InverseLaplaceTransform

Questions
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1

Hi !

I define a “triangular” function for t>0 :

b = a - c/t1;
tst[t_, t1_, a_, c_] = Piecewise[{{0, t < 0}, {a*t, 0 <= t <= t1}, {b*t + c, t1 < t <= -c/b}, {0, t > -c/b}}]

Then I compute Laplace Transform :

Ltst[s_, t1_, a_, c_] = LaplaceTransform[tst[t, t1, a, c], t, s, Assumptions -> t > 0 && t1 > 0 && a > 0 && c > 0]

And I want to check if I can fetch back the original function :

ILtst[t_, t1_, a_, c_] = InverseLaplaceTransform[Ltst[s, t1, a, c], s, t]

I successfully retrieve the original function for t>0, unfortunately, I get a function which is not zero for t<0.

In this simple case, it is not that disturbing because we can easily work around.
But I want to perform that work for 2 "triangular " functions which are zero for 0<t<T1 and 0<t<T1<T2.
Then the InverseLaplaceTransform provides a totally wrong result.

May I ask you how to force Mathematica to take into account the nullity of the functions in the corresponding ranges in the result of InverseLaplaceTransform?

Best Regards

Sylvain

2

Hi,

The result returned by InverseLaplaceTransform is valid only for t>0. The function is assumed to be 0 for t<0. For example,

In[149]:= f[t] == InverseLaplaceTransform[1/s, s, t]

Out[149]= f[t] == 1

it is implicit that f(t)=0 for t<0, and therefore, strictly speaking, f(t) is a step function. This needs to be taken into account when you interpret the result.

Youngjoo Chung

3

Dear Youngjoo,

Many thanks for those precisions.

Cheers!

Sylvain