over a Finite Interval ofTime,” tbid., 9, 679 (1969). (5 j H ) Morozan, T., “Stability ofStochastic Discrete Systems,” J M A A , 2 3 , l (1 968). (54H) Morozan, T., “Stability of Differential Systems with Random Parameters,” tbtd., 24, 669 (1968). (57H) Narendra, K . S., and Cho Y - S “Stability Analysis of Nonlinear and TimeVarying Discrete Systems,” SILM J:’Control, 6 , 625 (1968). (58H) Noonburg, V. W., “Roots of a Transcendental Equation Associated with a System of Differential-Difference Equations,” S I A M J . Appl. Moth., 17, 198 (1969). (59H) Othmer, H . G., and Scriven L. E. “Interactions of Reactions and Diffusion in Open Systems,” I N D .ENG.Cn& F U ~ D A M E N T 8, A L302 S , (1969). ( O H ) Pedley, T. J. “ O n the Instability of Viscous Flow in a Rapidly Rotating Pipe,” J . Fluid .A4eLh., 35, 97 (1969). (61H) Pis’men L. M.. “Existence and Stability of S. S. Regimes of,an Adiabatic Reactor with’an External Heat Exchanger and Cold Feed Bypass, Theor. Found. Chem. Eng., 2, 54 (1968). (62H) Raei3W G . “Stability Criteria for Control Systems with One Nonlinear Element, In;. J.’CoControl: 8, 659 (1968). (63H) Rosenblat S. “Centrifugal Instability of Time-Dependent Flows (Part 11,’’ J . Fluid Mech.,’33,’321 (1968). (64H) Sattinger, D. H., “Stability of Nonlinear Parabolic Systems,” J M A A , 24, 241 (1968). (65H) Siddigee, M . W. “Transient Stability of an A.C. Generator by Lyapunov’s Control, 8, 131 (1968). Direct Method,” Int. (66H) Siljak, D., “Nonlinear P t e m s : The Parameter Analysis and Design,” Wiley, 1968. (67H) Smith D. R and Conwa E. D., “An Atypical Example of Stability and Instability,:’ J M h , 26, 529 (18k9). (68H) Storev C. “Liapunov Methods in Chemical Engineering,” Brit. Chem. Eng., 13, i s 8 5 (1968). (69H) Tait K. E. “Stability Properties of Discrete-Continuous Feedback Control Systems 08 (1969). (211) Fogarty, L. E., and €Iowe, R . M., “Trajectory Optimization by a Direct Descent Process.” Simulution, 11. 145 (1968). (221) Friedman, A. “Optimal Control in Banach Space with Fixed End-Points,” J M A A , 24, 161 (i968). (231) Friedman, A,, “Differential Games of Pursuit in Banach Space,” ibid., 25, 93 (1969). (241) Fuller A. T. “Optimal Nonlinear Control of Systems with Pure Delay,” Inl. J , Conirol, 8, (45 (1968). (251) Gill A. and Sivan. R., “Optimal Control of Linear Systems with Quadratic Costs \$hick are Not Necessarily Positive Definite,” I E d E Trons. Auto. Control, 14, 83 (1969). (261) Griffin, R . E., and Sage 4. P. “Sensitivity Analysis of Fixed Point Linear Smoothing Algorithms,” Int.’ j. C o h a l , 8, 321 (1968). (271) Guardabassi, G., and Rinaldi, S., “Optimization of Discrete Systems with Delayed Control Variablrs, a Structural Approach,” Auto. Remote Control, 7,1063 I~.,-., lOhX).
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(27K) Reeve, P. J., “A Method of Approximating to Pure Time Delay,” Int. J. Control, 8, 53 (1968). (28K) Schmaedeke, 1%’. W.,“Approximate Solutions for Volterra Integral Equations of the First Kind,” J M A A , 23, 604 (1968). (29K) Schwartz, C., “Variational Principles for Integrals,” J . Cornput. Phys., 3, 512 (1969). (30K) ,She,t,mdeh, N. M., “ O n the Distribution of the Coefficients of Some Polynomials, SI.4M J . Appl. M a t h . , 16, 958 (1968). (31K) Thorp, J. S., and Lewine, R. N., “Exponential Approximation with Piecewise Linear Error Criteria,” J . Franklin Inst., 286, 308 (1968). (32K) Trauboth. H. H., “Recursive Formulas for the Evaluation of the Convolution Integral, J . ACM, 16, 63 (1969). (33K) Wittmeter-Kock, L., “A Method of Descent for Chebyshev Approximation,” B I T , 8, 328 (1968). NUMERICAL-LINEAR
ALGEBRA
(1L) Anderson, B. D. O., and Moore, J. B., “Algebraic Structure of Generalized Positive Real hlatrices,” S I A M J . Co~trol,6, 615 (1968). (2L) Ander,s,on, W. N., Jr., and Duffin, R. J., “Series and Parallel Addition of Matrices, J M A A , 26, 576 (1969). (3L) Banks. D . O., “Generalized Means of Eigenvalues,” ibid., 23, 409 (1968). (4L) Breuer, M . A., “Combinatorial Equivalence of (0, 1) Circulant Matrices,” J . Cornp. System Sci., 3, 8 (1969). (5L) Businger, P. A., “Matrices Which Can Be Optimally Scaled,” Numer. Math., 12, 346 (1968). (6L) Cryer, C. W., “Pivot Size in Gaussian Elimination,” ibid.: p 335. (7L) Delves, L. M., “Round-Off Errors in Variational Calculations,” J . Cumput. Phyr., 3, 17 (1968). (8L) Eisenfeld, J., “Quadratic Eigenvalue Problems,” J M A A , 23, 58 (1968). (9L) Fuller, A. T., “Conditions for a Matrix to Have Only Characteristic Roots with Negative Real Parts,” ibid.,p 71. (101.) Gargantini, I., “ O n the Computation of the Eigenvalues of a Tridiagonal Matrix,” Math. Comp., 23, 403 (1969). (11L) Gautschi, W.,“On the Condition of a Matrix Arising in the Numerical Inversion of the Laplace Transform,” ibid., p 109. (12L) Gear: C. W.,“A Simple Set of Test Matrices for Eigenvalue Programs,” ibid., p 119. (13L) Hall, C. A., and Porsching, T. A., “Computing the Maximal Eigenvalue and Eigenvector of a Positive hlatrix,” S I A M J . Numrr. A d . , 5 , 269 (1968). “Computing the Maximum Eigenvalue (141~)Hall, C . A , , and Porsching, T and Eigenvector of a Nonnegative I ucihle Matrix,” ibid., p 470. (15L) Hall, C. A . and Porsching, T. A., “ A Separation Theorem for Nonsymmetric hlatrices,” J.MhA, 23, 209 (1968). (164) Hall, C. A , , and Porsching, T. A . , “Generation of Positivc Test Matrices with Known Positive Spectra,” Comm. AC.M, 11, 559 (1968). (17L) Happ. H. H., “Foundations of Tcnsor h‘ctwork Theory,” J . Fronhlin Inxi,, 286, 561 (1968). (18L) Herriot, J. G., “Algorithms,” Comm. A C M , 11, 825 (1768). (19L) Hoffmann, B., “Matrices or Tensors?,” J . Franklin Inrt., 286, 557 (1968). (20L) Jameson, A,, “Solution of the Equation A X C XB = C by Inversion of a n M X M or N X N Matrix,” SIA.14 J . Appl. Math., 16, 1020 (1968). (21L) Kagiwada, H., and Kalaba, R., “Differential Systems for Eigenvalues of Fredholm Integral Equations,” J M A A , 23, 227 (1968). ses of T w o Commonly Occurring Maof Tridiagonal Matrices in the DescripD A M E N T A L S , 8, 169 (1969). (24L) Kotin, L., “ O n Positive and Periodic Solutions of Riccati Equations,” S I A M J . Appl. Math., 16, 1227 (1968). (25L) Ku, S. Y . , and Adler, R. J., “Computing. Polynomial Resultants,” Comm. A C M , 12, 23 (1969). (26L) Lee, R . Y . , “Turning Point Problems of Almost Diagonal Systcms,” J M A A , 24, 509 (1968). (27L) Lewis, T. 0. and Newman T. G . “Pseudoinvcrses of Positive Semidefinite Matrices,” SIAM’.J. iippl. hfnth.,’16, 70; (1968). (28L) Libby, P. A , , and Chen, K . K., “Application of Quasilinearization to a n Eigenvalue Problem Arising in Boundary-Layer Theory,” J . Cornput. Phys., 2, 356 (1968). (29L) Manherz, R . K., Jordan, B. M’., and Makimi, S. L., “Analog Methods for Computation of the Generalized Inverse,” IEEE Trans. Auto. Control, 13, 582 (1968). (301.) hiarshall. A. M’., and Olkin, I., “Scaling of Matrices to Achieve Specified Row and Column Sums,” A’umer. Moth., 12, 83 (1968). (31L) Martin, R. S., and Wilkinson, J. H., “Similarity Reduction of a General Matrix to Hcssenherg Form,” ibid., p. 349. (32L) Martin, R . S., and TL-ilkinson, J. H., “The Modified LR Algorithm for Complex Hessenberg hiatrices,” ibid., p 369, (33L) Martin, R . S., and Wilkinson, J. H., “The Implicit Q L Algorithm,” tbid., ”r -177.. (34L) Mehra, R . K.. “Computation of the Inverse Hessian Matrix Using Conjugate Gradient Methods,” P m . IEEE, 57, 225 (1969). (35L) biilne, R. D.: “An Oblique Matrix Pseudoinverse,” SIA.M J . AppI. Math., 16, 931 (1968). (36L) Milnes, H. \V. “A Note Concerning the Propertics of a Certain Class of Test Matrices,” Mn;h. Corn#., 22, 827 (1969). (37L) Parlett, B., “Global Convergence of the Basic Q R Algorithm on Hcssenberg Matrices,” ibid., p 803. (38L) Pease, M . C., “Inversion of Matrices by Partitioning,” J . A C M , 16, 302 (1969). (39L) Radke C E. “Classes of Matrices with Distinct, Real Charactcristic Values,” SIAM J . AiRi. &ath., 16, 1192 (1968). (40L) Rose, D. J., “An Algorithm for Solving a Special Class of Tridiagonal Systems ofLinear Equations,” Comm. A C M , 12, 234 (1969). (41L) Rubinstein M. and Rosen R . “Structural Analysis by Matrix Dccomposition,’’ J . Frhnklii Inst., 286, 341 (1‘968). (42L) Rutishauser, H. “Computational Aspects of F. L. Bauers Simultaneous Iteration Method,” .4‘umer. Math., 13, 4 (1969). (43L) Schwarz, H . R., “Tridiagonalization of a Symmetric Band Matrix,” ibid., 12, 231 (1968). (44L) Silvestri, A. J., Prater, C. D., and Wei, J., “On the Structure and Analysis
of Complex S stems of First-Order Chemical Reactions Containing Irreversible Steps-11,” C l e m . Ens. Sci., 23, 1191 (1968). (45L) Stewart G. W . “On the Continuity of the Generalized Inverse,” SIAM J. Appl. Math.,’17, 33 ’(1969). (46L) Stoer, J., “Lower Bounds of Matrices,” Numer. Math., 12, 146 (1968). (47L) Swy:t, R . A,, “A Recursive Relation for the Determinant of a Pentadiagonal Matrix, Comm. ACM, 12, 330 (1969). (48L) Tewarson, R. P., “On the Chebyshev Solution of Inconsistent Linear Equations,” B I T , 8, 232 (1968). (49L) Tewarson, R . P., “Projection Methods for Solving Sparse Linear Systems,” Comp. J., 12, 78 (1969). (50L) Van Ness J. E. “Inverse Iteration Method for Finding Eigenvectors,” I E E E Trans. Aha. Conbof, 14, 63 (1969). (51L) Vaughan, D. R., “A Negative Exponential Solution for the Matrix Riccati Equation,” ibid., p 72. (52L) Wing, G. M., “On a Generalization of a Method of Bellman and Latter for Obtaining Eigenvalue Bounds for Integral Operations,” J M A A , 23, 384 (1968). (53L) Yarlagadda R “An A plication of Tridiagonal Matrices to Network Synthesis,” SZAM J.‘kppI. Mal!., 16, 1146 (1968). (54L) Zielke, G., “Inversion of Modified Symmetric Matrices,” J. ACM, 15, 402 (1968). NUMERICAL-ORDINARY
DIFFERENTIAL EQUATIONS
(1M) Arbesman, R. W., and Kim, Y. G., “Generalized Relaxation Method in Chemical Kinetics,” IND. ENC.CHEM.FUNDAMENTALS, 8, 216 (1969). (2M) Benyon, P. R “A Review of Numerical Methods for Digital Simulation,” Simulation, 11, 219’\1968). (3M) Blue, J. L., “Spline Function Methods for Nonlinear Boundary-Value Problems,” Comm. ACM, 12, 327 (1969). (4M) Corrigan, T. E., and Beavers, W. O., “Dead Space Interaction in CSTR’s,” Chem. Eng. Sci., 23, 1003 (1968). (5M) Distefano, G . P., “Stability of Numerical Integration Techniques,” A.I.Ch.E. J., 14, 946 (1968). (6M) Dyer J. “Generalized Multistep Methods in Satellite Orbit Computation,” J . ACM,’15,’712 (1968). (7M) Ehle, B. L., “H$h Order A-Stable Methods for the Numerical Solution of Systems of O.D.E.’s, B I T , 8, 276 (1968). (8M) Hanson, R. J., “Simplification of Second Order Systems of Ordinary Differential Equations with a Turning Point,” SZAM J. Appl. Math., 16, 1059 (1968). (9M) Karim A. 1. A “A Theorem for the Stability of General Predictor-Corrector Methods d r the Sdlution of Systems of Differential Equations,” J . ACM, 15, 706 (1968). (10M) Kohfeld, J. J., and Thompson, G. T., “A Modification of Nordsieck’s Method Using a n “Off-Step” Point,” tbid., p 390. (11M) Kuiken, H. K., “Determination of the Intersection Points of Two Plane Curves by Means of Differential Equations,” Comm. ACM, 11, 502 (1968). (12M) Lester, W. A,, Jr., “De Vogelaere’s Method for the Numerical Integration of Second-Order Differential Equations without Explicit First Derivatives: Application to Coupled E uations Arising from the Schrodinger Equation,” J. Comput. Phys., 3, 322 (1962). (13M) Liniger, W “A Criterion for A-Stability of Linear Multistep Integration Formulae.” Corn;:. 3. 280 (1968). . (14M) Linz P. “Linear Multistep Methods for Volterra Integro-Differential Equations:” J: A C M , 16, 295 (1969). (15M) MacMillan, D. B., “Asymptotic Methods for Systems of Differential Equations in which Some Variables Have Very Short Response Times,” SIAM J. Appl. Math., 16, 704 (1968). (16M) Makinson, G. J., “Stable High Order I m licit Methods for the Numerical Solution of Systems of Differential Equations,” &np.”J., 11, 305 (1968). (17M) Martens H. R “A Comparative Study of Digital Integration Methods,” Simulation, 12,’ 87 (lg69). (18M) Oliver, J., “Inherent Instability in Systems of First-Order Linear Differential Equations,” J . Insf. Muih. Appl., 4, 399 (1968). (19M) Oliver, J. “An Error Estimation Technique for the Solution of Ordinary Differential EqLations in Chebyshev Series,” Comp. J., 12, 57 (1969). (20M) Rivlin T. J., “Polynomial Approximation and the T a u Method,” ibid., 11, 337 (1G68). (ZlM) Russell, T. W . F., and Buzzelli, D. T., “Reactor Analysis and Process Synthesis for a Class of Complex Reactions, IND.ENG.CHEM. PROCESS DESIGN DEVELOP 8 , 2 (1969). (22M) Sloan, I . H., “Method for the Numerical Solution of Linear Second-Order Differential Equations,” J . Comput. Phys., 3, 40 (1968). (23M) S t e t p , H . J “Improved Absolute Stability of Predictor-Corrector Schemes, Comp., 3,’586 (1968). (24M) Verner, J. H., “The Order of Some Implicit Runge-Kutta Methods,’’ Numer. Math., 13, 14 (1969). (25M) Waltson, D. E., and Waddell, E.,R., “Accelerating Convergence of OneStep Methods for the Numerical Solution of ODE’S,” Int. J . Comp. Math., 2, 23 (1968). (26M) Watt, J. M., “Consistency, Convergence and Stability of General Discretizations of the Initial Value Problem,” Numer. Math., 12, 11 (1968). I
NUMERICAL-PARTIAL DIFFERENTIAL EQUATIONS (1N) Albasiny, E. L., and Day, W. A., “The Numerical Solution of Linear OneDimensional Parabolic Problems by the Method of Moments,” J . Inst. Math. Appl., 4 , 140 (1968). (2N) Allgower E. and Guenther R “On the Numerical Solution of Higher Order Nonlinear Piradolic Equations:” d m p . , 3, 139 (1968). (3N) Alterman Z . S. and Isaacson E., “A Method for Calculating Frontal Motion,” J . Corn&. Phis., 4, 67 (1969)). (4N) Amsden, A. A., and Harlow F H., “Transport of Turbulence in Numerical Fluld Dynamics,” ibid., 3, 94 (i9i8). (5N) Angel E., “Discrete Invariant Imbedding and Elliptic Boundary-Value Problems’Over Irregular Regions,” J M A A , 23, 471 (1 968). (6N) Angel E., “A Building Block Technique for Elliptic Boundary-Value Problems Ove; Irregular Regions, ibid., 26, 75 (1969). (7N) Birtwistle, G. M., “The Explicit Solution of the Equation of Heat Conduction,” Comp. J . , 11, 317 (1968). (8N) Boynton F. P., and Thomson, A., “Numerical Computations of Steady, Supersonic, ‘Two-Dimensional Gas Flow in Natural Coordinates,” J. Comput. Phys., 3, 379 (1969). (9N) Bramble, J. H., Kellogg, R. B., and Thomee, V., “On the Rate of Convergence
of Some Difference Schemes for Second Order Elliptic Equations,” B I T , 8, 154 (1968). (10N) Cannon J. R and Hill, C. D., “A Finite-Difference Method for Degenerate Ellipti:-Paradblic Equations,” SIAM J . Numer. Anaf., 5 , 21 1 (1968). (11N) Chorin, A. J “Numerical Solution of the Navier-Stokes Equations,” Math. Comp., 22, 745 (1$69). (12N) Chorin, A. J., “ O n the Convergence of Discrete Approximations to the Navier-Stokes Equations,” ibid., 23, 341 (1969). (13N) Cooke K. L. and Krumme D. W. “Differential-Difference Equations and Nonlinear ’InitialiBoundary Vaiue PrAblems for Linear Hyperbolic Partial Differential Equations,” J M A A , 24, 372 (1968). (14N) Colwell, C. J., and Dranoff, J. S. “Nonlinear Equilibrium and Axial Mixing Effects in Intraparticle Diffusion-Cdntrolled Sorption by Ion Exchange Resin 8, 193 (1969). Beds,” IND.END.CHEM.FUNDAMENTALS, (15N) Crowley, W. P., “A Global Numerical Ocean Model: Part I,” J . Cornput. Phys., 3, 111 (1968). (16N) Davis, J. A., and Hageman, L. A. “An Iterative Method for Solving the Neutron Transport Equation in X-Y Geometry,” S I A M J . Appf. Math., 17, 149 (1969). (17N) Deiters, R. M and Tamiya N . “Improving the Analog Simulation o f Partial Differentia1”Equations b; HGbrid Computation,” Simulation, 11, 73 (1968). (18N) Dorney C. N. “Finite-Difference Approximation of the Exterior Problem for Poisson’s’Equatibn,’’ J . Comput. Phys., 2, 363 (1968). (19N) Dorr, F. W., “Remarks on the Iterative Solution of the Neumann Problem on a Rectangle by Successive Line Over-Relaxation,” Math. Comp., 23, 177 (1969). (20N) Du Pont T Kendall R . P and Rachford H. H Jr. “An Approximate Factorization’Prb’cedure fdr Solvlbg Self-Adjoint‘ Ellipti’. Difference Equations,’ SIAM J . Numer. Anaf., 5 , 559, 753 (1968). (21N) Evans, D. J., “The Use of Preconditioning in Iterative Methods for Solving Linear Equations with Symmetric Positive Definite Matrices,” J . Inst. M a t h . &I., 4, 295 (1968). (22N) Fagela-Alahastro E. B and Hellums J. D. “A Theoretical Study on Diffusion in Pulsating Fldw,” A:Z.Ch.E. J., 15,’164 (lb69). (23N) Fairweather, G., “A Note on a Generalization of a Method of Douglas,” Math. Comp., 23, 407 (1969). (24N) Finlayson, B. A., “Applications of the Method of Weighted Residuals and Variational Methods,” Brit. Chem. Eng.,14, 53, 179 (1969). (25N) Friedman, A., “Partial Differential Equations,’’ Halt, Rinehart & Winston, 1969. (26N) Gourlay A. R. “The Acceleration of the Peaceman-Rachford Method by Chebyshev Polyno&ials,” Comp. J., 11, 378 (1969). (27N) Gourlay, A. R., and Mitchell A. R . “High Accuracy A.D.I. Methods for Parabolic Equations with Variabld Coeffikents,” Numer. Math., 12, 180 (1968). (28N) Gourla A R. and Morris J. L. “A Multistep Formulation of the Optimized Lax-Genhrok Method f i r Noniinear Hyperbolic Systems in TWOSpace Variables,” Math. Comp., 22, 715 (1969). (29N) Greenspan, D., “Lectures 011 the Numerical Solution of Linear, Singular and Nonlinear Differential Equations,” Prentice-Hall, 1968. (30N) Greenspan, D., “Approximate Solution of Initial-Boundary Wave Equation Problems by Boundary-Value Techniques,” Comm. ACM, 11, 760 (1968). (31N) Hadjidimos, A., “ O n a Generalized Alternating Direction Implicit Method for Solving Laplace’s Equation,” Comp. J., 11, 324 (1968). (32N) Harlow, F. H., and Amsden, A. A. “Numerical Calculation of Almost Incompressible Flow,” J . Cornput. Phyr., 3, BO (1968). (33N) Harmathy, P. Z., “Simultaneous Moisture and Heat Transfer in Porous Systems with Particular Reference to Drying, IND. ENC. CHEH.FUNDAMENTALS 8, 9 2 (1969). (34N) Hedstrom, G . W., “The Rate of Convergence of Some Difference Schemes,” SIAM J . Numer. Anaf., 5, 363 (1968). (35N) Houghton, D. D., and Jones, W. L., “A Numerical Model for Linearized Gravity and Acoustic Waves,” J . Comput. Phys., 3, 339 (1969). (36N) Jaranden, I and Witherspoon, P. A. “Application of the Finite Element Method to Transient Flow in Porous Medii,” Soc. Petrol. Eng. J., 8, 241 (1968). (37N) Katszttis, T., “A Numerical Method for the Solution of Certain Neumann Problems, SIAM J . APPI. Math., 16, 723 (1968). (38N) Keast, P., “The Third Boundary Value Problem for Elliptic Equations,” Numer. Math., 12, 322 (1968). (39N) Kellogg, R . B., “A Nonlinear Alternating Direction Method,” Math. camp., 23, 23 (1969). (40N) Kreiss, H-O., “On the Numerical Solution of the Spherically Symmetric Diffusion Equation,” Numer. Math., 12, 223 (1968). (41N) Kusic, G. L., and Lavi, A., “Stability of Difference Methods for InitialValue Type Partial Differential Equations,” J. Comput. Phys., 3, 358 (1969). (42N) Kushner, H. J., “On the Numerical Solution of Degenerate Linear and Nonlinear Elliptic Boundary Value Problems,” S I A M J . Fumer. Anal., 5 , 664 (1968). (43N) Kushner, H. J., and Kleinman, A. J., “Numerical Methods for the Solution of the Degenerate Nonlinear Elliptic Equatlons Arising in Optimal Stochastic Control Theorv.” I E E E Trans. Auto. Control. 13. -144 I1 OAR) --,. (44N) de la Valle Poussin, F., “An Accelerated Relaxation Algorithm for Iterative Solution of Elliptic Equations,” S I A M J . Numer. Anal., 5 , 340 (1968). (45N) Liu, S., “Stable Explicit Difference Approximations to Parabolic Partial Differential Equations,” A.I.Ch.E. J., 15, 334 (1969). (46N) Lloyd, T., and McCallion, H., “Bounds for the Optimum Over-Relaxation Factor for the S.O.R. Solution of Laplace Type Equations over Irregular Regions,’’ Comp. J., 11, 329 (1968). (47N) Lynn, M. S., and Timlake W. P., “ T h e Use of Multiple Deflations in the Numerical Solution of Singula‘r Systems of Equations, with Applications to Potential Theory,” S U M J . Numer. And., 5, 303 (1968). (48N) ,Nomura, T., and Deiters, R . M . “Improving the Analog Simulation of Partial Differential-Equations by Hybrid Computing,” Simufation, 9,73 (1 968). (49N) Noronha, L. G., Po, C. Y . , and Womack, J. W. “ H brid Computation of the Dynamics of a Distributed System,” Comp. J., 11, 1’96 (7968). (50N) O’Brien V., “Form Factors for Deformed Spheroids in Stokes Flow,” A.Z.Ch.E. J.,’l4,870 (1968). (51N) Olbrich, W. E., and Wild J. D. “Diffusion from the Free Surface into a Liquid Film in Laminar Flow’ over befined Shapes,” Chcm. Enx. Sci., 24, 20 (1969). (52N). Osborne, M. R., “The Numerical Solution of a Periodic Parabolic Problem Subject to a Nonlinear Boundary Condition,” Numer. Moth., 12, 280 (1968). (53N). Osborne, M . R. “The Numerical Solution of the Heat Conduction Equation Subject to SeparatedBoundary Conditions,” ~ o m p J., . 12, 82 (1969). ~
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(54N) Parnaby, J. “Computers and Systems Design: Part 2, Digiial Computing Techniques,” C o h a l , 12, 621 (1968). (55N) Peck, R. E., and Kauh, J. Y.,“ Evaluation of Drying Schedules,” A.I.Ch.E. J., 15, 85 (1969). (56N) Petschek, A. G., and Hanson M . E. “Difference Equations for Two-Dimensional Elastic Flow,” J . Cornput. PLyx., 3, i07 (1 968). (57N) Pigford, R . L., Baker B and Blum, L). E., “An Equilibrium Theory of the Parametric Pump,” 1h.D. i N d : CHEM.FUNnAMESTALS, 8, 144 (1969). (58K) Price, H . S., Cavendish, J. C., and Varga, R . S., “Numerical Methods of Higher-Order Accuracy for Diffusion-Convection Equations,“ .Sot. Petretroi. Eng. J., 8, 241 (1968). (59N) Rachford, 11. €I., Jr., “Rounding Errors in Altcrnating Direction Methods for Parabnlic Problems,” S I A M J . ”.’Vumer. Anal., 5 , 407 (1968). ( 6 0 s ) Randall, T. J., “Kotc on a General Finite-Difference Formula for the Solution of Axially Symmetric Fields,” Comp. J., 11, 332 (1968). (GIN) Randall, T. J., “A Note on the Estimation of the Optimum Successive Overrelaxation Paramerer in Laplace’s Equation,” ihid., p 400. (62N) Reimer. M. “Finite Difference Forms Containing Derivatives of Higher Order,” S I . 4 h J.’.Vumer. Anal., 5, 725 (1968). (63N) Franklin, J. N., and Rodemich. E. R.. “Numerical Analysis of a n EllipticParabolic Partial Differential Equation,” ibid., p 680. (64s) Rolke R . \V.. and TVilhelm, R . H., “Kecuperativc Parametric Pumping,” I N D , ENG.‘CHEM. FUNDAMENTALS, 8, 235 (1969). (65N) Ruhe, A,, “On the Quadraiic Convergence of a Generalization of the Jacobi Method t o Aribitrary Matrices.” B I T , 8, 210 (1968). (66N) Smith J. “The Coupled Equation Approach to the Numerical Solution of the Bihz.rm&ic Equation by Finite Difierences. I,” SlA.ZI J . ‘Vumer. Anal., 5 , 323 (1968). (67N) Stewart, G . T.V., amd Lick D . W. “Numerical Solution of a Thin Plate Heat Transfer Problem,” Comm. AC:M, 11, i 3 9 (1968). (68iX) Stone H. L . “Iterative Solution of Implicit Approximations of Multidimensionh P a r t i h Differential Equations,“ S1A.M J . Numer. Anal., 5 , 530 (1968).
(69N) Straight F., and Baasel, M’.D . “ A Comparison of Theoretical and Experimental Resuits for the Internal Dissolution of Soluble Cylinders by Water in Laminar Flow,” A.I.Ch.E. J., 14, 722 (1968). (70N) Strang, G., “ O n the Construction and Comparison of Difference Schemes,” SIAM J . Numer. Bnal., 5, 506 (1968). (71h-) Sunouchi, H., “Perturbation Theory of Diff erence Schcmes,” Numer. .Math., 1 - -9 > 4. -5 4 f196RI.
(72N) Swartz, B. and ‘ivendroff, B., “Generalized Finite-Difference Schemes,” Moth. Comp., 2$, 37 (1969). (73K) Sweed, S . H., and Wilhelm, R. H., “Parametric Pnmping,” 1h.n. ENO. CHEM.FUNDAMENTALS, 8, 221 (1969). ( 7 4 s ) TFO,I,; C. “Generalized Solution of Freezing a Saturated Liquid in a Convex Container, A.>.Ch.E. J., 14, 720 (1968). (75N) Thuraisamy, V., “Approximate Soliltions for Mixed Boundary Value Problems by Finite Difference Methods,” ’Math. Comp., 23, 373 (1969). (76N) Tien, C . , and,,Sriniuasan S. “An Approximate Solution for Countercurrent Heat Exchangers, A.I.Ch.E.’J.,’15, 39 (1969). (77N) Van Leer B., “Stabilization of Differmcr Schemes for the Equations of Inviscid Com&ssible Flow by Artificial Ditfusion,” J . Cornput. PJlys., 3, 473 (1969). (78N) Verner, J. H., and Bernal, hl. J. M., “ O n Generalizations of the Theory of Consistent Ordering for Successive Over-Relaxation hfethods,” N u m e r . Math., 12, 215 (1968). Numerical Solution of Partial (79X) Von Rosenherg, D. Differential Equations,” (EON) Von Rosenberg, D . U., and Mount, D. E., “Improved Numcrical Solution of a Countercurrent Heat Exchanger,” Chcm. Enp. Qi., 24, 619 (1969). (BIN) Watts, J. I V , , and Von Rosenberg, D . U., “A Numerical Solution of a Trarisient Shock \Vave Problem,” ibid., p 49. (82N) Weinstein H . G., Stone, H . L., and Kwan, T. V., “Itcrativc Procedure for Solution of Syitems of Parabolic and Elliptic Equations in Thrce Dimensions,” IND, ENG.CHEM. FCNDAMENTALS, 8, 281 (1969). (83N) Williams, F. A,. “A Nonlinear Diffusion Problem Relevant to Desalination by Reverse Osmosis,” SI.4.M J . Ap$l. Math., 17, 59 (1969). (84N) Vichnevetsky, R., “Generalized Finite-Ilifference Approximations for the Parallel Solution of Initial Value Problems,” Simulation, 12, 233 (1969). NUMERICAL-BOUNDARY
VALUE PROBLEMS
( 1 0 ) Bailev P. B.. and Shampine, L. F., “On Shooting Methods for Two-Point Boundar$’Value Problems,” J M A A , 23, 235 (1968). ( 2 0 ) Bellman R. “Invariant Imbedding and Multipoint Boundary-Value Problems,” ;bid., ’24,461 (1968). ( 3 0 ) Bickley, TV. G ‘,Piecewise Cubic Interpolation and Two-Point Boundary Problems,” Cornp. ?., 11,206 ( 1 9 6 8 ) . ( 4 0 ) Chandra, J., and Fleishman, B. A,, “Positivity and Comparison Results for Nonlinear Boundary Value Problems and Related Periodic Solutions,” J.MA.4 24, 545 (1968). ( 5 0 ) Chorlton, F., “Boundary Value Problems in Science and Engineering,” Van Nostrand-Reinhold, 1969. ( 6 0 ) Ciarlet, P. G., Schultz, hl. hl.,and Varga, R. S., “Numerical Methods of High-Order Accuracy for Nonlinear Boundary Value Problems. 111. Cig.enva1ue Problems,” Kurner. .tfofh., 12, 120 (1968). ( 7 0 ) Ciarlet P. G Schultz h4. M . and Varga, R . S., “Numerical Methods of High O r d i r Acc$racy far ’Nonlinekr Boundary Value Problems. IV. Periodic Boundary Conditions,” ibid., p 266. ( 8 0 ) Ciarlet, P. G., Schultz, M . M,:and Varga, R . S., “Numerical Methods of High-Order Accuracy for Nonlinear Boundary Value Problcms,” ibid., 13, 51 (1969). (90) Falkenherg J. C. “A Method for Integration of Unstable Systems of Ordinary Differential Eduatiohs Subject to Two-Point Boundary Conditions,” B I T , 8, 86 (1968). hod Applied t o Convective Instability
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+
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NUMERICAL-ROOT
LOCATION ure for Solving Nonlinear Equations in n Theorems for Partitioned Matrices,”
(3Q) Bypyden C. G . “A New Method of Solving Konlincar Simultaneous Equations, Corn;. 3., 12: 94 (1969). (4Q) Brown, K. M . , and Dennis, J. E., ,It.., “ O n Newton-Likc Iteration Functions: General Convergence Theorems and a Specific Algorithm,” N u m e r . .$4ath., 12, 186 (1968). (5Q) Kine, R . F., and Phillip> D . L., “The Logarithmic Error and Newton’s Method for the Square Root, Comm. ACh’, 12, 87 (1969). ( 6 9 ) Miranker, W. L., L‘ParallelMethods for Approximating the Root of a Function, I B d i J . Res. Uroelofi., 13, 297 (1969). ( 7 4 ) [Moore, R.,fVI., “Approximations to Nonlinear Operator Equations and Newton s Method, ”Jtmer. Math., 12, 2 3 (1968). ( 8 4 ) Paspuali: A,, “ O n thc Convergence of Nonlinear Simultaneous Displacements, J . Corn#. S’slzrn Sci.,3, 1 (1969). (9Q) Sterbenz, P. H., and Fike C. T . “Optimal Starting Approximations for Nebton’s Method,” Math. Corn;., 23, 3?3 (1969).
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