This may be a start:

DynamicModule[{fdm, n, m, \[Alpha], \[Beta], h, a, b, A, B},
 fdm[n_] := (m = n - 1; \[Alpha] = -4 + h^2; \[Beta] = 
    6 - 2 h^2 + h^4; h = 1/n;
   Do[Do[a[i, j] = 0, {i, 1, m}], {j, 1, m}];
   Do[b[j, 1] = 0, {j, 1, m}];
   b[m, 1] = -9 - 3 \[Alpha] - 5 h^3;
   b[1, 1] = 3 + 2 h - h^2; b[2, 1] = -1;
   b[m - 1, 1] = -3;
   b[2, 1] = If[n == 4, -4, -1];
   Do[a[j, j] = \[Beta], {j, 1, m - 1}];
   Do[a[j, j] = \[Beta] - 3, {j, m, m}];
   Do[a[j, j + 1] = \[Alpha], {j, 1, m - 1}];
   Do[a[j + 1, j] = \[Alpha], {j, 1, m - 1}];
   Do[a[j + 1, j] = \[Alpha] + 1, {j, m - 1, m}];
   Do[a[j, j + 2] = 1., {j, 1, m - 2}];
   Do[a[j + 2, j] = 1., {j, 1, m - 2}];
   A = Array[a, {m, m}];
   B = Array[b, {m, 1}];
   Flatten[LinearSolve[A, B]]); 
 u = DSolveValue[{u''''[x] + u''[x] + u[x] == 0, u[0] == 1, 
    u'[0] == 2, u[1] == 3, u'''[1] == 5}, u, x];
 Panel[SetterBar[Dynamic[n], Range[4, 20]], 
  Dynamic[{ListPlot[fdm[n], Joined -> True, 
     PlotMarkers -> Automatic],
    Plot[u[x], {x, 0, 1}]}]]]

You can adjust the details of the plots to compare them better. I don't know what you have in mind exactly.

DynamicModule assumes you give a list of local variables:

DynamicModule[{fdm, n, m, \[Alpha], \[Beta], h, a, b, A, B},
 fdm[n_] := (m = n - 1; \[Alpha] = -4 + h^2; \[Beta] = 
    6 - 2 h^2 + h^4; h = 1/n; 
   Do[Do[a[i, j] = 0, {i, 1, m}], {j, 1, m}];
   Do[b[j, 1] = 0, {j, 1, m}];
   b[m, 1] = -9 - 3 \[Alpha] - 5 h^3;
   b[1, 1] = 3 + 2 h - h^2; b[2, 1] = -1;
   b[m - 1, 1] = -3;
   b[2, 1] = If[n == 4, -4, -1];
   Do[a[j, j] = \[Beta], {j, 1, m - 1}];
   Do[a[j, j] = \[Beta] - 3, {j, m, m}];
   Do[a[j, j + 1] = \[Alpha], {j, 1, m - 1}];
   Do[a[j + 1, j] = \[Alpha], {j, 1, m - 1}];
   Do[a[j + 1, j] = \[Alpha] + 1, {j, m - 1, m}];
   Do[a[j, j + 2] = 1., {j, 1, m - 2}];
   Do[a[j + 2, j] = 1., {j, 1, m - 2}];
   A = Array[a, {m, m}];
   B = Array[b, {m, 1}];
   Flatten[LinearSolve[A, B]]);
 Panel[SetterBar[Dynamic[n], Range[4, 20]], 
  Dynamic[ListPlot[{fdm[n]}, Joined -> True, 
    PlotMarkers -> Automatic]]]]

You may consider using SparseArray to define your matrix A:

SparseArray[{
  ({i_, i_} /; i < m) -> \[Beta],
  {m, m} -> \[Beta] - 3,
  ({i_, j_} /; Abs[i - j] == 1 && i < m) -> \[Alpha],
  {m, m - 1} -> \[Alpha] + 1,
  ({i_, j_} /; Abs[i - j] == 2) -> 1.}, {m, m}]