UnitStep function

By dividing r[t] by the UnitStep function we obtain the correct behavior the solution is... 

All DiracDelta and UnitStep functionality is now autoloaded. The package Calculus DiracDelta is obsolete. 

We find that the integration looks like an exponential decay except that now a new function has appeared—the UnitStep function. To see how the UnitStep function behaves in time we can plot it as shown here ... 

The UnitStep function is everywhere zero until it comes to t = 0 and then it goes to a value of unity, which maintains ad infinitum. We can rewrite the solution as a function of time ... 

We can modify the solution to include this behavior. The critical radius can be expressed as a multiple of the initial radius at time zero. We want the function to literally "blow-up" when we reach this radius. The critical condition can be expressed with the UnitStep function. It will be of unit value until the critical condition is reached, whereupon it will go to zero. Recall the behavior of the UnitStep given in terms of r[t] and the critical condition of 2r[0] ... 

By dividing r[t] by the UnitStep function we obtain the correct behavior the solution is meaningful up to the critical radius but not beyond and is shown in what follows in the In statement and two graphs ... 

The upset begins at time tl, and it instantaneously rises to a value of qp. The upset stays at the level qp from tl until a time just less than t2. At time t2 it falls instantaneously back to zero. This kind of function is called a UnitStep. For the sake of making a graph of this type of disturbance, we plot the UnitStep function and 1 — UnitStep function using the parameters 1 and 6 in order to have a pulse of unit height that is five time-units wide ... 

How can we model such a problem To do the analysis we need to introduce and become comfortable with two new functions the Dirac-Delta function and the UnitStep (or Heaviside) function. The Dirac-Delta function is infinitely intense and infinitesimally narrow—like a pulse of laser light. We can imagine it arising in the following way. We begin by considering a pulse that is quite broad, such as the function that is plotted here ... 

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