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

Abstract

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.

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

“…It is well to keep in mind that the parametric equations of the delta confirm that its area has unit value, that they comply with the fundamental property and that they yield the correct Laplace [1] and Fourier transforms [2].…”

“…This is the parametric Dirac delta, a more rigorous derivation of which was presented in [2] where it was clearly established that its value is 0 for x a < and x a > and its value is infinity at the single point: x a = . ≤ ≤ + the value of the deltas is infinity; in order to avoid problems, instead of the value 1 in Equation (7) a value of 1.00000001 was used for plotting purposes.…”