Dirac delta representation by exact parametric equations.

Chicurel-Uziel, Enrique

doi:10.1016/j.jsv.2007.03.087

Cited by 12 publications

(7 citation statements)

References 5 publications

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“…The parametric representation of the Dirac delta as proposed in [1] has been reviewed with the purpose of clarifying the concept.…”

Section: Discussionmentioning

confidence: 99%

“…The purpose of this paper is to present a more rigorous derivation of the parametric Dirac delta than that which was presented in [1], and also to illustrate its application in partial differential equations and nonlinear problems.…”

Section: Introductionmentioning

confidence: 99%

“…The particular parameterization presented in [1] permits this differentiation to be carried out by means of elementary calculus and the resulting pair of parametric equations are exact and closed.…”

Section: Introductionmentioning

confidence: 99%

“…The delta equations have the same function values as those specified in the definition; the area involved has a unit value; they comply with the fundamental property and yield the correct Laplace and Fourier transforms [1]. In the solution of differential equations, they are handled exclusively by calculus and algebra, both at an elementary level.…”

Section: Introductionmentioning

confidence: 99%

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Parametric Dirac Delta to Simplify the Solution of Linear and Nonlinear Problems with an Impulsive Forcing Function

Chicurel-Uziel

Godínez-Rojano²

2013

JAMP

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The Laplace transform is a very useful tool for the solution of problems involving an impulsive excitation, usually represented by the Dirac delta, but it does not work in nonlinear problems. In contrast with this, the parametric representation of the Dirac delta presented here works both in linear and nonlinear problems. Furthermore, the parametric representation converts the differential equation of a problem with an impulsive excitation into two equations: the first equation referring to the impulse instant (absent in the conventional solution) and the second equation referring to post-impulse time. The impulse instant equation contains fewer terms than the original equation and the impulse is represented by a constant, just as in the Laplace transform, the post-impulse equation is homogeneous. Thus, the solution of the parametric equations is considerably simpler than the solution of the original equation. The parametric solution, involving the equations of both the dependent and independent variables in terms of the parameter, is readily reconverted into the usual equation in terms of the dependent and independent variables only. This parametric representation may be taught at an earlier stage because the principle on which it is based is easily visualized geometrically and because it is only necessary to have a knowledge of elementary calculus to understand it and use it.

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“…The parametric representation of the Dirac delta as proposed in [1] has been reviewed with the purpose of clarifying the concept.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Parametric Dirac Delta to Simplify the Solution of Linear and Nonlinear Problems with an Impulsive Forcing Function

Chicurel-Uziel

Godínez-Rojano²

2013

JAMP

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“…According to distribution theory, the Dirac delta is the result of differentiating the Heaviside unit step. The particular parametrization presented in [1] permits this differentiation to be carried out by means of elementary calculus and the resulting pair of parametric equations is exact and closed.…”

Section: Introductionmentioning

confidence: 99%

Parametrization to Improve the Solution Accuracy of Problems Involving the Bi-Dimensional Dirac Delta in the Forcing Function

Chicurel-Uziel¹,

Godínez²

2015

JAMP

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The representation of the Dirac delta, obtained by differentiating the parametric equation of the unit step with a riser, is used to solve two examples referring to problems of a different physical nature, each with the product of two deltas as a forcing function. Each problem was solved by an entirely different procedure. In comparison with non-parametric solutions, the present solutions are both more accurate and truer representations of the physics involved.

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Time Response of First- and Second-order Dynamical Systems

Ángeles

2011

Mechanical Engineering Series

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Dirac delta representation by exact parametric equations.

Cited by 12 publications

References 5 publications

Parametric Dirac Delta to Simplify the Solution of Linear and Nonlinear Problems with an Impulsive Forcing Function

Parametric Dirac Delta to Simplify the Solution of Linear and Nonlinear Problems with an Impulsive Forcing Function

Parametrization to Improve the Solution Accuracy of Problems Involving the Bi-Dimensional Dirac Delta in the Forcing Function

Time Response of First- and Second-order Dynamical Systems

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