Laplace Transforms (Part 10. Convolution Integrals)

Convolution Integrals

On occasion we will run across transforms of the form,

that can’t be dealt with  easily using partial fractions.  We would like a way to take the inverse transform of such a transform.  We can use a convolution integral to do this.

Convolution Integral

If f(t) and g(t) are piecewise continuous function on  then the convolution integral of f(t) and g(t) is,

A nice property of convolution integrals is.

Or,

The following fact will allow us to take the inverse transforms of a product of transforms.

Fact

Let’s work a quick example to see how this can be used.

Example 1  Use a convolution integral to find the inverse transform of the following transform.                                                           Solution First note that we could use #11 from out table to do this one so that will be a nice check against our work here. Now, since we are going to use a convolution integral here we will need to write it as a product whose terms are easy to find the inverse transforms of.  This is easy to do in this case.                                                   So, in this case we have,                Using a convolution integral h(t) is,                                             This is exactly what we would have gotten by using #11 from the table.

Convolution integrals are very useful in the following kinds of problems.

Example 2  Solve the following IVP                                 Solution First, notice that the forcing function in this case has not been specified.  Prior to this section we would not have been able to get a solution to this IVP.  With convolution integrals we will be able to get a solution to this kind of IVP.  The solution will be in terms of g(t) but it will be a solution. Take the Laplace transform of all the terms and plug in the initial conditions.                                         Notice here that all we could do for the forcing function was to write down G(s) for its transform.   Now, solve for Y(s).                                           We factored out a 4 from the denominator in preparation for the inverse transform process.  To take inverse transforms we’ll need to split up the first term and we’ll also rewrite the second term a little.                                           Now, the first two terms are easy to inverse transform.  We’ll need to use a convolution integral on the last term.  The two functions that we will be using are,                                                We can shift either of the two functions in the convolution integral.  We’ll shift g(t) in our solution.  Taking the inverse transform gives us,                              So, once we decide on a g(t) all we need to do is to an integral and we’ll have the solution.

As this last example has shown, using convolution integrals will allow us to solve IVP’s with general forcing functions.  This could be very convenient in cases where we have a variety of possible forcing functions and don’t which one we’re going to use.  With a convolution integral all that we need to do in these cases is solve the IVP once then go back and evaluate an integral for each possible g(t).  This will save us the work of having to solve the IVP for each and every g(t).