In[]:=
fx[t_]=Piecewise[{{0,t<=0},{(8/3)*(6*t^3-6*t^2+1),t<1/2},{(8/3)*(-2*(t-1)^3),t<1},{0,t>=1}}]
Out[]=
0
t≤0
8
3
(1-6
2
t
+6
3
t
)
t<
1
2
-
16
3
3
(-1+t)
t<1
0
True
In[]:=
fy[t_]=Piecewise[{{0,t<=0},{(16/9)*Sqrt[3]-(16/3)*t+(64/27)*t^3,t<Sqrt[3]/2},{0,t>Sqrt[3]/2}}]
Out[]=
0
t≤0
16
3
3
-
16t
3
+
64
3
t
27
t<
3
2
0
True
In[]:=
Ly=LaplaceTransform[fx[t],t,s]
Out[]=
8
-s

(12-48
s/2

+36
s

-12
s

s+
s

3
s
)
3
4
s
In[]:=
Lx=LaplaceTransform[fy[t],t,s]
Out[]=
16
-
3
s
2

-8+8
3
s
2

-4
3
s-3
3
s
2

2
s
+
3
3
s
2

3
s
9
4
s
In[]:=
dL=InverseLaplaceTransform[Lx*Ly,s,t]
Out[]=
128
27
3
t-
3
2
t
2
-2
3
3
t
+
11
4
t
6
+
3
3
4
t
2
-
9
5
t
10
-
2
6
t
15
+
2
7
t
35
+
1
2
3
4
(-1+t)
HeavisideTheta[-1+t]-
3
10
5
(-1+t)
HeavisideTheta[-1+t]+
2
105
7
(-1+t)
HeavisideTheta[-1+t]-2
3
4
-
1
2
+t
HeavisideTheta-
1
2
+t+
6
5
5
-
1
2
+t
HeavisideTheta-
1
2
+t-
8
105
7
-
1
2
+t
HeavisideTheta-
1
2
+t-
6
-1-
3
2
+t
HeavisideTheta-1-
3
2
+t
5
3
-
2
105
7
-1-
3
2
+t
HeavisideTheta-1-
3
2
+t+
4
6
-
1
2
-
3
2
+t
HeavisideTheta-
1
2
-
3
2
+t
5
3
+
8
105
7
-
1
2
-
3
2
+t
HeavisideTheta-
1
2
-
3
2
+t-
2
3
-
3
2
+t
HeavisideTheta-
3
2
+t
3
-
1
3
4
-
3
2
+t
HeavisideTheta-
3
2
+t+
2
5
3
5
-
3
2
+t
HeavisideTheta-
3
2
+t+
2
15
6
-
3
2
+t
HeavisideTheta-
3
2
+t-
1
5
3
6
-
3
2
+t
HeavisideTheta-
3
2
+t-
2
35
7
-
3
2
+t
HeavisideTheta-
3
2
+t
In[]:=
Integrate[dL,{t,0,1/2+Sqrt[3]/2}]
Out[]=
22681-8
3
22680
In[]:=
dRect[t_]=Integrate[fx[tau]*fy[t-tau],{tau,-Infinity,Infinity},Assumptions->{t>=0,t<=1/2+Sqrt[3]/2}]
Out[]=
4(157+105
3
-161t-252
2
t
+140
3
t
)
2835
t
1
2
(1+
3
)
64t(210
3
-315t-420
3
2
t
+385
3
t
+315
3
3
t
-189
4
t
-28
5
t
+12
6
t
)
2835
0<t≤
1
2
-
16(31+105
3
-308t-1680
3
t+2478
2
t
+2520
3
2
t
-2380
3
t
-1680
3
3
t
+700
4
t
+420
3
4
t
+84
5
t
-112
6
t
+16
7
t
)
2835
1
2
<t≤
3
2
-
4(-1765-1017
3
+525t+336
3
t+7476
2
t
+4536
3
2
t
-13020
3
t
-6720
3
3
t
+6160
4
t
+5040
3
4
t
-3024
5
t
-448
6
t
+192
7
t
)
2835
1<t<
1
2
(1+
3
)
-
4(-821+663
3
-4067t-6384
3
t+16212
2
t
+14616
3
2
t
-20860
3
t
-13440
3
3
t
+8960
4
t
+6720
3
4
t
-2688
5
t
-896
6
t
+256
7
t
)
2835
3
2
<t≤1
0
True
In[]:=
Integrate[dRect[t],{t,0,1/2+Sqrt[3]/2}]
Out[]=
22681-8
3
22680
In[]:=
Integrate[fx[t],{t,0,1}]
Out[]=
1
In[]:=
Integrate[fy[t],{t,0,Sqrt[3]/2}]
Out[]=
1