Nonlinear Singular Perturbation Phenomena

Chang, K. W.; Howes, F. A.

doi:10.1007/978-1-4612-1114-3

Cited by 179 publications

(55 citation statements)

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“…In the first place, both of them are singularly perturbed [17,21] and for this reason, their modes of stability loss have the shapes of boundary effects, usually localized in boundary zones. Secondly, as a rule, these modes represent damping out oscillating harmonics.…”

Section: Equations Of Critical States Of the Ds Inside Inclined Bore-mentioning

confidence: 99%

Critical Buckling of Drill Strings in Cylindrical Cavities of Inclined Bore-Holes

Musa¹,

Gulyayev²,

Shlyun³

et al. 2016

JMEA

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Notwithstanding the fact that the problem of drill string buckling (Eulerian instability) inside the cylindrical cavity of an inclined bore-hole attracts attention of many specialists, it is far from completion. This peculiarity can be explained by the complexity of its mathematic model which is described by singularly perturbed equations. Their solutions (eigen modes) have the shapes of boundary effects or buckles (harmonic wavelets) localized in zones of the bore-hole that are not specified in advance. Therefore, the problem should be stated in the domain of entire length of the drill string or in some separated part including an expected zone of its buckling. In the paper, a mathematic model for computer analysis of incipient buckling of a drill string in cylindrical channel of an inclined bore-hole is elaborated. The constitutive equation is deduced with allowance made for action of gravity, contact, and friction forces. Computer simulation of the drill string buckling is performed for different values of the bore-hole inclination angle, its length, friction coefficient, and clearance. The eigen values (critical loads) are found and modes of stability loss are constructed. The numerical results for the case when the inclination angle equals friction angle coincide with ones obtained analytically.

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Section: Equations Of Critical States Of the Ds Inside Inclined Bore-mentioning

confidence: 99%

Critical Buckling of Drill Strings in Cylindrical Cavities of Inclined Bore-Holes

Musa¹,

Gulyayev²,

Shlyun³

et al. 2016

JMEA

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“…where the functions v i , 0 ≤ i ≤ 4 are defined to be the solutions for the following first order problems 2) and the last function v 5 satisfies the second order problem…”

Section: Error Estimatesmentioning

confidence: 99%

High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts

Salama¹,

Alamery²

2016

Turk J Math

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In this paper, we propose and analyze a high-order uniform method for solving boundary value problems (BVPs) for singularly perturbed nonlinear delay differential equations with small shifts (delay and advance). Such types of BVPs play an important role in the modeling of various real life phenomena, such as the variational problem in control theory and in the determination of the expected time for the generation of action potentials in nerve cells. To obtain parameter-uniform convergence, the present method is constructed on a piecewise-uniform Shishkin mesh. The error estimate is discussed and it is shown that the method is uniformly convergent with respect to the singular perturbation parameter. Moreover, a bound of the global error is also derived. The effect of small shifts on the solution behavior is shown by numerical computations. Several numerical examples are presented to support the theoretical results, and to demonstrate the efficiency and the high-order accuracy of the proposed method.

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“…The truncation error of F in approximating F ε in terms of y is defined to be F y−F ε y ∞ . It is clear that (F y) 0 = (F ε y)(0) and (F y) N = (F ε y) (1). We shall bound |(F y) i − (F ε y)(x i )|, for i = 1, 2, .…”

Section: A Central Difference Scheme On a Shishkin Meshmentioning

confidence: 99%

“…See for example Chang and Howes [1], D'Annunzio [2], Fife [4], Herceg [7], Herceg and Petrović [8] and Lorenz [9].…”

Section: Introductionmentioning

confidence: 99%

A uniformly convergent method for a singularly perturbed semilinear reaction–diffusion problem with multiple solutions

Sun

Stynes

1996

Math. Comp.

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Abstract. This paper considers a simple central difference scheme for a singularly perturbed semilinear reaction-diffusion problem, which may have multiple solutions. Asymptotic properties of solutions to this problem are discussed and analyzed. To compute accurate approximations to these solutions, we consider a piecewise equidistant mesh of Shishkin type, which contains O(N ) points. On such a mesh, we prove existence of a solution to the discretization and show that it is accurate of order N −2 ln 2 N , in the discrete maximum norm, where the constant factor in this error estimate is independent of the perturbation parameter ε and N . Numerical results are presented that verify this rate of convergence.

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Nonlinear Singular Perturbation Phenomena

Cited by 179 publications

References 0 publications

Critical Buckling of Drill Strings in Cylindrical Cavities of Inclined Bore-Holes

Critical Buckling of Drill Strings in Cylindrical Cavities of Inclined Bore-Holes

High-order uniformly convergent method for nonlinear singularly perturbed delay differential equations with small shifts

A uniformly convergent method for a singularly perturbed semilinear reaction–diffusion problem with multiple solutions

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