Lakshmikantham Applied Nonlinear Analysis

Proceedings of an International Conference on Applied Nonlinear Analysis, Held at the University of Texas at Arlington, Arlington, Texas, April 20-22, 1978
1. Auflage 2014
ISBN: 978-1-4832-7206-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 746 Seiten, Web PDF

Applied Nonlinear Analysis contains the proceedings of an International Conference on Applied Nonlinear Analysis, held at the University of Texas at Arlington, on April 20-22, 1978. The papers explore advances in applied nonlinear analysis, with emphasis on reaction-diffusion equations; optimization theory; constructive techniques in numerical analysis; and applications to physical and life sciences. In the area of reaction-diffusion equations, the discussions focus on nonlinear oscillations; rotating spiral waves; stability and asymptotic behavior; discrete-time models in population genetics; and predator-prey systems. In optimization theory, the following topics are considered: inverse and ill-posed problems with application to geophysics; conjugate gradients; and quasi-Newton methods with applications to large-scale optimization; sequential conjugate gradient-restoration algorithm for optimal control problems with non-differentiable constraints; differential geometric methods in nonlinear programming; and equilibria in policy formation games with random voting. In the area of constructive techniques in numerical analysis, numerical and approximate solutions of boundary value problems for ordinary and partial differential equations are examined, along with finite element analysis and constructive techniques for accretive and monotone operators. In addition, the book explores turbulent fluid flows; stability problems for Hopf bifurcation; product integral representation of Volterra equations with delay; weak solutions of variational problems, nonlinear integration on measures; and fixed point theory. This monograph will be helpful to students, practitioners, and researchers in the field of mathematics.

Autoren/Hrsg.

Lakshmikantham, V.

Inhaltsverzeichnis

1;Front Cover
;1
2;Applied Nonlinear
Analysis;4
3;Copyright Page
;5
4;Table of Contents
;6
5;LIST OF CONTRIBUTORS;14
6;PREFACE;20
7;PART 1: INVITED ADDRESSES
AND RESEARCH REPORTS;22
7.1;CHAPTER 1. ON CONTRACTING INTERVAL ITERATION
FOR NONLINEAR PROBLEMS IN;24
7.1.1;I. INTRODUCTION;24
7.1.2;II.
ITERATION;25
7.1.3;III. PROPERTIES OF THE ITERATION;28
7.1.4;REFERENCES;32
7.2;CHAPTER 2. A CONSTRUCTIVE METHOD FOR LINEAR AND NONLINEAR STOCHASTIC
PARTIAL DIFFERENTIAL EQUATIONS;34
7.2.1;I. LINEAR DETERMINISTIC PARTIAL DIFFERENTIAL EQUATIONS;34
7.2.2;II. LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS;36
7.2.3;III. LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS;37
7.2.4;IV. NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS;37
7.2.5;V. NONLINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS;43
7.2.6;ACKNOWLEDGMENTS;44
7.2.7;REFERENCES;44
7.3;CHAPTER 3. A NON LINEAR INTEGRAL AND A BANG-BANG THEOREM;46
7.3.1;INTRODUCTION;46
7.3.2;RESULTS;49
7.3.3;REFERENCES;66
7.4;CHAPTER 4. COPTIMAL CONTROL OF DIFFUSION-REACTION SYSTEMS;68
7.4.1;REFERENCES;80
7.5;CHAPTER 5. A NONLINEAR GENERALIZATION OF
THE HEAT EQUATION ARISING IN PLASMA PHYSICS;82
7.5.1;ACKNOWLEDGMENTS;86
7.5.2;REFERENCES;86
7.6;CHAPTER 6. PERIODIC SOLUTIONS OF DELAY
DIFFERENTIAL EQUATIONS ARISING IN SOME MODELS OF EPIDEMICS;88
7.6.1;REFERENCES;98
7.7;CHAPTER 7. COMPARISON THEOREMS FOR SYSTEMS
OF REACTION-DIFFUSION EQUATIONS;100
7.7.1;I. INTRODUCTION;100
7.7.2;II. NOTATION;101
7.7.3;III.
COMPARISON THEOREMS;102
7.7.4;IV. EXAMPLES;105
7.7.5;REFERENCES;108
7.8;CHAPTER 8. SEQUENTIAL CONJUGATE GRADIENT-RESTORATION ALGORITHM FOR OPTIMAL CONTROL PROBLEMS
WITH NONDIFFERENTIAL CONSTRAINTS’;110
7.8.1;REFERENCES;112
7.9;CHAPTER 9. ROTATING SPIRAL WAVES AND OSCILLATIONS IN REACTION-DIFFUSION
EQUATIONS;116
7.9.1;I. INTRODUCTION;116
7.9.2;II. ROTATING SPIRAL WAVES;117
7.9.3;III. MULTI-SPECIES INTERACTIONS;121
7.9.4;REFERENCES;129
7.10;CHAPTER 10. SOME APPLICATIONS OF ROTHE'S METHOD
TO PARABOLIC AND RELATED EQUATIONS;132
7.10.1;REFERENCES;141
7.11;CHAPTER 11. A COARSE-RESOLUTION ROAD MAP TO METHODS FOR APPROXIMATING SOLUTIONS OF TWO-POINT
BOUNDARY-VALUE PROBLEMS;144
7.12;CHAPTER 12. CONE-VALUED PERIODIC SOLUTIONS OF ORDINARY
DIFFERENTIAL EQUATIONS;148
7.12.1;I. AUXILIARY RESULTS;149
7.12.2;II.
PERIODIC SOLUTIONS VIA THE POINCARE OPERATOR;151
7.12.3;III.
EXISTENCE WITHOUT UNIQUENESS;153
7.12.4;IV
GALERKIN APPROXIMATIONS;159
7.12.5;REFERENCES;162
7.13;CHAPTER 13. THE BISTABLE NONLINEAR DIFFUSION EQUATION:
BASIC THEORY AND SOME APPLICATIONS;164
7.13.1;I. INTRODUCTION;164
7.13.2;II. ASYMPTOTIC THEORY OF THE BISTABLE NONLINEAR DIFFUSION
EQUATION;166
7.13.3;III. AN EXAMPLE FROM POPULATION GENETICS;168
7.13.4;IV. STATIONARY PATTERN FORMATION FOR SYSTEMS;170
7.13.5;V. SHARP FRONTS AND SINGULAR PERTURBATIONS;176
7.13.6;REFERENCES;179
7.14;CHAPTER 14. PRODUCT INTEGRAL REPRESENTATION OF SOLUTIONS
TO SEMILINEAR VOLTERRA EQUATIONS WITH DELAY;182
7.14.1;REFERENCES;193
7.15;CHAPTER 15. ANGLE-BOUNDED OPERATORS AND UNIQUENESS OF PERIODIC SOLUTIONS OF
CERTAIN ORDINARY DIFFERENTIAL EQUATIONS;196
7.15.1;INTRODUCTION;196
7.15.2;SECTION II;201
7.15.3;REFERENCES;203
7.16;CHAPTER 16. COMPARTMENTAL MODELS OF BIOLOGICAL SYSTEMS:
LINEAR AND NONLINEAR;206
7.16.1;I. INTRODUCTION;206
7.16.2;II. MODELS AND THE MODELING PROCESS IN BIOLOGY;207
7.16.3;III. COMPARTMENTAL SYSTEMS;209
7.16.4;IV. NONLINEAR COMPARTMENTAL SYSTEMS;213
7.16.5;V. IDENTIFICATION AND THE INVERSE PROBLEM;214
7.16.6;V I . COMPLEXITY AND STABILTY;224
7.16.7;REFERENCES;225
7.17;CHAPTER 17. NEW OPTIMIZATION PROBLEMS FOR DYNAMIC MLTLTICONTROLLER
DECISION THEORY;228
7.17.1;I. INTRODUCTION;228
7.17.2;II. PROFIT MAXIMIZATION OF A
FIRM – AN EXAMPLE;230
7.17.3;III.
DISCUSSION;232
7.17.4;ACKNOWLEDGMENTS;233
7.17.5;REFERENCES;233
7.18;CHAPTER 18. STABILITY TECHNIQUE AND THOUGHT PROVOCATIVE DYNAMICAL SYSTEMS
II;236
7.18.1;SECTION I;236
7.18.2;SECTION
II;237
7.18.3;SECTION III;238
7.18.4;REFERENCES;239
7.19;CHAPTER 19. REACTION-DIFFUSION EQUATIONS IN ABSTRACT
CONES;240
7.19.1;I. INTRODUCTION;240
7.19.2;II. PRELIMINARY RESULTS;243
7.19.3;III.
MAIN RESULTS;246
7.19.4;IV. APPLICATIONS;257
7.19.5;REFERENCES;261
7.20;CHAPTER 20. NUMERICAL SOLUTION OF NEURO-MUSCULAR
SYSTEMS;266
7.20.1;I. INTRODUCTION;266
7.20.2;II. REFORMULATION OF THE PROBLEM;270
7.20.3;III. COMPUTATION OF HIGHER ORDER DERIVATIVES;271
7.20.4;V. COMPUTATION OF THE DERIVATIVES
OF;274
7.20.5;VI. COMPUTATION OF SUCCESSIVE DERIVATIVES OF V (u(t));279
7.20.6;VI. IMPLEMENTATION OF THE NUMERICAL ANALYSIS;282
7.20.7;VIII. PROGRAM DESCRIPTION;284
7.20.8;REFERENCES;286
7.21;CHAPTER 21. SEPARATRICES FOR DYNAMICAL SYSTEMS;288
7.21.1;REFERENCES;291
7.22;CHAPTER 22. STABILITY PROBLEMS FOR HOPF
BIFURCATION;294
7.22.1;I. INTRODUCTION;294
7.22.2;II.
PRELIMINARIES;296
7.22.3;III. ASYMPTOTIC STABILITY AND COMPLETE INSTABILITY OF THE ORIGIN
FOR µ = 0, RECOGNIZED BY MEANS OF REDUCED SYSTEMS;297
7.22.4;IV. ATTRACTING AND REPULSING CLOSED ORBITS;301
7.22.5;V. APPLICATION TO THE FITZHUGH NERVE CONDUCTION EQUATIONS;305
7.22.6;REFERENCES;307
7.23;CHAPTER 23. AN ITERATIVE METHOD FOR APPROXIMATING SOLUTIONS
TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS;308
7.23.1;INTRODUCTION;308
7.23.2;I. FINITE DIFFERENCE APPROXIMATION;309
7.23.3;II. QUASI-SOLUTIONS;310
7.23.4;III.
DESCRIPTION OF ITERATION PROCESS;312
7.23.5;IV. PROOFS;314
7.23.6;REFERENCES;317
7.24;CHAPTER 24. ON THE EXISTENCE OF INVARIANT MEASURES;320
7.24.1;I. INTRODUCTION;320
7.24.2;II. KRILOFF-BOGOLIUBOFF THEOREM;321
7.24.3;III. ABSOLUTELY CONTINUOUS INVARIANT MEASURES;322
7.24.4;I V . THE QUADRATIC MAP;325
7.24.5;V. APPLICATIONS;326
7.24.6;REFERENCES;327
7.25;CHAPTER 25. THE ROLE OF DIRECT FEEDBACK
IN THE CARDIAC PACEMAKER;330
7.25.1;I. INTRODUCTION;330
7.25.2;ACKNOWLEDGMENTS;341
7.25.3;REFERENCES;341
7.26;CHAPTER 26. THE CURRENT STATE OF THE
N-BODY PROBLEM;344
7.26.1;ABSTRACT;344
7.27;CHAPTER
27. STABILITY OF McSHANE SYSTEMS;346
7.27.1;I. INTRODUCTION;346
7.27.2;II. PRELIMINARIES;347
7.27.3;III. A COMPARISON THEOREM;350
7.27.4;IV. STABILITY RESULTS;352
7.27.5;REFERENCES;354
7.28;CHAPTER 28. CONSTRUCTIVE TECHNIQUES FOR ACCRETIVE
AND MONOTONE OPERATORS;356
7.28.1;ACKNOWLEDGMENT;364
7.28.2;REFERENCES;364
7.29;CHAPTER 29. SUPOR Q:
A BOUNDARY PROBLEM SOLVER FOR ODE's;368
7.29.1;ABSTRACT;368
7.29.2;REFERENCES;371
7.30;CHAPTER 30. SOME RECENT DEVELOPMENTS IN STABILITY
OF GENERAL SYSTEMS;372
7.30.1;INTRODUCTION;372
7.30.2;I. SET-THEORETIC FRAMEWORK AND NOTATIONS;374
7.30.3;II.
STABILITY;374
7.30.4;III. LIAPUNOV–TYPE
FUNCTION;378
7.30.5;IV.
EXTENSIONS;385
7.30.6;REFERENCES;389
7.31;CHAPTER 31. ON CERTAIN SOLUTIONS OF AN
INTEGRODIFFERENTIAL EQUATION;394
7.31.1;REFERENCES;401
7.32;CHAPTER 32. A GREEN1S FORMULA FOR WEAK SOLUTIONS
OF VARIATIONAL PROBLEMS;402
7.32.1;I. INTRODUCTION;402
7.32.2;II. GREEN'S FORMULA;403
7.32.3;III. APPLICATIONS;404
7.32.4;IV. PSEUDOPARABOLIC EQUATION;406
7.32.5;V. NONLINEAR PROBLEMS;407
7.33;CHAPTER 33. APPLICATION OF FIXED POINT THEOREMS
IN APPROXIMATION THEORY;410
7.33.1;REFERENCES;414
7.34;CHAPTER 34. EQUIVALENCE OF CONJUGATE GRADIENT METHODS
AND QUASI-NEWTON METHODS;416
7.34.1;ABSTRACT;416
7.35;CHAPTER 35. APPROXIMATE SOLUTION OF ELLIPTIC BOUNDARY VALUE PROBLEMS BY SYSTEMS
OF ORDINARY DIFFERENTIAL EQUATIONS;418
7.35.1;INTRODUCTION;418
7.35.2;I. MONOTONE APPROXIMATION OF SOLUTIONS TO [P(h)];419
7.35.3;II. EXTREMAL SOLUTIONS FOR INFINITE SYSTEMS;421
7.35.4;111. PROOF OF THEOREM 1;424
7.35.5;REFERENCES;425
7.36;CHAPTER 36. ASYMPTOTIC BEHAVIOR OF A CLASS
OF DISCRETE-TIME MODELS IN POPULATION GENETICS;428
7.36.1;I. INTRODUCTION;428
7.36.2;II. DEFINITION OF THE WAVE SPEED;434
7.36.3;III. THE PROPAGATION OF GENETIC CHANGE;440
7.36.4;REFERENCES;442
8;PART 2:
CONTRIBUTED PAPERS;444
8.1;CHAPTER 37. THE VOLUME OF DISTRIBUTION
IN SINGLE-EXIT COMPARTMENTAL SYSTEMS;446
8.1.1;I. INTRODUCTION;446
8.1.2;II. DEVELOPMENT OF THE MATHEMATICAL MODEL;447
8.1.3;III.
BASIC FACTS ABOUT THE MODEL;449
8.1.4;IV. IDENTIFICATION OF THE TRANSFER COEFFICIENTS;450
8.1.5;V. ESTIMATION OF VOLUME FROM EXPERIMENTAL OBSERVATIONS;453
8.1.6;VI. AN APPLICATION;455
8.1.7;VII. FINAL REMARKS;456
8.1.8;ACKNOWLEDGMENTS;457
8.1.9;REFERENCES;457
8.2;CHAPTER 38. ON IDENTIFICATION OF COMPARTMENTAL SYSTEMS;460
8.2.1;I. INTRODUCTION;460
8.2.2;II. GENERAL COMPARTMENTAL MATRICES;461
8.2.3;III. SINGLE INPUT-OBSERVATION COMPARTMENTAL (SIOC) SYSTEMS;463
8.2.4;IV. SPECIAL SYSTEMS;465
8.2.5;REFERENCES;468
8.3;CHAPTER 39. PRECONDITIONING FOR CONSTRAINED OPTIMIZATION PROBLEMS WITH APPLICATIONS
ON BOUNDARY VALUE PROBLEMS;470
8.3.1;ABSTRACT;470
8.4;CHAPTER 40. EVALUATION OF QUASI-LINEAR TECHNIQUES
FOR NONLINEAR PROCESSES WITH RANDOM INPUTS;472
8.4.1;I. INTRODUCTION;472
8.4.2;II. MODEL FORMULATION;473
8.4.3;III. EVALUATION OF EQUIVALENT GAIN, C;476
8.4.4;IV. COMPARISON OF THE TWO APPROACHES;481
8.4.5;VI. APPENDIX A;485
8.4.6;REFERENCES;486
8.5;CHAPTER 41. TWO PROBLEMS IN NONLINEAR FINITE
ELEMENT ANALYSIS~;488
8.5.1;I. INTRODUCTION;488
8.5.2;II. ANALYSIS AND METHOD;489
8.5.3;III. NUMERICAL RESULTS;494
8.5.4;REFERENCES;497
8.6;CHAPTER 42. FIXED POINT THEORY AND INWARDNESS CONDITIONS;500
8.6.1;REFERENCES;502
8.7;CHAPTER 43. A DIRECT COMPUTATIONAL METHOD FOR ESTIMATING
THE PARAMETERS OF A NONLINEAR MODEL;506
8.7.1;I. INTRODUCTION;506
8.7.2;II. DCM (DIRECT COMPUTATIONAL METHOD);507
8.7.3;111. FILTERING SCHEME;508
8.7.4;IV. LINEAR LEAST SQUARES METHOD FOR
ESTIMATING;511
8.7.5;V. NUMERICAL EXAMPLES;511
8.7.6;REFERENCES;517
8.8;CHAPTER 44. A NOTE ON THE ASYMPTOTIC BEHAVIOR
OF NONLINEAR SYSTEMS;520
8.8.1;REFERENCES;525
8.9;CHAPTER 45. A SECOND-STAGE EDDY-VISCOSITY CALCULATION
FOR THE FLAT-PLATE TURBULENT BOUNDARY LAYER;528
8.9.1;I, INTRODUCTION;528
8.9.2;II.
PODT-SAS AND THE SMALL-EDDY VISCOSITY;529
8.9.3;III. CALCULATIONS;530
8.9.4;IV. RESULTS;532
8.9.5;V. CONCLUSIONS;533
8.9.6;VI. PROGNOSIS;535
8.9.7;ACKNOWLEDGMENTS;535
8.9.8;REFERENCES;536
8.10;CHAPTER 46. NONLINEAR OPTIMIZATION AND EQUILIBRIA
IN POLICY FORMATION GAMES WITH RANDOM VOTING;540
8.10.1;I. INTRODUCTION;540
8.10.2;II.
ELECTIONS WITH PROBABILISTIC VOTING;541
8.10.3;III.
MEDIAN RANDOM VOTER RESULTS;543
8.10.4;IV. CONCLUSION;546
8.10.5;REFERENCES;546
8.11;CHAPTER 47. ON THE BOUNDED SOLUTIONS OF A NONLINEAR CONVOLUTION
EQUATION;550
8.11.1;ABSTRACT;550
8.12;CHAPTER 48. SOMZ UNRESOLVED QUESTIONS PERTAINING TO THE MATHEMATICAL ANALYSIS OF
FLUORESCENCE DECAY DATA;552
8.12.1;I. INTRODUCTION;552
8.12.2;II. FLUORESCENCE DECAY ANALYSIS; A DESCRIPTION OF THE PROBLEM;553
8.12.3;III. METHOD OF ANALYSIS;554
8.12.4;IV. DESCRIPTION OF THE DATA;556
8.12.5;V. ANALYSIS OF THE DATA;557
8.12.6;VI. NON-RANDOM ERRORS;559
8.12.7;VII. UNRESOLVED PROBLEMS;561
8.12.8;REFERENCES;562
8.13;CHAPTER 49. SEPARATION AND MONOTONICITY RESULTS
FOR THE ROOTS OF THE MOMENT PROBLEM;564
8.13.1;I. INTRODUCTION;564
8.13.2;II. MOMENT PROBLEM;565
8.13.3;III. THE SUBEIGENVALUE PROBLEM;566
8.13.4;IV. NUMERICAL EXAMPLE;572
8.13.5;REFERENCES;573
8.14;CHAPTER 50. SYSTEM IDENTIFICATION OF MODELS EXHIBITING EXPONENTIAL, HARMONIC AND RESONANT
MODES;576
8.14.1;I. INTRODUCTION;576
8.14.2;II.
GENERALIZED DISCRETE MOMENT SEQUENCE;579
8.14.3;III. HANKEL MATRICES OF GENERALIZED DISCRETE MOMENT SEQUENCES;580
8.14.4;IV. GENERATION OF
g.d.m.s.;582
8.14.5;V. IDENTIFICATION PROCEDURE;583
8.14.6;VI. NUMERICAL EXAMPLES;585
8.14.7;REFERENCES;588
8.15;CHAPTER 51. DIFFERENTIAL EQUATION ALGORITHMS FOR MINIMIZING
A FUNCTION SUBJECT TO NONNEGATIVE CONSTRAINTS;590
8.15.1;I. INTRODUCTION;590
8.15.2;II. OPTIMIZATION WITH POSITIVE CONSTRAINTS;591
8.15.3;III. OPTIMIZATION WITH NONNEGATIVE CONSTRAINTS;593
8.15.4;REFERENCES;595
8.16;CHAPTER 52. STABILITY OF A NONLINEAR DELAY DIFFERENCE EQUATION IN POPULATION
DYNAMICS;598
8.16.1;I. INTRODUCTION;598
8.16.2;II. POPULATION WITH NONOVERLAPPING GENERATIONS;599
8.16.3;III. AGE-STRUCTURED POPULATION;601
8.16.4;I V . A FIN WHALE POPULATION;603
8.16.5;ACKNOWLEDGMENTS;605
8.16.6;REFERENCES;605
8.17;CHAPTER 53. BILINEAR APPROXIMATION AND HARMONIC ANALYSIS
OF ANALYTIC CONTROL/ANALYTIC STATE SYSTEMS;608
8.17.1;INTRODUCTION;608
8.17.2;I. BILINEAR APPROXIMATION;608
8.17.3;II. HARMONIC ANALYSIS OF ANALYTIC CONTROL/BILINEAR SYSTEMS;613
8.17.4;III. DIFFUSION-REACTION AND OUTPUT FUNCTIONALS;621
8.17.5;CONCLUDING REMARKS;623
8.17.6;REFERENCES;624
8.18;CHAPTER 54. PERSISTENT SETS VIA LYAPUNOV FUNCTIONS;626
8.18.1;ABSTRACT;626
8.19;CHAPTER 55. SPATIAL HETEROGENEITY AND THE STABILITY
OF PREDATOR-PREY SYSTEMS: POPULATION CYCLES;628
8.19.1;I. INTRODUCTION;628
8.19.2;II.
THE SIMPLEST MODEL;630
8.19.3;III. AN 'AGE DEPENDENT'
MODEL;631
8.19.4;IV. FUNCTIONAL RESPONSE OF PREDATORS;632
8.19.5;V. DISCUSSION;634
8.19.6;ACKNOWLEDGEMENTS;635
8.19.7;APPENDIX;635
8.19.8;REFERENCES;637
8.20;CHAPTER 56. CAUCHY SYSTEM FOR THE NONLINEAR BOUNDARY
VALUE PROBLEM OF A SHALLOW ARCH;640
8.20.1;I. INTEGRO-DIFFERENTIAL EQUATION FOR A SHALLOW ARCH;640
8.20.2;II. BOUNDARY CONDITIONS;641
8.20.3;III. DERIVATION OF AN INITIAL VALUE PROBLEM;642
8.20.4;IV. NUMERICAL METHOD;646
8.20.5;V. REFERENCES;646
8.21;CHAPTER 57. A SUMMARY OF RECENT EXPERIMENTSTO COMPUTE THE TO
POLOGICAL DEGREE;648
8.21.1;I. THE CONCEPT AND THE COMPUTATION FORMULA;649
8.21.2;11. THE ALGORITHM;650
8.21.3;III. PERFORMANCE OF THE ALGORITHM AND SCOPE OF APPLICATION;651
8.21.4;IV. NUMERICAL RESULTS;652
8.21.5;REFERENCES;653
8.22;CHAPTER 58. COMPUTATION OF EIGENVALUES/EIGENFUNCTIONS
FOR TWO POINT BOUNDARY VALUE PROBLEMS;656
8.22.1;I. INTRODUCTION;656
8.22.2;II. STATEMENT OF PROBLEM;657
8.22.3;III. FINAL BOUNDARY CONDITION FUNCTION;659
8.22.4;IV. NONLINEAR EQUATION SOLVER;664
8.22.5;V. ERROR TOLERANCES;666
8.22.6;VI. ORTHONORMALIZATION;668
8.22.7;VII. EXAMPLES;670
8.22.8;REFERENCES;676
8.23;CHAPTER 59. A CONTINUUM MODEL APPROPRIATE FOR NONLINEAR ANALYSIS OF THE SOLIDIFICATION OF A PURE METAL;678
8.23.1;I. INTRODUCTION;678
8.23.2;II. SURFACE FREE ENERGY AND THE NONLINEAR SOLIDIFICATION MODEL;680
8.23.3;III. THE GOVERNING NONDIMENSIONALIZED EQUATIONS FOR A PROTOTYPE
PROBLEM;684
8.23.4;REFERENCES;688
8.24;CHAPTER 60. QUALITATIVE DYNAMICS FROM ASYMPTOTIC
EXPANSIONS;690
8.24.1;REFERENCES;694
8.25;CHAPTER 61. A SECOND STAGE EDDY-VISCOSITY MODEL FOR TURBULENT FLUID FLOWS: OR,
A UNIVERSAL STATISTICAL TOOL?;696
8.25.1;I. INTRODUCTION;696
8.25.2;II.
PODT-SAS HISTORY;698
8.25.3;III.
LIST OF CONJECTURED FIELDS OF APPLICABILITY OF PODT-SAS;705
8.25.4;IV. A SECOND-STAGE EDDY-VISCOSITY MODEL FOR
TURBULENTS:;706
8.25.5;V. CLOSURE;706
8.25.6;REFERENCES;707
8.26;CHAPTER 62. FIXED POINT ITERATIONS USING INFINITE MATRICES;710
8.26.1;REFERENCES;723
8.27;CHAPTER 63. A NUMERICAL METHOD FOR SOLVING
THE HAMILTON-JACOBI INITIAL VALUE PROBLEM;726
8.27.1;ABSTRACT;726
8.28;CHAPTER 64. DIFFERENTIAL GEOMETRIC METHODS
IN NONLINEAR PROGRAMMING;728
8.28.1;I. INTRODUCTION;728
8.28.2;II. FEASIBILITY-IMPROVING GRADIENT ACUTE PROJECTION METHODS;729
8.28.3;III. CONTINUOUS ANALOGUE OF
FIGAP METHOD;737
8.28.4;REFERENCES;739
8.29;CHAPTER 65. LIMITING EQUATIONS AND TOTAL STABILITY;742
8.29.1;I .;742
8.29.2;II.;743
8.29.3;REFERENCES;746