### 1.9 Obtain Laplace transform for a piecewise functions

Problem: Obtain the Laplace transform for the function deﬁned in the following ﬁgure.

Function f(t) to obtain its Laplace transform

Comment: Mathematica solution was easier than Matlab’s. In Matlab the deﬁnition of the Laplace transform is applied to each piece separately and the result added. Not ﬁnding the piecewise maple function to access from inside MATLAB did not help.

Mathematica

 Remove["Global`*"]; f[t_] := Piecewise[{{0,t<0}, {t,t>= 0 && t< T}, {T, t>T}}] Simplify[LaplaceTransform[f[t],t,s] ,{T>0}] Out[]= (1 - E^((-s)*T))/s^2

Matlab

 clear all; syms T t s; syms s positive; I1 = int(t*exp(-s*t),t,0,T); I2 = int(T*exp(-s*t),t,T,Inf); result = simple(I1+I2); pretty(result) 1 - exp(-T s) ------------- 2 s

Maple

 With Maple, had to use Heaviside else Laplace will not obtain the transform of a piecewise function. restart; assume(T>0): interface(showassumed=0): f:=t->piecewise(t<0,0,t>=0 and tT,T): r:=convert(f(t),Heaviside): r:=inttrans[laplace](r,t,s); ${\frac {1-{{e}^{-sT}}}{{s}^{2}}}$