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Semi-Analytical and Semi-Numerical Methods in Celestial Mechanics

Slawomir Breiter
1997 International Astronomical Union Colloquium  
The role of numerical analysis tools in analytical theories is briefly discussed.  ...  Four main ideas behind semi-analytical theories are identified and reviewed: series contraction, analytical support of numerical integration, piecewise solutions, and numerical evaluation of action-angle  ...  Transforming the equations of motion by averaging, one obtains a slowly evolving system; the system which would cause some problems in further analytical treatment, but which can be integrated numerically  ... 
doi:10.1017/s0252921100046881 fatcat:22ojbnkjfrfttpqceobkgqt4r4

Page 135 of Mathematical Reviews Vol. 51, Issue 1 [page]

1976 Mathematical Reviews  
Howes, On approximate and true solutions of a nonlinear singular perturbation problem (p. 820); Stig-Olof Londen, On the variation of the solutions of a nonlinear integral equation (p. 821); D. L.  ...  Miller, Some fundamental theory of Volterra integral Ae vee (pp. 568-579); J. A. Nohel and D. F.  ... 

Page 2621 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
For some problems the solution of this system can be found in a special form. Then a first integral F can be calculated by integration.  ...  In fact, no general theory is developed, but to show how this perturbation 2000d:34113 method works, the method is applied to the van der Pol equation, a Mathieu equation, and an equation for a harmonic  ... 

Page 1174 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
These exclusions correspond to the small divisors in the perturbation series and are the reason that one can only prove that the perturbed system behaves like the unperturbed one on some subset of all  ...  On the one hand is the theory of completely integrable systems. These are Hamiltonian systems for which there exist canonical variables in which the equa- tions of motion are explicitly integrable.  ... 

Page 1702 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
MA); Ono, Kinya (1-BOST; Boston, MA) On the application of geometric singular perturbation theory to some classical two-point boundary value problems.  ...  Summary: “In this tutorial, we illustrate how geometric singular perturbation theory provides a complementary dynamical systems- based approach to the method of matched asymptotic expansions for some classical  ... 

Page 136 of Mathematical Reviews Vol. , Issue 84a [page]

1984 Mathematical Reviews  
Authors’ summary: “We generalize the theory of asymptotic integration of singularly perturbed problems, developed earlier for parabolic, hyperbolic, and elliptic equations [Lomov, Dokl. Akad.  ...  Lutz, Robert; Sari, Tewfik 84a:34057 Applications of nonstandard analysis to boundary value problems in singular perturbation theory.  ... 

Page 2668 of Mathematical Reviews Vol. , Issue 84g [page]

1984 Mathematical Reviews  
In some physical problems, the perturbations of impulsive type are more realistic. In this way, the notion of a measure differential equation becomes very natural.  ...  The problem of when an indefinite integral of an almost periodic function is almost periodic and related questions are discussed in Chapter 6.  ... 

On generators of new methods of the perturbation theory

E.A. Grebenikov
1996 Symposium - International astronomical union  
In this article are discussed classical and modern interpretations of the perturbation theory methods.  ...  We call Ζ(ζ,/,μ) and zo the generators of the perturbation theory for problem (1) and equation (4) the generating equation for equation (1).  ...  differs from the classical perturbation theory in an essential point: the determination of perturbation /,μ) from the equation (7) does not require the preliminary solving of a generating equation (  ... 
doi:10.1017/s0074180900127536 fatcat:pvn35vooobf4hjwnlgikqpmqyy

Page 1699 of Mathematical Reviews Vol. 57, Issue 5 [page]

1979 Mathematical Reviews  
S. 57 #13105 Some results and problems of the theory of the stability of the ~— of linear almost periodic equations with aftereffect. (Russian Differential equations with deviating argument (Proc.  ...  There is also an introductory chapter on the theory of linear integral equations.  ... 

Page 2416 of Mathematical Reviews Vol. , Issue 86f [page]

1986 Mathematical Reviews  
Lomov, Integration of singularly perturbed problems by means of series that converge not only in the asymptotic sense but also in the usual sense (Russian) (pp. 230-235, 445); A. K.  ...  Vencovské, On a problem in the theory of linear differential equations with quasiperiodic coefficients (pp. 214-217, 444); E.  ... 

Nonlinear waves in integrable and nonintegrable systems (Mathematical Modeling and Computation 16) By Jianke Yang

Karima Khusnutdinova
2015 Bulletin of the London Mathematical Society  
Such nearly -integrable equations naturally appear in the studies of various physical problems.  ...  In this chapter, soliton perturbation theory based on asymptotic multiple -scales expansions is developed for the perturbed NLS equation.  ...  The fifth chapter moves further away from integrability, and deals with equations which can not be viewed as perturbations of integrable models.  ... 
doi:10.1112/blms/bdu100 fatcat:ryukenp735b6zmx5edlx2kssim

Page 4089 of Mathematical Reviews Vol. , Issue 83j [page]

1983 Mathematical Reviews  
Mitropol’skii], Integral manifolds in the theory of differential equations (pp. 749-775); F. Neuman, Global theory of linear differential equations of the nth order (pp. 777-794); M.  ...  Kozlov, The method of vector Ljapunov functions in the qualitative theory of differential equations in Banach spaces (pp. 685-707); J.  ... 

Page 5375 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
Some of the main motivations for studying the theory of those singularly perturbed equations originate from elasticity theory.  ...  Singular perturbations in the theory of thin elastic rods, 3.  ... 

On the Singular Perturbations for Fractional Differential Equation

Abdon Atangana
2014 The Scientific World Journal  
The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative.  ...  We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations.  ...  Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.  ... 
doi:10.1155/2014/752371 pmid:24683357 pmcid:PMC3934389 fatcat:7n5p7ll4hvbrnde5lxzutifywy

Page 1428 of Mathematical Reviews Vol. 50, Issue 5 [page]

1975 Mathematical Reviews  
Advances in differential and integral equations. 10389 A collection of papers presented at the Conference on Qualita- tive Theory of Nonlinear Differential and Integral Equations held at the University  ...  Gustafson, Examples in the oscillation theory of higher order equations (p. 203); Frank Hoppensteadt, Asymptotic ex- pansions for singularly perturbed systems (pp. 203-204); J. D.  ... 
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