The dynamics of a current collection system for an electric locomotive

Ockendon, J. R.; Tayler, A. B.

doi:10.1098/rspa.1971.0078

Cited by 371 publications

(92 citation statements)

References 2 publications

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“…Following the ideas used by Ockendon and Tayler [24] and Fox et al [11] in the case of the original pantograph equation (1), we start by observing that if q = 1 then Eq. (13) is reduced to the equation y ′ = 0, which has constant solutions only.…”

Section: Perturbative Proofmentioning

confidence: 99%

“…Historically, 1 the term "pantograph" dates back to the seminal paper of 1971 by Ockendon and Tayler [24], where such equations 2 emerged in a mathematical model for the dynamics of an overhead current collection system on an electric locomotive (with the physically relevant value q < 1). At about the same time, a systematic analysis of solutions to the pantograph equation was started by Fox et al [11], where various analytical, perturbation, and numerical techniques were discussed at length (for both q < 1 and q > 1).…”

Section: Introductionmentioning

confidence: 99%

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On Bounded Solutions of the Balanced Generalized Pantograph Equation

Bogachev

Derfel

Molchanov

et al. 2008

The IMA Volumes in Mathematics and Its Applications

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The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in early 1970s by T. Kato in connection with the analysis of the pantograph equation, y ′ (x) = ay(qx) + by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a 2 y ′′ (x)+a 1 y ′ (x)+y(x) = ∞ 0 y(αx) µ(dα), where a 1 ≥ 0, a 2 ≥ 0, a 2 1 +a 2 2 > 0, and µ is a probability measure. Namely, setting K := ∞ 0 ln α µ(dα), we prove that if K ≤ 0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K > 0 then such a solution exists. The result in the critical case, K = 0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with "multiplication" jumps. The paper also includes three "elementary" proofs for the simple prototype equation y ′ (x) + y(x) = 1 2 y(qx) + 1 2 y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.

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Section: Perturbative Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Bounded Solutions of the Balanced Generalized Pantograph Equation

Bogachev

Derfel

Molchanov

et al. 2008

The IMA Volumes in Mathematics and Its Applications

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show abstract

“…If n = 0, then (1.1) is the pantograph equation. This equation has been studied extensively by several researchers (for example, [2,7,8]) owing to its wide range of applications, including light absorption in the Milky Way [1], current collection for an electric train [3,9], probability theory [4], and more recently a cell growth model [5]. Most of these studies focused on the asymptotic behaviour of solutions as x → ∞ for initial-value problems; however, the cell growth model brought boundary-value problems to the fore.…”

Section: Introductionmentioning

confidence: 99%

An Eigenvalue Problem Involving a Functional Differential Equation Arising in a Cell Growth Model

Brunt¹,

Vlieg-Hulstman²

2010

ANZIAM J.

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We interpret a boundary-value problem arising in a cell growth model as a singular Sturm-Liouville problem that involves a functional differential equation of the pantograph type. We show that the probability density function of the cell growth model corresponds to the first eigenvalue and that there is a family of rapidly decaying eigenfunctions.2000 Mathematics subject classification: primary 34K06; secondary 34K10.

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“…The name pantograph came from the work of Ockendon and Tayler [1] on the collection of current by the pantograph head of an electric locomotive.…”

Section: Introductionmentioning

confidence: 99%

An improved Morgan-voyce collocation method for numerical solution of multi-pantograph equations

İlhan¹

2017

ntmsci

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Abstract:In this article, an improved collocation method based on the Morgan-Voyce polynomials for the approximates solution of multi-pantograph equations is introduced. The method is based upon the improvement of Morgan-Voyce polynomial solutions with the aid of the residual error function. First, the Morgan-Voyce collocation method is applied to the multi-pantograph equations and then Morgan-Voyce polynomial solutions are obtained. Second, an error problem is constructed by means of the residual error function and this error problem is solved by using the Morgan-Voyce collocation method. By summing the Morgan-Voyce polynomial solutions of the original problem and the error problem, we have the improved Morgan-Voyce polynomial solutions. When the exact solution of problem is not known, the absolute error can then be approximately computed by the Morgan-Voyce polynomial solution of the error problem. Numerical examples that the pertinent features of the method are presented. We have applied all of the numerical computations on computer using a program written in MATLAB.

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The dynamics of a current collection system for an electric locomotive

Cited by 371 publications

References 2 publications

On Bounded Solutions of the Balanced Generalized Pantograph Equation

On Bounded Solutions of the Balanced Generalized Pantograph Equation

An Eigenvalue Problem Involving a Functional Differential Equation Arising in a Cell Growth Model

An improved Morgan-voyce collocation method for numerical solution of multi-pantograph equations

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