Partial Differential Equations

I/Monotone Convergence and Positive Operators.- 1. Introduction.- 2. Monotone Operators.- 3. Monotonicity.- 4. Convergence.- 5. Differential Equations with Initial Conditions.- 6. Two-Point Boundary Conditions.- 7. Nonlinear Heat Equation.- 8. The Nonlinear Potential Equation.- Bibliography and Comments.- II/Conservation.- 1. Introduction.- 2. Analytic and Physical Preliminaries.- 3. The Defining Equations.- 4. Limiting Differential Equations.- 5. Conservation for the Discrete Approximation.- 6. Existence of Solutions for Discrete Approximation.- 7. Conservation for Nonlinear Equations.- 8. The Matrix Riccati Equation.- 9. Steady-State Neutron Transport with Discrete Energy Levels.- 10. Analytic Preliminaries.- 11. Reflections, Transmission, and Loss Matrices.- 12. Existence and Uniqueness of Solutions.- 13. Proof of Conservation Relation.- 14. Proof of Nonnegativity.- 15. Statement of Result.- Bibliography and Comments.- III / Dynamic Programming and Partial Differential Equations.- 1. Introduction.- 2. Calculus of Variations as a Multistage Decision Process.- 3. A New Formalism.- 4. Layered Functionals.- 5. Dynamic Programming Approach.- 6. Quadratic Case.- 7. Bounds.- Bibliography and Comments.- IV / The Euler-Lagrange Equations and Characteristics.- 1. Introduction.- 2. Preliminaries.- 3. The Fundamental Relations of the Calculus of Variations.- 4. The Variational Equations.- 5. The Eulerian Description.- 6. The Lagrangian Description.- 7. The Hamiltonian Description.- 8. Characteristics.- Bibliography and Comments.- V / Quasilinearization and a New Method of Successive Approximations.- 1. Introduction.- 2. The Fundamental Variational Relation.- 3. Successive Approximations.- 4. Convergence.- Bibliography and Comments.- VI / The Variation of Characteristic Values and Functions.- 1. Introduction.- 2. Variational Problem.- 3. Dynamic Programming Approach.- 4. Variation of the Green's Function.- 5. Justification of Equating Coefficients.- 6. Change of Variable.- 7. Analytic Continuation.- 8. Analytic Character of Green's Function.- 9. Alternate Derivation of Expression for ?(x).- 10. Variation of Characteristic Values and Characteristic Functions.- 11. Matrix Case.- 12. Integral Equations.- Bibliography and Comments.- VII / The Hadamard Variational Formula.- 1. Introduction.- 2. Preliminaries.- 3. A Minimum Problem.- 4. A Functional Equation.- 5. The Hadamard Variation.- 6. Laplace-Beltrami Operator.- 7. Inhomogeneous Operator.- Bibliography and Comments.- VIII / The Two-Dimensional Potential Equation.- 1. Introduction.- 2. The Euler-Lagrange Equation.- 3. Inhomogeneous and Nonlinear Cases.- 4. Green's Function.- 5. Two-Dimensional Case.- 6. Discretization.- 7. Rectangular Region.- 8. Associated Minimization Problem.- 9. Approximation from Above.- 10. Discussion.- 11. Semidiscretization.- 12. Solution of the Difference Equations.- 13. The Potential Equation.- 14. Discretization.- 15. Matrix-Vector Formulation.- 16. Dynamic Programming.- 17. Recurrence Equations.- 18. The Calculations.- 19. Irregular Regions.- Bibliography and Comments.- IX / The Three-Dimensional Potential Equation.- 1. Introduction.- 2. Discrete Variational Problems.- 3. Dynamic Programming.- 4. Boundary Conditions.- 5. Recurrence Relations.- 6. General Regions.- 7. Discussion.- Bibliography and Comments.- X / The Heat Equation.- 1. Introduction.- 2. The One-Dimensional Heat Equation.- 3. The Transform Equation.- 4. Some Numerical Results.- 5. Multidimensional Case.- Bibliography and Comments.- XI / Nonlinear Parabolic Equations.- 1. Introduction.- 2. Linear Equation.- 3 The Non-negativity of the Kernel.- 4. Monotonicity of Mean Values.- 5. Positivity of the Parabolic Operator.- 6. Nonlinear Equations.- 7. Asymptotic Behavior.- 8. Extensions.- Bibliography and Comments.- XII / Differential Quadrature.- 1. Introduction.- 2. Differential Quadrature.- 3. Determination of Weighting Coefficients.- 4. Numerical Results for First Order Problems.- 5. Systems of Nonlinear Partial Differential Equations.- 6. Higher Order Problems.- 7. Error Representation.- 8. Hodgkin-Huxley Equation.- 9. Equations of the Mathematical Model.- 10. Numerical Method.- 11. Conclusion.- Bibliography and Comments.- XIII / Adaptive Grids and Nonlinear Equations.- 1. Introduction.- 2. The Equation ut =-uux.- 3. An Example.- 4. Discussion.- 5. Extension.- 6. Higher Order Approximations.- Bibliography and Comments.- XIV / Infinite Systems of Differential Equations.- 1. Introduction.- 2. Burgers' Equation.- 3. Some Numerical Examples.- 4. Two-Dimensional Case.- 5. Closure Techniques.- 6. A Direct Method.- 7. Extrapolation.- 8. Difference Approximations.- 9. An Approximating Algorithm.- 10. Numerical Results.- 11. Higher Order Approximation.- 12. Truncation.- 13. Associated Equation.- 14. Discussion of Convergence of u (N).- 15. The Fejer Sum.- 16. The Modified Truncation.- Bibliography and Comments.- XV / Green's Functions.- 1. Introduction.- 2. The Concept of the Green's Function.- 3. Sturm-Liouville Operator.- 4. Properties of the Green's Function for the Sturm-Liouville Equation.- 5. Properties of the ? Function.- 6. Distributions.- 7. Symbolic Functions.- 8. Derivative of Symbolic Functions.- 9. What Space Are We Considering?.- 10. Boundary Conditions.- 11. Properties of Operator L.- 12. Adjoint Operators.- 13. n-th Order Operators.- 14. Boundary Conditions for the Sturm-Liouville Equation.- 15. Green's Function for Sturm-Liouville Operator.- 16. Solution of the Inhomogeneous Equation.- 17. Solving Non-Homogeneous Boundary Conditions.- 18. Boundary Conditions Specified on Finite Interval [a, b].- 19. Scalar Products.- 20. Use of Green's Function to Solve a Second-order Stochastic Differential Equation.- 21. Use of Green's Function in Quantum Physics.- 22. Use of Green's Functions in Transmission Lines.- 23. Two-Point Green's Functions - Generalization to n-point Green's Functions.- 24. Evaluation of Arbitrary Functions for Nonhomogeneous Boundary Conditions by Matrix Equations.- 25. Mixed Boundary Conditions.- 26. Some General Properties.- 1. Nonnegativity of Green's Functions and Solutions.- 2. Variation-Diminishing Properties of Green's Functions.- Notes.- XVI / Approximate Calculation of Green's Functions.- XVII / Green's Functions for Partial Differential Equations.- 1. Introduction.- 2. Green's Functions for Multidimensional Problems in Cartesian Coordinates.- 3. Green's Functions in Curvilinear Coordinates.- 4. Properties of ? Functions for Multi-dimensional Case.- XVIII / The Ito Equation and a General Stochastic Model for Dynamical Systems.- XIX / Nonlinear Partial Differential Equations and the Decomposition Method.- 1. Parametrization and the An Polynomials.- 2. Inverses for Non-simple Differential Operators.- 3. Multidimensional Green's Functions by Decomposition Method.- 4. Relationships Between Green's Functions and the Decomposition Method for Partial Differential Equations.- 5. Separable Systems.- 6. The partitioning Method of Butkovsky.- 7. Computation of the An.- 8. The Question of Convergence.