equ=(∂_(x,x))T[x]==3*0.2*(T[x]-20);
bound=T[0]==0;bound2=T[1]==0;

Tapr=a*x*(1-x);

Plot[Evaluate[Table[Tapr,{a,0,10}]],{x,0,1}]

equApr=equ/.{T[x]->Tapr,T''[x]->(∂_(x,x))Tapr}

Residuum=equApr/.Equal[left_,right_]:>-left+right

Residuum==0

Plot[Evaluate[Table[Residuum,{a,5.4,6,0.1}]],{x,0,1}]

MVR=0==(∫_0^1)w*Residuum\[DifferentialD]x

Collocation method

CM=0==(∫_0^1)DiracDelta[x-0.5]*Residuum\[DifferentialD]x

resCM=Solve[CM,a][[1]]

Tapr/.resCM

Plot[Evaluate[Tapr/.resCM],{x,0,1}]

Plot[Evaluate[Residuum/.resCM],{x,0,1}]

Least Square Method

Residuum

Plot[Evaluate[(∂_a) Residuum],{x,0,1}]

LSM=0==(∫_0^1)((∂_a) Residuum)*Residuum\[DifferentialD]x

resLSM=Solve[LSM,a][[1]]

Tapr/.resLSM

Plot[Evaluate[Tapr/.resLSM],{x,0,1}]

Plot[Evaluate[Residuum/.resLSM],{x,0,1}]

Galerkin’s Method

Residuum

Plot[Evaluate[(∂_a) Tapr],{x,0,1}]

GM=0==(∫_0^1)((∂_a) Tapr)*Residuum\[DifferentialD]x

resGM=Solve[GM,a][[1]]

Tapr/.resGM

Plot[Evaluate[{Tapr/.resCM,Tapr/.resLSM,Tapr/.resGM}],{x,0,1}]

Plot[Evaluate[{Residuum/.resCM,Residuum/.resLSM,Residuum/.resGM}],{x,0,1}]