5.3: Heaviside and Dirac Delta Functions
The Laplace transform technique becomes truly useful when solving odes with discontinuous or impulsive inhomogeneous terms, these terms commonly modeled using Heaviside or Dirac delta functions. We will discuss these functions in turn, as well as their Laplace transforms.
Heaviside Function
The Heaviside or unit step function (see Fig. \(\PageIndex{1}\)) , denoted here by \(u_c(t)\), is zero for \(t < c\) and is one for \(t ≥ c\); that is, \[\label{eq:1}u_c(t)=\left\{\begin{array}{rl}0,&t<c; \\ 1,&t\geq c.\end{array}\right.\]
The precise value of \(u_c(t)\) at the single point \(t = c\) shouldn’t matter.
The Heaviside function can be viewed as the step-up function. The step-down function—one for \(t < c\) and zero for \(t ≥ c\)—is defined as \[\label{eq:2}1-u_c(t)=\left\{\begin{array}{rl}1,&t<c; \\ 0,&t\geq c.\end{array}\right.\]
The step-up, step-down function—zero for \(t < a\), one for \(a ≤ t < b\), and zero for \(t ≥ b\)—is defined as \[\label{eq:3}u_a(t)-u_b(t)=\left\{\begin{array}{rl}0,&t<a; \\ 1,&a\leq t<b; \\ 0,&t\geq b.\end{array}\right.\]
The Laplace transform of the Heaviside function is determined by integration:
\[\begin{aligned}\mathcal{L}\{u_c(t)\}&=\int_0^{\infty}e^{-st}u_c(t)dt \\ &=\int_c^{\infty}e^{-st}dt \\ &=\frac{e^{-cs}}{s},\end{aligned}\] and is given in line 12 of Table 5.1.1.
The Heaviside function can be used to represent a translation of a function \(f(t)\) a distance \(c\) in the positive \(t\) direction. We have \[u_c(t)f(t-c)=\left\{\begin{array}{rl}0,&t<c; \\ f(t-c),&t\geq c.\end{array}\right.\nonumber\]
The Laplace transform is \[\begin{aligned}\mathcal{L}\{u_c(t)f(t-c)\}&=\int_0^{\infty}e^{-st}u_c(t)f(t-c)dt \\ &=\int_c^{\infty}e^{-st}f(t-c)dt \\ &=\int_0^{\infty}e^{-s(t'+c)}f(t')dt' \\ &=e^{-cs}\int_0^{\infty}e^{-st'}f(t')dt' \\ &=e^{-cs}F(s),\end{aligned}\] where we have changed variables to \(t' = t − c\). The translation of \(f(t)\) a distance \(c\) in the positive \(t\) direction corresponds to the multiplication of \(F(s)\) by the exponential \(e^{−cs}\). This result is shown in line 13 of Table 5.1.1.
Piecewise-defined inhomogeneous terms can be modeled using Heaviside functions. For example, consider the general case of a piecewise function defined on two intervals:
\[f(t)=\left\{\begin{array}{rl}f_1(t),&\text{if }t<c; \\ f_2(t),&\text{if }t\geq c.\end{array}\right.\nonumber\]
Using the Heaviside function \(u_c\), the function \(f(t)\) can be written in a single line as \[f(t)=f_1(t)+(f_2(t)-f_1(t))u_c(t).\nonumber\]
This example can be generalized to piecewise functions defined on multiple intervals.
As a concrete example, suppose the inhomogeneous term is represented by a linearly increasing function, which then turns into a constant function, as sketched in Fig. \(\PageIndex{2}\). Explicitly, \[f(t)=\left\{\begin{array}{rl}t,&\text{if }t<1; \\ 1,&\text{if }t\geq 1.\end{array}\right.\nonumber\]
We can rewrite \(f(t)\) using the Heaviside function \(u_1(t)\):
\[f(t)=t-u_1(t)(t-1);\nonumber\] and we can take the Laplace transform of the function \(f(t)\) by writing \[F(s)=\mathcal{L}\{t\}-\mathcal{L}\{u_1(t)(t-1)\}.\nonumber\]
The Laplace transform of the first term is found from line 4 of the table, and the Laplace transform of the second term is found from a combination of line 13 and line 4. We have \[F(s)=\frac{1}{s^2}-\frac{e^{-s}}{s^2}.\nonumber\]
Dirac Delta Function
The Dirac delta function, denoted as \(\delta (t)\), is defined by requiring that for any function \(f(t)\), \[\int_{-\infty}^{\infty}f(t)\delta(t)dt=f(0).\nonumber\]
The usual view of the shifted Dirac delta function \(\delta (t − c)\) is that it is zero everywhere except at \(t = c\), where it is infinite, and the integral over the Dirac delta function is one. The Dirac delta function is technically not a function, but is what mathematicians call a distribution. Nevertheless, in most cases of practical interest, it can be treated like a function, where physical results are obtained following a final integration.
There are many ways to represent the Dirac delta function as a limit of a welldefined function. For our purposes, the most useful representation makes use of the step-up, step-down function of \(\eqref{eq:3}\) (see Fig. \(\PageIndex{3}\)):
\[\delta(t-c)=\underset{\epsilon\to 0}{\lim}\frac{1}{2\epsilon}(u_{c-\epsilon}(t)-u_{c+\epsilon}(t)).\nonumber\]
Before taking the limit, the well-defined step-up, step-down function is zero except over a small interval of width \(2\epsilon\) centered at \(t = c\), over which it takes the large value \(1/2\epsilon\). The integral of this function is one, independent of the value of \(\epsilon\).
The Laplace transform of the Dirac delta function is easily found by integration using the definition of the delta function:
\[\begin{aligned}\mathcal{L}\{\delta (t-c)\}&=\int_0^{\infty}e^{-st}\delta (t-c)dt \\ &=e^{-cs}.\end{aligned}\]
This result is shown in line 14 of Table 5.1.1.