Why is the LAPLACE command in the Symbolic Toolbox able to handle ABS(t+1) but unable to process ABS(t-1)?

I am using the following commands:

syms t s    

laplace(abs(t))

 ans =

 1/s^2 

laplace(abs(t+1))

 ans =

 1/s^2+1/s

laplace(abs(t-1))

 ans =

 laplace(abs(t-1),t,s)

Answer 1

[MathWorks Support Team]

The problem seems to stem from Maple not knowing whether or not to assume 't' is greater or less than one. For example, if we do the following, we do receive an answer:

maple('assume (t>1)')

 ans =

      ''

laplace(abs(t-1))

 ans =

 1/s^2*exp(-s)*exp(s)*(1-s)

So, as a workaround you might try the following:

If we have the following as our problem:

 Int(0,INF,'abs(t-1)*exp(-s*t)')

We break the integral up into the following two parts:

 part1 = integral(1,INF,'(t-1)*exp(-s*t)')

and

 part2 = integral(0,1,'(1-t)*exp(-s*t)')

Where our entire integral is the sum of part1 and part2.

For Part 1, a substitution of the following will give a fairly straightforward Laplace transform:

 Tau = t-1

For Part 2, we can use MAPLE to symbolically integrate, and then evaluate at the endpoints. For example:

integral = maple('int','(1-t)*exp(-s*t)','t')

 integral =

 -1/s*(1/s*(-exp(-s*t)*s*t-exp(-s*t))+exp(-s*t))

end1 = subs(integral,1)

 end1 =

 -1/s*(1/s*(-exp(-s)*s-exp(-s))+exp(-s))

end2 = subs(integral,0)

 end2 =

 -1/s*(-1/s+1)

answer = end1 - end2

 answer =

 -1/s*(1/s*(-exp(-s)*s-exp(-s))+exp(-s))+1/s*(-1/s+1)

This will allow you to calculate the transform of abs(t-1) and similar functions.