Differential Equations with Small Parameters and Relaxation Oscillations / by E. F. Mishchenko, N. Kh. Rozov
(Mathematical Concepts and Methods in Science and Engineering ; 13)
| データ種別 | 電子ブック |
|---|---|
| 出版情報 | Boston, MA : Springer US : Imprint: Springer , 1980 |
| 本文言語 | 英語 |
| 大きさ | X, 228 p : online resource |
書誌詳細を非表示
| 内容注記 | 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 |
|---|---|
| 著者標目 | *Mishchenko, E. F. author Rozov, N. Kh author SpringerLink (Online service) |
| 件 名 | LCSH:Science FREE:Science, general FREE:Science, general |
| 分 類 | DC23:500 |
| 巻冊次 | ISBN:9781461590477 
|
| ISBN | 9781461590477 |
| URL | http://dx.doi.org/10.1007/978-1-4615-9047-7 |
目次／あらすじ