ClearAll[Matrix];

(*------------------------------------------------------------------*)

h = 0.1;
Xy = Range[-1.0, 1.0, h];
t = Length[Xy] + 2;
Cstala = (1/(h^2)) + (1/(2*h));
Astala = (1/(h^2)) - (1/(2*h));
Bstala = -2/(h^2);

(*------------------------------------------------------------------*)
\

(*  f(x) = 2x + 1, interval : -1 , 1 , step : 0.1  *)
alfaZero = 0;
alfaN = 0;
betaZero = 0;
betaN = 0;
n = t - 1;
A = Table[0, {i, n - 1}];
B = Table[0, {i, n - 1}];
CVector = Table[0, {i, n - 1}];
F = Table[0, {i, n - 1}];
TableAlfa = Table[0, {i, n + 1}];
TableBeta = Table[0, {i, n + 1}];
TableAlfa[[1]] = alfaZero;
TableAlfa[[n + 1]] = alfaN;
TableBeta[[1]] = betaZero;
TableBeta[[n + 1]] = betaN;

(*------------------------------------------------------------------*)
\

(*   Fill the Vectors A, B, C and F for function f(x) = 2x + 1 in \
range (-1,1) with step 0.1 *)
For[i = 1, i <= n - 1, i++,
  	A[[i]]  = 1*Astala;
  	B[[i]] = (-1)*(Bstala + Xy[[i]]);
  	CVector[[i]] = 1*Cstala;
  	F[[i]]  = 2*Xy[[i]] + 1; (* f(x) = 2x + i *)
  ];


(*------------------------------------------------------------------*)

(*Fill the alfa and beta vectors---------------------------------the \
vectors A,B,C,F are indexing from 1 to N-1\[Rule]size is N-1 the \
vectors alfa and beta are indexing from 0 to N\[Rule]size is N+1 So \
that is the reason why when we take alfa[i] and A[i] we need to index \
A[i-1],because it has indexes ones \
back---------------------------------*)
For[i = 2, i < n + 1, i++,
  TableAlfa[[i]] = (-1*
      CVector[[i - 1]])/(A[[i - 1]]*TableAlfa[[i - 1]] - B[[i - 1]]);
  TableBeta[[i]] = (F[[i - 1]] -
      A[[i - 1]]*TableBeta[[i - 1]])/(A[[i - 1]]*TableAlfa[[i - 1]] -
      B[[i - 1]]);
  ];

(*------------------------------------------------------------------*)

(*Calculating resaults depended on alfa and beta \
values------------------------------------------------------------------\
*)
U = Table[0, {i, n + 1}];
(*Firstly,calculate U[n]*)
U[[n + 1]] = (TableAlfa[[n + 1]]*TableBeta[[n]] +
     TableBeta[[n + 1]])/(1 - TableAlfa[[n + 1]]*TableAlfa[[n]]);
(*Then,calculate backward U*)
For[i = n, i > 0, i--,
  U[[i]] = TableAlfa[[i]]*U[[i + 1]] + TableBeta[[i]];];

(*------------------------------------------------------------------*)