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

Enrique J. Chicurel-Uziel, Francisco A. Godínez-Rojano

Instituto de Ingeniería, Instituto de Investigaciones en Materiales,.

Universidad Nacional Autónoma de México, México D.F., México.

**DOI: **10.4236/jamp.2013.17003
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Instituto de Ingeniería, Instituto de Investigaciones en Materiales,.

Universidad Nacional Autónoma de México, México D.F., México.

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.

Keywords

Dirac Delta; Parametric Representation; Nonlinear Differential Equations; Impulsive Problems

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Chicurel-Uziel, E. and Godínez-Rojano, F. (2013) Parametric Dirac Delta to Simplify the Solution of Linear and Nonlinear Problems with an Impulsive Forcing Function. *Journal of Applied Mathematics and Physics*, **1**, 16-25. doi: 10.4236/jamp.2013.17003.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] |
E. Chicurel-Uziel, “Dirac Delta Representation by Exact Parametric Equations. Application to Impulsive Vibration Systems,” Journal of Sound and Vibration, Vol. 305, No. 11-12, 2007, pp. 134-150. http://dx.doi.org/10.1016/j.jsv.2007.03.087 |

[2] | A. D. Snider, “Partial Differential Equations,” Prentice Hall, Upper Saddle River, 1999, p. 35. |

[3] | E. Kreyszig, “Advanced Engineering Mathematics,” Wiley, New York, 2006, p. 234. |

[4] | I. Stakgold, “Green’s Functions and Boundary Value Problems,” Wiley, New York, 1998, pp. 57-58. |

[5] | E. Butkov, “Mathematical Physics,” Addison Wesley, Reading, 1976, p. 114. |

[6] | M. D. Greenberg, “Advanced Engineering Mathematics,” Prentice Hall, Upper Saddle River, 1998, p. 269. |

[7] | R. F. Hoskins, “Generalized Functions,” John Wiley & Sons, Chichester, 1979, p. 42. |

[8] | E. Chicurel-Uziel, “Parameterization to Avoid the Gibbs Phenomenon,” In: N. Mastorakis, M. Demiralp and V. M. Mladenov, Eds., Computers and Simulation in Modern Science, Vol. IV, WSEAS Press, Chapter 17, 2010, pp. 186-195. |

[9] | R. Haberman, “Applied Partial Differential Equations,” Pearson Prentice Hall, Upper Saddle River, 2004. |

[10] | K. D. Cole, J. V. Beck, A. Haji-Sheikh and Bahman Litkouhi, “Heat Conduction Using Green’s Functions,” 2nd Edition, CRC Press, Taylor & Francis Group, 2011, pp. 28-29. |

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