ClearAll["Global`*"];
M=1;α=1.25;a=0;b=1;
λ1=
(t-a)*
α-1
(b-s)
-
α-1
(t-s)
Gamma[α]
;λ2=
t*
α-1
(b-s)
Gamma[α]
;
bb[0]=
Exp[3]-1
Exp[3]+2
*t;
ww[0]=bb[0]/.ts;
TableD2=
1
Gamma[2-α]
*Integrate[
1-α
(s-τ)
*(D[ww[k]/.sτ,{τ,2}]),{τ,a,s},GenerateConditionsFalse];DD[k]=Normal[Series[Simplify[D2+D[ww[k],s]-2*ww[k]],{s,0,M}]];H[k+1]=Simplify[Integrate[λ1*DD[k],{s,a,t},GenerateConditionsFalse]];G[k+1]=Simplify[Integrate[λ2*DD[k],{s,t,b},GenerateConditionsFalse]];bb[k+1]=Simplify[bb[k]+β*(G[k+1]+H[k+1])];ww[k+1]=Expand[bb[k+1]/.ts],{k,0,M};
u1=Expand[bb[M]/.st]
0.864164t+0.0847467tβ-0.76272
1.25
t
β+0.677974
2.25
t
β
ExpandFullSimplify
1
Gamma[2-α]
*Integrate[
1-α
(t-τ)
*(D[u1/.tτ,{τ,2}]),{τ,a,t},GenerateConditionsFalse]+D[u1,t]-2*u1
0.864164-1.72833t-0.779418β-0.9534
0.25
t
β-0.169493tβ+1.72833
1.
t
β+3.05088
1.25
t
β-1.35595
2.25
t
β
root=Integrate[
2
%
,{t,0,1}]
0.248927-0.822165β+0.711862
2
β
Plot[%,{β,0,1}]
0.2
0.4
0.6
0.8
1.0
0.05
0.10
0.15
0.20
0.25
NSolve[root,β,Reals]
{}