Methods of Nonlinear Analysis

Name: Methods of Nonlinear Analysis
Brand: Academic Press
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BELLMAN(Herausgeber*in)

Academic Press

Erschienen am 14. Mai 2014

281 Seiten

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Front Cover
Methods of Nonlinear Analysis
Copyright Page
Preface
Contents
Contents of Volume I
Chapter 9. Upper and Lower Bounds via Duality
9.1. Introduction
9.2. Guiding Idea
9.3. A Simple Identity
9.4. Quadratic Functional: Scalar Case
9.5. min u J = max v H
9.6. The Functional l t o [u'2 + g(u)] dt
9.7. Geometric Aspects
9.8. Multidimensional Case
9.9. The Rayleigh-Ritz Method
9.10. Alternative Approach
9.11. J(u) = lto [u'2 + f(t)u2] dt
General f(t)
9.12. Geometric Aspects
Miscellaneous Exercises
Bibliography and Comments
Chapter 10. Caplygin's Method and Differential Inequalities
10.1. Introduction
10.2. The Caplygin Method
10.3. The Equation u' & a(t)u + f (t)
10.4. The Linear Differential Inequality L(u) & f ( t )
10.5. Elementary Approach
10.6. An Integral Identity
10.7. Strengthening of Previous Result
10.8. Factorization of the Operator
10.9. Alternate Proof of Monotonicity
10.10. A Further Condition
10.11. Two-point Boundary Conditions
10.12. Variational Approach
10.13. A Related Parabolic Partial Differential Equation
10.14. Nonnegativity of u(t, s)
10.15. Limiting Behavior
10.16. Limiting Behavior: Energy Inequalities
10.17. Monotonicity of Maximum
10.18. Lyapunov Functions
10.19. Factorization of the nth-order Linear Operator
10.20. A Result for the nth-order Linear Differential Equation
10.21. An Example
10.22. Linear Systems
10.23. Partial Differential Equation-I
10.24. Partial Differential Equation-II
Miscellaneous Exercises
Bibliography and Comments
Chapter 11. Quasilinearization
11.1. Introduction
11.2. The Riccati Equation
11.3. Explicit Representation
11.4. Successive Approximations and Monotone Convergence
11.5. Maximum Interval of Convergence
11.6. Dini's Theorem and Uniform Convergence
1 1.7. Newton-Raphson-Kantorovich Approximation
11.8. Quadratic Convergence
11.9. Upper Bounds
11.10. u' = g(u, t )
11.11. Random Equation
11.12. Upper and Lower Bounds
1 I. 13. Asymptotic Behavior
11.14. Multidimensional Riccati Equation
11.15. Two-point Boundary Value Problems
11.16. Maximum Interval of Convergence
11.1 7. Quadratic Convergence
11.18. Discussion
11.19. Computational Feasibility
11.20. Elliptic Equations
11.21. Parabolic Equations
1 1.22. Minimum and Maximum Principles
Miscellaneous Exercises
Bibliography and Comments
Chapter 12. Dynamic Programming
12.1. Introduction
12.2. Multistage Processes
12.3. Continuous Version
12.4. Multistage Decision Processes
12.5. Stochastic and Adaptive Processes
12.6. Functional Equations
12.7. Infinite Stage Process
12.8. Policy
12.9. Approximation in Policy Space
12.10. Discussion
12.11. Calculus of Variations as a Multistage Decision Process
12.12. A New Formalism
12.13. The Principle of Optimality
12.14. Quadratic Case
12.15. Multidimensional Case
12.1 6. Computational Feasibility
12.17. Stability
12.18. Computational Feasibility: General Case-I
12.19. Computational Feasibility: General Case-II
12.20. The Curse of Dimensionality
12.21. Constraints
12.22. Two-point Boundary Value Problems
12.23, Rigorous Aspects
12.24. The Equation ut = .p(x, u,)
12.24. The Equation ut = f(x, ux)
12.25. The Equation ut = q(x, us=)
12.25. The Equation ut = f(x, uxx)
12.26. Generalized Semigroup Theory and Nonlinear Equations
12.27. Ill-conditioned Systems
12.28. Adaptive Polygonal Approximation
12.29. Partial Differential Equations
12.30. Successive Approximations to Reduce Dimensionality
Miscellaneous Exercises
Bibliography and Comments
Chapter 13. Invariant Imbedding
13.1. Introduction
13.2. Maximum Altitude
13.3. Functional Equation Approach
13.4. Invariant Imbedding
13.5. Inhomogeneous Medium
13.6. Computational Aspects
13.7. Two-point Boundary Value Problems
13.8. A Simple Transport Process
13.9. Invariant Imbedding: Particle Counting
13.10. Perturbation Analysis
13.11. Linear Systems
13.1 2. Linear Transport Process
13.13. Reflection and Transmission Matrices
13.14. Invariant Imbedding
13.15. Equivalence of Analytic Formulations
13.16. Conservation Relation
13.17. Internal Fluxes
13.18. Asymptotic Behavior
13.19. Multidimensional Case
13.20. Asymptotic Phase
13.21. Wave Propagation
13.22. The Bremmer Series
13.23. Localization Principles
13.24. Localization for Wave Motion
13.25. The Classical Solution to Wave Propagation through an Inhomogeneous Medium
13.26. Riccati Equation
13.27. Invariant Imbedding
13.28. Functional Equations
13.29. Random Walk
13.30. Classical Formulation
13.31. Invariant Imbedding
13.32. Determination of u(a, a + 1 )
Miscellaneous Exercises
Bibliography and Comments
Chapter 14. The Theory of Iteration
14.1. Introduction
14.2. Iteration
14.3. Abel-Schröder Functional Equation
14.4. Formal Analysis
14.5. Koenig's Representation
14.6. Majorization
14.7. Refined Asymptotic Behavior
14.8. Stochastic Effects
14.9. Stochastic Iteration
14.10. Stochastic Matrices
14.11. Random Initial Values
14.12. Imbedding
14.13. Differential Equations
14.14. Associated Partial Differential Equation
14.15. Multidimensional Case
14.16. A Counterexample
14.17. Statement of Result
14.18. Differential Equations
14.19. Commensurable Characteristic Roots
14.20. Two-point Boundary Values
14.21. Constant Right-hand Incident Flux
14.22. Associated Partial Differential Equation
14.23. Branching Processes
14.24. Closure of Operations
14.25. Number of Arithmetic Operations
Miscellaneous Exercises
Bibliography and Comment
Chapter 15. Infinite Systems of Ordinary Differential Equations and Truncation
15.1. Introduction
15.2. Ordinary Differential Equations and Difference Methods
15.3. Stability Considerations
15.4. Round-off Error
15.5. Linear Partial Differential Equations
15.6. Partial Discretization
15.7. Preservation of Properties
15.8. Orthonormal Expansions and Infinite Systems
15.9. Truncation
15.10. Associated Equation
15.11. Discussion of Convergence of u(n)
15.12. The Fejer Sum
15.13. Modified Partial Differential Equation
15.14. Modified Truncation
15.15. Infinite Systems of Ordinary Differential Equations
15.16. Proof of Theorem
15.17. The "Principe des Réduites
15.18. Random Walk
15.19. The "Principe des RCduites
15.20. Monotone Convergence
15.21. Existence of a Solution
15.22. Closure of the Process
Miscellaneous Exercises
Bibliography and Comment
Chapter 16. Integral and Differential Quadrature
16.1. Introduction
16.2. Laplace Transform
16.3. Simplifying Properties of the Laplace Transform
16.4. Examples
16.5. Complex Inversion Formula
16.6. Abelian and Tauberian Results
16.7. Interpolation Technique
16.8. Selective Calculation
16.9. Impossibility
16.10. Quadrature Techniques
16.11. Numerical Inversion of the Laplace Transform
16.12. Explicit Inverse of (xik)
16.13. Example of Ill-conditioning
16.14. Nonlinear Equations-I
16.15. Nonlinear Equations-II
16.16. Quasilinearization
16.1 7. Tychonov Regularization
16.18. Self-consistent Methods
16.19. An Imbedding Technique
16.20. Nonlinear Summability
16.21. Time-dependent Transport Process
16.22. Radiative Transfer
16.23. Radiative Transfer via Invariant Imbedding
16.24. Time-dependent Case
16.25. Error Analysis
16.26. Differential Quadrature
16.27. Application to Partial Differential Equations
16.28. Identification Problems
16.29. Quasilinearization
16.30. Differential Quadrature
Miscellaneous Exercises
Bibliography and Comment
Author Index
Subject Index

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