Partial Differential Equations

The purpose of this book is to present some new methods in the treatment of partial differential equations. Some of these methods lead to effective numerical algorithms when combined with the digital computer. Also presented is a useful chapter on Green's functions which generalizes, after an introduction, to new methods of obtaining Green's functions for partial differential operators. Finally some very new material is presented on solving partial differential equations by Adomian's decomposition methodology. This method can yield realistic computable solutions for linear or non­ linear cases even for strong nonlinearities, and also for deterministic or stochastic cases - again even if strong stochasticity is involved. Some interesting examples are discussed here and are to be followed by a book dealing with frontier applications in physics and engineering. In Chapter I, it is shown that a use of positive operators can lead to monotone convergence for various classes of nonlinear partial differential equations. In Chapter II, the utility of conservation technique is shown. These techniques are suggested by physical principles. In Chapter III, it is shown that dyn~mic programming applied to variational problems leads to interesting classes of nonlinear partial differential equations. In Chapter IV, this is investigated in greater detail. In Chapter V, we show. that the use of a transformation suggested by dynamic programming leads to a new method of successive approximations.

Contenu

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. Pr…