Introduction to asymptotic methods /
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| Author / Creator: | Awrejcewicz, J. (Jan) |
|---|---|
| Imprint: | Boca Raton, FL : Chapman & Hall/CRC, 2006. |
| Description: | xviii, 251 p. : ill. ; 25 cm. |
| Language: | English |
| Series: | CRC series--modern mechanics and mathematics ; [5] |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/6094888 |
Table of Contents:
- Introduction
- 1. Elements of mathematical modeling
- 1.1. Structure of a mathematical model
- 1.2. Examples of reducing problems to a dimensionless form
- 1.3. Mathematical model adequacy and properties. Regular and singular perturbations
- 2. Expansion of functions and mathematical methods
- 2.1. Expansions of elementary functions into power series
- 2.1.1. Newton's binomial
- 2.1.2. Taylor and Maclaurin series
- 2.1.3. Estimation of approximating functions values
- 2.1.4. Estimation of approximating values of definite integrals
- 2.1.5. Approximating solution of Cauchy problem for ordinary differential equations
- 2.2. Mathematical methods of perturbations
- 2.2.1. Perturbation along a parameter and coordinates. Classical method of a small parameter
- 2.2.2. Comparison of infinitely small and infinitely large functions. Scaling functions. Magnitude order quantities
- 2.2.3. Asymptotical sequences and decompositions
- 2.2.4. Nonuniform asymptotic decompositions
- 2.2.5. Operations on asymptotic decompositions
- 2.2.6. Asymptotic series. Comparison of asymptotic and convergent series. Advantages of application of asymptotic series and decompositions
- Exercises
- 3. Regular and singular perturbations
- 3.1. Introduction. Asymptotic approximation with respect to a parameter
- 3.2. Nonuniformities of a classical perturbation approach
- 3.3. Method of "elongated" parameters
- 3.4. Method of deformed variables
- 3.5. Method of scaling and full approximation
- 3.5.1. Deformation of one independent variable
- 3.5.2. Deformation of two independent variables
- 3.5.3. Method of full approximation
- 3.6. Multiple scale methods
- 3.6.1. Introduction
- 3.6.2. Derivative decomposition along one and two variables
- 3.6.3. Application to the problems of vibrations
- 3.7. Variations of arbitrary constants
- 3.8. Averaging methods
- 3.8.1. Methods of Van der Pol and Krylov-Bogolubov-Mitropolskiy (KBM)
- 3.8.2. Duffing's problem and the averaging procedure
- 3.9. Matching asymptotic decompositions
- 3.9.1. Fundamental notions and terminology
- 3.9.2. Example with a boundary layer
- 3.9.3. Fundamental rules and order of matching
- 3.9.4. Construction of matched asymptotic expansion
- 3.9.5. Example with a singularity
- 3.9.6. On the choice of internal variables
- 3.10. On the sources of nonuniformities
- 3.11. On the influence of initial conditions
- 3.12. Analysis of strongly nonlinear dynamical problems
- 3.13. A few perturbation parameters
- Exercises
- 4. Wave-impact processes
- 4.1. Definition of a cylinder-like piston wave
- 4.1.1. Defining the problem, its solution and analysis
- 4.1.2. Nonlinear solution in the vicinity of the piston
- 4.1.3. Nonlinear solution in the vicinity of the front of the impact wave
- 4.1.4. Methods of strained coordinates and renormalization
- 4.1.5. Effectiveness of various asymptotic methods
- 4.2. One-dimensional nonstationary nonlinear waves
- 4.2.1. Formulation of the problem and its solution
- 4.2.2. Renormalization method and singularities
- 4.2.3. Analytical method of characteristics
- 4.2.4. Multiple scales method
- 5. Pade approximations
- 5.1. Determination and characteristics of Pade approximations
- 5.2. Application of Pade approximations
- 5.2.1. Simple examples
- 5.2.2. Supersonic flow round a thin cone in circumsonic regime
- 5.2.3. Damping of the ball-shaped waves of pressure in a free space and in a tube
- 5.2.4. Analysis of the "blow-up" phenomenon
- 5.2.5. Homoclinic orbits
- 5.2.6. Vibrations of nonlinear system with nonlinearity close to sign (x)
- Exercises
- 6. Averaging of ribbed plates
- 6.1. Averaging in the theory of ribbed plates
- 6.2. Kantorovich-Vlasov-type methods
- 6.2.1. Kantorowich-Vlasov method (KVM)
- 6.2.2. Vindiner method (VM)
- 6.2.3. Method of variational iterations (MVI)
- 6.2.4. Agranowsky-Baglay-Smirnov method (ABSM)
- 6.2.5. Combined method (CM)
- 6.2.6. Kantorovich-Vlasov method with the amendment
- 6.2.7. Vindiner method with the amendment
- 6.2.8. Vindiner method and variational iterations
- 6.3. Transverse vibrations of rectangular plates
- 6.4. Deflections of rectangular plates
- 7. Chaos foresight
- 7.1. The analyzed system
- 7.2. Melnikov-Gruendler's approach
- 7.3. Melnikov-Gruendler function
- 7.4. Numerical results
- 8. Continuous approximation of discontinuous systems
- 8.1. An illustrative example
- 8.2. Higher dimensional systems
- 9. Nonlinear dynamics of a swinging oscillator
- 9.1. Parametrical form of canonical transformations
- 9.2. Function derivative
- 9.3. Invariant normalization of Hamiltonians
- 9.4. Algorithm of invariant normalization with the help of parametric transformations
- 9.5. Algorithm of invariant normalization for asymptotical determination of the Poincare series
- 9.6. Examples of asymptotical solutions
- 9.7. A swinging oscillator
- 9.8. Normal form
- 9.9. Normal form integral
- References
- Index
