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

Explore Solution 12.31.

For any  a > 0  we define  [Graphics:../Images/LaplaceConvolutionMod_gr_80.gif]  on the interval  [0, a]  as follows.

[Graphics:../Images/LaplaceConvolutionMod_gr_81.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_82.gif]

 

 

 

 

Plot  [Graphics:../Images/LaplaceConvolutionMod_gr_83.gif]  for  [Graphics:../Images/LaplaceConvolutionMod_gr_84.gif].

[Graphics:../Images/LaplaceConvolutionMod_gr_85.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_86.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_87.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_88.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_89.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_90.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_91.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_92.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_93.gif]

Find  [Graphics:../Images/LaplaceConvolutionMod_gr_94.gif]  to obtain the Laplace transform of  [Graphics:../Images/LaplaceConvolutionMod_gr_95.gif].  

[Graphics:../Images/LaplaceConvolutionMod_gr_96.gif]

[Graphics:../Images/LaplaceConvolutionMod_gr_97.gif]

 

 

 

 

Aside. We can check this with Mathematica's  result using the LaplaceTransform package.

[Graphics:../Images/LaplaceConvolutionMod_gr_98.gif]





[Graphics:../Images/LaplaceConvolutionMod_gr_99.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell