Matched Asymptotic Exapnsion

An example

ϵ x'' (t)  +  γ x' (t) = 
        0, 
x (0) = 0, x (1) = 1

e =. ; 
    sol =  DSolve[{e x''[t]  +  x'[t] == 0, x[0] == 0, 
          x[1] == 1}, x[t], t]

y[t_, f_] := 
      Simplify[x[t]  /.  sol[[1]]   /.  e -> f]

y[t, ϵ]

Plot[{y[t, 10], y[t, 1], y[t, 0.1], y[t, 0.01]}, {t, 0, 
        1}, 
PlotRange -> {{0, 1}, {0, 1}}, 
      AspectRatio -> 1, AxesLabel -> {t, x}]

Another Example

ϵ x'' (t)  + (1 + ϵ) x' (t) = 
        0, 
x (0) = 0, x (1) = (e^(-1))

e =. ;  
    DSolve[{e x''[t]  +  (1 + e) x'[t] + x[t] == 0, 
        x[0] == 0, x[1] == Exp[-1]}, x[t], t]

y[t_, f_] := 
      Simplify[x[t]  /.  %5[[1]]   /.  e -> f]

y[t, \[RawEscape]ϵ]

Plot[{y[t, 0.1], y[t, 0.01], y[t, 0.001]}, {t, 0, 
        1}, 
PlotRange -> {{0, 1}, {0, 1}}]

Yet Another Example

ϵ x'' (t)  +  2 x' (t) = 
        1, 
x (0) = 0, x (1) = 1

e =. ;  
    DSolve[{e x''[t]  +  2  x'[t] == 1, x[0] == 0, 
        x[1] == 1}, x[t], t]

y[t_, f_] := 
      Simplify[x[t]  /.  %1[[1]]   /.  e -> f]

y[t, g]

Plot[{y[t, 1], y[t, 0.1], y[t, 0.01]}, {t, 0, 
        1}, 
PlotRange -> {{0, 1}, {0, 1}}]

Inner and outer solutions for ϵ=0.1

Plot[{1/2 (t   +  1 ), 
1/
            2 (1 - Exp[-2/0.1 t]), 1 (1 - Exp[-2/0.1 t]), 
        1/3 (1 - Exp[-2/0.1 t])} , {t, 0, 1}, 
      PlotRange -> {{0, 1}, {0, 1}}]

Sum of Inner and outer solutions for ϵ=0.1

Plot[1/2  -  1/2 Exp[-2/0.1 t]  +  t/2, {t, 0, 1}, 
      PlotRange -> {{0, 1}, {0, 1}}]