Unit step and impulse function equations to simplify the solution of engineering problems
Abstract
A unit step equation is proposed that when differentiated by elementary calculus yields the impulse function and when the resulting impulse function equation is integrated by elementary calculus yields the proposed unit step equation. Using these two equations, a two stage methodology is presented for the simplification of the solution of problems involving the impulse function.Downloads
References
B. Engquist, A. K. Tornberg and R. Tsai, Discretization of Dirac delta functions in level set methods, Journal of Computational Physics archive, Vol. 207 Issue 1, July( 2005), pages 28-51.
X. Xu, Nonlinear trigonometric approximation and the Dirac delta function, Journal of Computational and Applied Mathematics 209 (2006) 234 – 245.
P. Smereka, The numerical approximation of a delta function with application to level set methods. Journal of Computational Physics 211 (2006) 77–90.
C. Min and F. Gibou, Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions, Journal of Computational Physics archive, Vol. 227, Issue 22, Nov. (2008),
pages 9686-9695.
M. Jauregui and C. Tsallis, New representations of π and Dirac delta using the nonextensive-statisticalmechanics q-exponential function, Journal of Mathematical Physics 51, 063304, (2010).
A. Chevreuil, A. Plastino, and C. Vignat, On a conjecture about Dirac´s delta representation using qexponentials, Journal of Mathematical Physics 51, 093502,( 2010).
Y.T. Li and R. Wong, Integral and Series Representations of the Dirac Delta Function, Communications on Pure and Applied Analysis, (2013). arXiv:1303.1943[math.CA].
L. Schwartz, Théorie des distributions, Paris, Hermann, (1966).
M. A. Al-Gwaiz, Theory of distributions, CRC Press, (2012).
E. Chicurel-Uziel, Dirac delta representation by exact parametric equations. Application to impulsive vibration systems, J. of Sound and Vibration, Vol. 305,134, (2007).
E. Chicurel-Uziel and F. A. Godínez-Rojano, F. A.,(2015) Parametric Dirac delta to simplify the solution of linear and nonlinear problems with an impulsive forcing function, Journal of Applied Mathematics and Physics, Vol. 1, 16-25. http://dx.doi.org/10.4236/jamp.(2013).17003
E. Chicurel-Uziel and F. A. Godínez-Rojano, (2015) Parametrization to improve the Solution Accuracy of Problems Involving the Bi-Dimensional Dirac Delta in the Forcing Function, Journal of Applied Mathematics and Physics, 3,1168-1177. http://dx.doi.org/10.4236/jamp.(2015).39144
R. Haberman, Applied Partial Differential Equations, 4th Ed., Pearson, Prentice Hall, Upper Saddle River, New Jersey, US, (2004), pages 51-53.
Copyright (c) 2018 Journal of Information Sciences and Computing Technologies
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
TRANSFER OF COPYRIGHT
JISCT is pleased to undertake the publication of your contribution to Journal of Information Sciences and Computing Technologies
The copyright to this article is transferred to JISCT(including without limitation, the right to publish the work in whole or in part in any and all forms of media, now or hereafter known) effective if and when the article is accepted for publication thus granting JISCT all rights for the work so that both parties may be protected from the consequences of unauthorized use.