In[]:=
ClearAll[u1,u2,a,b,c,d,F1,F2,m];​​m=4;​​λ=-1;​​​​​​u1[0]=a*π*
3
t
+b*π*t;​​u2[0]=c*π*
2
t
+d;​​​​​​For[n=0,n≤m,n++,oldu2=u2[n];​​F2[n]=-
2
(π+4*
3
t
)
*u1[n]-6*
2
t
*u2[n]+12*Integrate[
2
(t-s)
*u1[n]+(t-s)*oldu2,{s,0,t}];​​u2[n+1]=u2[n]+λ*Integrate[D[u2[n],t]-F2[n],{t,0,t}];​​​​F1[n]=-3*
2
t
*u1[n]+(π-2*
3
t
)*oldu2+6*Integrate[(t-s)*u1[n]+
2
(t-s)
*oldu2,{s,0,t}];​​u1[n+1]=u1[n]+λ*Integrate[D[u1[n],t]-F1[n],{t,0,t}]];​​​​​​​​eq1=Collect[u1[1],t]​​eq2=Collect[u2[1],t]​​
Out[]=
dπt+
1
3
c
2
π
3
t
Out[]=
d-
1
2
b
3
π
2
t
-
1
4
a
3
π
4
t
+
4bπ
5
-
8b
2
π
5
5
t
+
4aπ
7
-
8a
2
π
7
7
t
-2bπ
8
t
-
8
5
aπ
10
t
In[]:=
eq3=eq1/.{c1,d1,t0}​​eq4=eq2/.{c1,d1,t0}​​eq5=eq1/.c1,d1,t
1
4
​​eq6=eq2/.c1,d1,t
1
4

Out[]=
0
Out[]=
1
Out[]=
π
4
+
2
π
192
Out[]=
1-
aπ
655360
-
bπ
32768
-
a
3
π
1024
-
b
3
π
32
+
4aπ
7
-
8a
2
π
7
16384
+
4bπ
5
-
8b
2
π
5
1024
In[]:=
Solveeq3+eq5
Sqrt[2]
2
,eq4+eq61+
Sqrt[2]
2
,{a,b}
Out[]=
SolveFalse,2-
aπ
655360
-
bπ
32768
-
a
3
π
1024
-
b
3
π
32
+
4aπ
7
-
8a
2
π
7
16384
+
4bπ
5
-
8b
2
π
5
1024
1+
1
2
,{a,b}