Convolution for the Laplace Transform
Chapter 12 Fourier Series and the Laplace Transform
12.10 Convolution for the Laplace Transform
This section is a continuation of our development of the Laplace Transforms in Section 12.5, Section 12.6, Section 12.7, Section 12.8 and Section 12.9.
If we let denote the transforms of , respectively, then the inverse of the product is given by the function . It is called the convolution of and can be regarded as s generalized product of . Convolution will assist us in solving integral equations.
Theorem 12.24 (Convolution
denote the Laplace transforms of
respectively. Then the product is
the Laplace transform of the convolution
is denoted by , and
has the integral representation
Example 12.29. Show that .
If we let , and , then , and , respectively. Now,
applying the convolution theorem, we get
Explore Solution 12.29.
Example 12.30. Use the convolution theorem to solve the integral equation .
A graph of the solution.
the convolution theorem, we obtain
Solving for , we
have , and
and then ,
and then we get
Finally, the solution is
obtained using facts from Table 12.2: and , and
Explore Solution 12.30.
Engineers and physicists sometimes
consider forces that produce large effects but that are applied over
a very short time interval. The force acting at the time
an earthquake starts is an example. This phenomenon leads
to the idea of a unit impulse function . Let's
consider the small positive constan a. The
The unit impulse function is obtained by
letting the interval width go to zero, or
Figure 11.29 shows the graph for . Although is called the Dirac Delta function, it is not an ordinary function. To be precise it is a distribution, and the theory of distributions permits manipulations of as though it were a function. Here, we will treat as a function and investigate its properties.
Figure 12.29. Graphs of for .
Example 12.31. Show that .
By definition, the Laplace transform of
the above equation and using L'Hôpital's rule, we
Explore Solution 12.31.
We now turn to the unit impulse
function. First, we consider the
by integrating :
Taking the limit as results
in the important fact that
where is the unit step function that was introduced in Section 12.7. The situation is illustrated in Figure 12.30.
Figure 12.30. The integral of is , which becomes when for .
We demonstrate the response of a system to the unit impulse function in Example 12.32.
12.32. Solve the initial value
Taking transforms results
and the solution is
Figure 12.31. A graph of the solution .
Remark. The condition is not satisfied by the "solution" . Recall that all solutions involving the use of the Laplace transform are to be considered zero for values of , hence the graph of is given above in Figure 12.31. Note that has a jump discontinuity of magnitude at the origin. This discontinuity occurs because either or must have a jump discontinuity at the origin whenever the Dirac delta function, , occurs as part of the input or driving function.
Explore Solution 12.32.
The convolution method can be used to solve initial value problems. The tedious mechanical details of problem solving can be facilitated with computer software such as Maple, Matlab, or Mathematica.
Theorem 12.25 (Initial value Problem -
IVP convolution method). The unique solution to the
initial value problem
with and , is given by
where is the solution to the homogeneous equation
with , and
has the Laplace transform given by .
Example 12.33. Use convolution
to solve the initial value problem
A graph of the solution.
First, we obtain the
the solution by solving with
and . Taking
the Laplace transform yields , and
we can rearrange the terms to obtain . Solving
gives , and
it follows that
Second, we observe that and , and
also that . We
compute the portion of
the solution with a convolution:
Now we can compute the solution using the convolution method in
Explore Solution 12.33.
Exercises for Section 12.10. The Laplace Transform: Convolution
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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
(c) 2012 John H. Mathews, Russell W. Howell