scholarly journals Hill Problem Analytical Theory to the Order Four: Application to the Computation of Frozen Orbits around Planetary Satellites

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Martin Lara ◽  
Jesús F. Palacián

Frozen orbits of the Hill problem are determined in the double-averaged problem, where short and long-period terms are removed by means of Lie transforms. Due to the perturbation method we use, the initial conditions of corresponding quasi-periodic solutions in the nonaveraged problem are computed straightforwardly. Moreover, the method provides the explicit equations of the transformation that connects the averaged and nonaveraged models. A fourth-order analytical theory is necessary for the accurate computation of quasi-periodic frozen orbits.

2004 ◽  
Vol 02 (01) ◽  
pp. 71-85 ◽  
Author(s):  
YUJI LIU ◽  
WEIGAO GE

In this paper, we study four-point boundary value problems for a fourth-order ordinary differential equation of the form [Formula: see text] with one of the following boundary conditions: [Formula: see text] or [Formula: see text] Growth conditions on f which guarantee existence of at least three positive solutions for the problems (E)–(B1) and (E)–(B2) are imposed.


2018 ◽  
Vol 21 (04) ◽  
pp. 1850027 ◽  
Author(s):  
KULDIP SINGH PATEL ◽  
MANI MEHRA

In this paper, a compact scheme with three time levels is proposed to solve the partial integro-differential equation that governs the option prices in jump-diffusion models. In the proposed compact scheme, the second derivative approximation of the unknowns is approximated using the value of these unknowns and their first derivative approximations, thereby allowing us to obtain a tridiagonal system of linear equations for a fully discrete problem. Moreover, the consistency and stability of the proposed compact scheme are proved. Owing to the low regularity of typical initial conditions, a smoothing operator is employed to ensure the fourth-order convergence rate. Numerical illustrations concerning the pricing of European options under the Merton’s and Kou’s jump-diffusion models are presented to validate the theoretical results.


1972 ◽  
Vol 22 ◽  
pp. 115-128 ◽  
Author(s):  
John Evans

Following their discovery of the “Burnt Palace” at Beycesultan in the mid 1950's, Seton Lloyd and James Mellaart drew attention to a number of features of its architecture which seemed to indicate links with the palace architecture of Minoan Crete, and discussed the possible significance of these similarities (Lloyd and Mellaart, 1956 118–123, 1965 61, 62). Whatever this may be in terms of relationships between the two areas in the second millennium B.C., however, it seems clear that they cannot throw any light on the first appearance of palaces in Crete. The problems of the origin and development of the Cretan Bronze Age palaces are complex, and though they have been much discussed since the first excavations in the early years of the century, a major obstacle to progress has always been the lack of precise evidence, or even of any evidence at all, for the early stages of the process. As they stand, most of the palaces are the product of a series of rebuildings and remodellings over a long period, and it is not always clear just what they were like when first erected. Most frustrating of all, however, is the lack of evidence bearing on the question of whether they were preceded, during the Early Bronze Age, by buildings which were in any respect analogous in form and function. It has long been clear that the sites of some of the major Middle and Late Minoan palaces were occupied during the Early Minoan period, but at Phaistos and Knossos at any rate extensive clearing and levelling in preparation for the erection of the Middle Minoan palaces has obliterated practically all traces of the Early Minoan buildings. At Phaistos Branigan has hinted that the fragments of walls found by Pernier (1935, pl. VI) on the highest point of the hill might have belonged to a building of some consequence, possibly similar to the Early Minoan II mansion known as the House on the Hill at Vasiliki (Branigan 1970, p. 41). Branigan thinks that in addition to the rooms mentioned by Pernier, there may be traces of a corridor similar to that in the Vasiliki building. Only the bottom two courses of the walls survive, so that it is difficult to say much about their construction, though it seems to be poorer than that of the walls of some Early Minoan private houses later found by Levi on another part of the site.


2021 ◽  
pp. 121
Author(s):  
S.S. Kritskaia

We solve one boundary problem of fourth order with initial conditions, that appears, for example, when one solves the problem about lateral oscillations of elastic-viscous-relaxating rod of variable profile with variable momentum of inertia with freely supported ends.


2018 ◽  
Vol 15 (142) ◽  
pp. 20170976 ◽  
Author(s):  
Laurent Duchemin ◽  
Christophe Eloy ◽  
Eric Badel ◽  
Bruno Moulia

Plants have developed different tropisms: in particular, they reorient the growth of their branches towards the light (phototropism) or upwards (gravitropism). How these tropisms affect the shape of a tree crown remains unanswered. We address this question by developing a propagating front model of tree growth. Being length-free, this model leads to self-similar solutions after a long period of time, which are independent of the initial conditions. Varying the intensities of each tropism, different self-similar shapes emerge, including singular ones. Interestingly, these shapes bear similarities to existing tree species. It is concluded that the core of specific crown shapes in trees relies on the balance between tropisms.


1866 ◽  
Vol 3 (29) ◽  
pp. 495-501 ◽  
Author(s):  
Harry G. Seeley

Summary1. The brown Boulder-clay corresponds with the brown Clay of the Eastern counties, and is oldest.2. Then the hill gravel, the blue Boulder-clay, and perhaps the shell bed of March corresponds to the contorted drift.3. As elevation progressed, the Fenlands would become one great fiord, ramifying at the Cambridge end up the southern valleys in the Chalk. First the coarse gravels of low levels were formed, and finally, during a long period—for the flints are wonderfully worn—the fine gravel of the plains. After which the country was elevated, and the sea denuded the superficial beds and retired. This corresponds to the Upper Boulder-clay and coarse gravel of the Norfolk section.4. Now rivers cut their channels, and there commenced luxuriant vegetable growth, which corresponds with the excavation of the Mundesley river and the Mundesley peat.5. Then a depression, during which was formed the Buttery-clay the Fens. This corresponds with the Upper Sands and Gravel of Mundesley.6. And, finally, comes the second peat and the present state of nature.I believe this succession is true for a far larger area than the Fenlands, perhaps for all Great Britain.


1970 ◽  
Vol 34 (1) ◽  
pp. 71-82
Author(s):  
M Alhaz Uddin ◽  
MA Sattar

The unified Krylov-Bogoliubov-Mitropolskii (KBM) method is used for determining theanalytical approximate solution of a fourth order weakly nonlinear differential system with strongdamping and slowly varying coefficients when a pair of eigen-values of the unperturbed equationis a multiple (approximately or perfectly) of the other pair or pairs. In a damped case, one of thenatural frequencies of the linearized equation may be a multiple of the other. The analytical firstorder approximate solution for different initial conditions shows a good coincidence with thoseobtained by the numerical procedure. The method is illustrated by an example.Key words: Perturbation method; Weak nonlinearity; Oscillatory process; Strong damping; Varying coefficientsDOI: 10.3329/jbas.v34i1.5493Journal of Bangladesh Academy of Sciences, Vol.34, No.1, 71-82, 2010


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