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  • Produktbild: Partial Differential Equations
  • Produktbild: Partial Differential Equations
Band 15

Partial Differential Equations New Methods for Their Treatment and Solution

Aus der Reihe Mathematics and Its Applications

Fr. 137.00

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

19.10.2011

Verlag

Springer Netherland

Seitenzahl

290

Maße (L/B/H)

23.5/15.5/1.7 cm

Gewicht

492 g

Auflage

Softcover reprint of the original 1st ed. 1985

Sprache

Englisch

ISBN

978-94-010-8804-6

Beschreibung

Rezension

`
As a rich source of ideas the text should be highly recommended to mature students in appropriate fields and to research workers.
'

Zentralblatt für Mathematik (1986)

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

19.10.2011

Verlag

Springer Netherland

Seitenzahl

290

Maße (L/B/H)

23.5/15.5/1.7 cm

Gewicht

492 g

Auflage

Softcover reprint of the original 1st ed. 1985

Sprache

Englisch

ISBN

978-94-010-8804-6

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

Email: ProductSafety@springernature.com

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  • Produktbild: Partial Differential Equations
  • Produktbild: 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 Itô 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.