Module

for

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  [Graphics:Images/LaplaceConvolutionMod_gr_1.gif]  denote the transforms of  [Graphics:Images/LaplaceConvolutionMod_gr_2.gif],  respectively, then the inverse of the product  [Graphics:Images/LaplaceConvolutionMod_gr_3.gif]  is given by the function  [Graphics:Images/LaplaceConvolutionMod_gr_4.gif].  It is called the convolution of  [Graphics:Images/LaplaceConvolutionMod_gr_5.gif]  and can be regarded as s generalized product of   [Graphics:Images/LaplaceConvolutionMod_gr_6.gif].  Convolution will assist us in solving integral equations.

 

Theorem 12.24 (Convolution Theorem).  Let [Graphics:Images/LaplaceConvolutionMod_gr_7.gif] and [Graphics:Images/LaplaceConvolutionMod_gr_8.gif] denote the Laplace transforms of [Graphics:Images/LaplaceConvolutionMod_gr_9.gif] and [Graphics:Images/LaplaceConvolutionMod_gr_10.gif], respectively.  Then the product  [Graphics:Images/LaplaceConvolutionMod_gr_11.gif]  is the Laplace transform of the convolution of  [Graphics:Images/LaplaceConvolutionMod_gr_12.gif]  and [Graphics:Images/LaplaceConvolutionMod_gr_13.gif],  and is denoted by  [Graphics:Images/LaplaceConvolutionMod_gr_14.gif],  and has the integral representation  

            [Graphics:Images/LaplaceConvolutionMod_gr_15.gif]    

Proof.

 

Example 12.29.  Show that  [Graphics:Images/LaplaceConvolutionMod_gr_16.gif].  

Solution.

    If we let  [Graphics:Images/LaplaceConvolutionMod_gr_17.gif],  and  [Graphics:Images/LaplaceConvolutionMod_gr_18.gif],  then  [Graphics:Images/LaplaceConvolutionMod_gr_19.gif],  and  [Graphics:Images/LaplaceConvolutionMod_gr_20.gif],  respectively.  Now, applying the convolution theorem, we get

        [Graphics:Images/LaplaceConvolutionMod_gr_21.gif]  

Explore Solution 12.29.

 

Example 12.30.  Use the convolution theorem to solve the integral equation  [Graphics:Images/LaplaceConvolutionMod_gr_35.gif].  

                          [Graphics:Images/LaplaceConvolutionMod_gr_36.gif]

                                              A graph of the solution.

Solution.

    Letting  [Graphics:Images/LaplaceConvolutionMod_gr_37.gif]  and using  [Graphics:Images/LaplaceConvolutionMod_gr_38.gif]  in the convolution theorem, we obtain  

            [Graphics:Images/LaplaceConvolutionMod_gr_39.gif]
        
            [Graphics:Images/LaplaceConvolutionMod_gr_40.gif].   

Solving for  [Graphics:Images/LaplaceConvolutionMod_gr_41.gif],  we have  [Graphics:Images/LaplaceConvolutionMod_gr_42.gif],  and then  [Graphics:Images/LaplaceConvolutionMod_gr_43.gif], and then  [Graphics:Images/LaplaceConvolutionMod_gr_44.gif], and then we get

            [Graphics:Images/LaplaceConvolutionMod_gr_45.gif]  .

Finally, the solution  [Graphics:Images/LaplaceConvolutionMod_gr_46.gif]  is obtained using facts from Table 12.2:  [Graphics:Images/LaplaceConvolutionMod_gr_47.gif]  and  [Graphics:Images/LaplaceConvolutionMod_gr_48.gif],  and the computation  

            [Graphics:Images/LaplaceConvolutionMod_gr_49.gif]

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  [Graphics:Images/LaplaceConvolutionMod_gr_61.gif].  Let's consider the small positive constan a.  The function  [Graphics:Images/LaplaceConvolutionMod_gr_62.gif]  is defined by

            [Graphics:Images/LaplaceConvolutionMod_gr_63.gif].     

    The unit impulse function is obtained by letting the interval width go to zero, or

            [Graphics:Images/LaplaceConvolutionMod_gr_64.gif].  

    Figure 11.29 shows the graph  [Graphics:Images/LaplaceConvolutionMod_gr_65.gif]  for  [Graphics:Images/LaplaceConvolutionMod_gr_66.gif].  Although [Graphics:Images/LaplaceConvolutionMod_gr_67.gif] 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 [Graphics:Images/LaplaceConvolutionMod_gr_68.gif] as though it were a function.  Here, we will treat [Graphics:Images/LaplaceConvolutionMod_gr_69.gif] as a function and investigate its properties.

                  [Graphics:Images/LaplaceConvolutionMod_gr_70.gif][Graphics:Images/LaplaceConvolutionMod_gr_71.gif][Graphics:Images/LaplaceConvolutionMod_gr_72.gif]

                                  Figure 12.29.  Graphs of  [Graphics:Images/LaplaceConvolutionMod_gr_73.gif]  for  [Graphics:Images/LaplaceConvolutionMod_gr_74.gif].

 

Example 12.31.  Show that  [Graphics:Images/LaplaceConvolutionMod_gr_75.gif].  

Solution.

    By definition, the Laplace transform of [Graphics:Images/LaplaceConvolutionMod_gr_76.gif] is

            [Graphics:Images/LaplaceConvolutionMod_gr_77.gif]  

Letting  [Graphics:Images/LaplaceConvolutionMod_gr_78.gif]  in the above equation and using L'Hôpital's rule, we obtain  

            [Graphics:Images/LaplaceConvolutionMod_gr_79.gif]  

Explore Solution 12.31.

 

 

    We now turn to the unit impulse function.  First, we consider the function  [Graphics:Images/LaplaceConvolutionMod_gr_100.gif]  obtained by integrating  [Graphics:Images/LaplaceConvolutionMod_gr_101.gif]:  

            [Graphics:Images/LaplaceConvolutionMod_gr_102.gif]  

Taking the limit as   [Graphics:Images/LaplaceConvolutionMod_gr_103.gif]  results in the important fact that

            [Graphics:Images/LaplaceConvolutionMod_gr_104.gif],

where  [Graphics:Images/LaplaceConvolutionMod_gr_105.gif]is the unit step function that was introduced in Section 12.7.  The situation is illustrated in Figure 12.30.

                   [Graphics:Images/LaplaceConvolutionMod_gr_106.gif][Graphics:Images/LaplaceConvolutionMod_gr_107.gif]

                   Figure 12.30.  The integral of [Graphics:Images/LaplaceConvolutionMod_gr_108.gif] is [Graphics:Images/LaplaceConvolutionMod_gr_109.gif], which becomes [Graphics:Images/LaplaceConvolutionMod_gr_110.gif] when for [Graphics:Images/LaplaceConvolutionMod_gr_111.gif].

 

 

    We demonstrate the response of a system to the unit impulse function in Example 12.32.

 

Example 12.32.   Solve the initial value problem

            [Graphics:Images/LaplaceConvolutionMod_gr_112.gif]  
            with  
            [Graphics:Images/LaplaceConvolutionMod_gr_113.gif].  

Solution.

    Taking transforms results in  [Graphics:Images/LaplaceConvolutionMod_gr_114.gif]  so that

            [Graphics:Images/LaplaceConvolutionMod_gr_115.gif],  
and the solution is

            [Graphics:Images/LaplaceConvolutionMod_gr_116.gif]

                          [Graphics:Images/LaplaceConvolutionMod_gr_117.gif]

                              Figure 12.31.  A graph of the solution  [Graphics:Images/LaplaceConvolutionMod_gr_118.gif].

Remark.  The condition  [Graphics:Images/LaplaceConvolutionMod_gr_119.gif]  is not satisfied by the "solution" [Graphics:Images/LaplaceConvolutionMod_gr_120.gif].  Recall that all solutions involving the use of the Laplace transform are to be considered zero for values of  [Graphics:Images/LaplaceConvolutionMod_gr_121.gif],  hence the graph of  [Graphics:Images/LaplaceConvolutionMod_gr_122.gif] is given above in Figure 12.31.  Note that  [Graphics:Images/LaplaceConvolutionMod_gr_123.gif]  has a jump discontinuity of magnitude  [Graphics:Images/LaplaceConvolutionMod_gr_124.gif]  at the origin.  This discontinuity occurs because either [Graphics:Images/LaplaceConvolutionMod_gr_125.gif] or [Graphics:Images/LaplaceConvolutionMod_gr_126.gif]  must have a jump discontinuity at the origin whenever the Dirac delta function, [Graphics:Images/LaplaceConvolutionMod_gr_127.gif], 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[Graphics:Images/LaplaceConvolutionMod_gr_140.gif],  Matlab[Graphics:Images/LaplaceConvolutionMod_gr_141.gif],  or  Mathematica[Graphics:Images/LaplaceConvolutionMod_gr_142.gif].

 

Theorem 12.25 (Initial value Problem - IVP convolution method). The unique solution to the initial value problem  

            [Graphics:Images/LaplaceConvolutionMod_gr_143.gif],  

with  [Graphics:Images/LaplaceConvolutionMod_gr_144.gif]  and   [Graphics:Images/LaplaceConvolutionMod_gr_145.gif],  is given by  

            [Graphics:Images/LaplaceConvolutionMod_gr_146.gif],  

where  [Graphics:Images/LaplaceConvolutionMod_gr_147.gif]  is the solution to the homogeneous equation  

            [Graphics:Images/LaplaceConvolutionMod_gr_148.gif],  

with  [Graphics:Images/LaplaceConvolutionMod_gr_149.gif],  and  

            [Graphics:Images/LaplaceConvolutionMod_gr_150.gif]  has the Laplace transform given by  [Graphics:Images/LaplaceConvolutionMod_gr_151.gif].  

Proof.

 

Example 12.33. Use convolution to solve the initial value problem

            [Graphics:Images/LaplaceConvolutionMod_gr_152.gif]  
            with
            [Graphics:Images/LaplaceConvolutionMod_gr_153.gif].  

                          [Graphics:Images/LaplaceConvolutionMod_gr_154.gif]
                                    A graph of the solution.

Solution.

    First, we obtain the portion  [Graphics:Images/LaplaceConvolutionMod_gr_155.gif]  of the solution by solving  [Graphics:Images/LaplaceConvolutionMod_gr_156.gif]  with [Graphics:Images/LaplaceConvolutionMod_gr_157.gif] and [Graphics:Images/LaplaceConvolutionMod_gr_158.gif].  Taking the Laplace transform yields  [Graphics:Images/LaplaceConvolutionMod_gr_159.gif],  and we can rearrange the terms to obtain  [Graphics:Images/LaplaceConvolutionMod_gr_160.gif].  Solving for [Graphics:Images/LaplaceConvolutionMod_gr_161.gif] gives  [Graphics:Images/LaplaceConvolutionMod_gr_162.gif],  and it follows that  

            [Graphics:Images/LaplaceConvolutionMod_gr_163.gif]  

Second, we observe that  [Graphics:Images/LaplaceConvolutionMod_gr_164.gif]  and  [Graphics:Images/LaplaceConvolutionMod_gr_165.gif],  and also that  [Graphics:Images/LaplaceConvolutionMod_gr_166.gif].  We compute the portion  [Graphics:Images/LaplaceConvolutionMod_gr_167.gif]  of the solution with a convolution:   

            [Graphics:Images/LaplaceConvolutionMod_gr_168.gif]  

Now we can compute the solution using the convolution method in Theorem 12.25:  

            [Graphics:Images/LaplaceConvolutionMod_gr_169.gif]  

Explore Solution 12.33.

 

Exercises for Section 12.10.  The Laplace Transform: Convolution

 

Library Research Experience for Undergraduates

Laplace Transform

 

 

 

 

 Return to the Complex Analysis Modules

 

 

Return to the Complex Analysis Project

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2012 John H. Mathews, Russell W. Howell