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Differential Equations with Small Parameters and Relaxation Oscillations
Band 13

Differential Equations with Small Parameters and Relaxation Oscillations

Aus der Reihe Mathematical Concepts and Methods in Science and Engineering Band 13

49,99 €

inkl. MwSt, Versandkostenfrei

Beschreibung

Details

Einband

Taschenbuch

Erscheinungsdatum

27.12.2012

Verlag

Springer Us

Seitenzahl

228

Maße (L/B/H)

22,9/15,2/1,4 cm

Gewicht

360 g

Auflage

Softcover reprint of the original 1st edition 1980

Sprache

Englisch

ISBN

978-1-4615-9049-1

Beschreibung

Details

Einband

Taschenbuch

Erscheinungsdatum

27.12.2012

Verlag

Springer Us

Seitenzahl

228

Maße (L/B/H)

22,9/15,2/1,4 cm

Gewicht

360 g

Auflage

Softcover reprint of the original 1st edition 1980

Sprache

Englisch

ISBN

978-1-4615-9049-1

Herstelleradresse

Springer-Verlag GmbH
Tiergartenstr. 17
69121 Heidelberg
DE

Email: ProductSafety@springernature.com

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  • Differential Equations with Small Parameters and Relaxation Oscillations
  • I. Dependence of Solutions on Small Parameters. Applications of Relaxation Oscillations.- 1. Smooth Dependence. Poincaré’s Theorem.- 2. Dependence of Solutions on a Parameter, on an Infinite Time Interval.- 3. Equations with Small Parameters Multiplying Derivatives.- 4. Second-Order Systems. Fast and Slow Motion. Relaxation Oscillations.- 5. Systems of Arbitrary Order. Fast and Slow Motion. Relaxation Oscillations.- 6. Solutions of the Degenerate Equation System.- 7. Asymptotic Expansions of Solutions with Respect to a Parameter.- 8. A Sketch of the Principal Results.- II. Second-Order Systems. Asymptotic Calculation of Solutions.- 1. Assumptions and Definitions.- 2. The Zeroth Approximation.- 3. Asymptotic Approximations on Slow-Motion Parts of the Trajectory.- 4. Proof of the Asymptotic Representations of the Slow-Motion Part.- 5. Local Coordinates in the Neighborhood of a Junction Point.- 6. Asymptotic Approximations of the Trajectory on the Initial Part of a Junction.- 7. The Relation between Asymptotic Representations and Actual Trajectories in the Initial Junction Section.- 8. Special Variables for the Junction Section.- 9. A Riccati Equation.- 10. Asymptotic Approximations for the Trajectory in the Neighborhood of a Junction Point.- 11. The Relation between Asymptotic Approximations and Actual Trajectories in the Immediate Vicinity of a Junction Point.- 12. Asymptotic Series for the Coefficients of the Expansion Near a Junction Point.- 13. Regularization of Improper Integrals.- 14. Asymptotic Expansions for the End of a Junction Part of a Trajectory.- 15. The Relation between Asymptotic Approximations and Actual Trajectories at the End of a Junction Part.- 16. Proof of Asymptotic Representations for the Junction Part.- 17. Asymptotic Approximations of the Trajectory on the Fast-Motion Part.- 18. Derivation of Asymptotic Representations for the Fast-Motion Part.- 19. Special Variables for the Drop Part.- 20. Asymptotic Approximations of the Drop Part of the Trajectory.- 21. Proof of Asymptotic Representations for the Drop Part of the Trajectory.- 22. Asymptotic Approximations of the Trajectory for Initial Slow-Motion and Drop Parts.- III. Second-Order Systems. Almost-Discontinuous Periodic solutions.- 1. Existence and Uniqueness of an Almost-Discontinuous Periodic Solution.- 2. Asymptotic Approximations for the Trajectory of a Periodic Solution.- 3. Calculation of the Slow-Motion Time.- 4. Calculation of the Junction Time.- 5. Calculation of the Fast-Motion Time.- 6. Calculation of the Drop Time.- 7. An Asymptotic Formula for the Relaxation-Oscillation Period.- 8. Van der Pol’s Equation. Dorodnitsyn’s Formula.- IV. Systems of Arbitrary Order. Asymptotic Calculation of Solutions.- 1. Basic Assumptions.- 2. The Zeroth Approximation.- 3. Local Coordinates in the Neighborhood of a Junction Point.- 4. Asymptotic Approximations of a Trajectory at the Beginning of a Junction Section.- 5. Asymptotic Approximations for the Trajectory in the Neighborhood of a Junction Point.- 6. Asymptotic Approximation of a Trajectory at the End of a Junction Section.- 7. The Displacement Vector.- V. Systems of Arbitrary Order. Almost-Discontinuous Periodic Solutions.- 1. Auxiliary Results.- 2. The Existence of an Almost-Discontinuous Periodic Solution. Asymptotic Calculation of the Trajectory.- 3. An Asymptotic Formula for the Period of Relaxation Oscillations.- References.