restart;
PDEtools[declare](f(x, t), prime = x, prime = t);
               f(x, t) will now be displayed as f

derivatives with respect to x of functions of one variable will 

   now be displayed with '

derivatives with respect to t of functions of one variable will 

   now be displayed with '

"de1 := (1 - p)*diff(f(x, t), t) + p*(2*f(x, t)^2*diff(f(x, t), 

   x) - diff(f(x, t), x)^2 + 1/2*f(x, t)*diff(f(x, t), x $ 2))"

      "ibvc := f(x, 0) + tan(x);

        n := 3;

        F := unapply(add(b[k](x, t)*p^k, k = 0 .. n), x);"

   F := proc (x) options operator, arrow; b[0](x, t)+b[1](x, 

      t)*p+b[2](x, t)*p^2+b[3](x, t)*p^3 end proc

        "DE1 := series(eval(de1, f = F), p = 0, n + 1)"

          "CO := map(coeffs, eval([ibvc], f = F), p)"

for k from 0 to n do IBVC1 := select(has, CO, b[k]); slv := dsolve({coeff(DE1, p, k), op(IBVC1)}); b[k] := unapply(rhs(slv), x, t); end do;
'F(x)' = F(x) + O(p^(n + 1));

Error, invalid input: rhs received {b[0](x, t) = -tan(x)}, which is not valid for its 1st argument, expr
                                               2               3
F(x) = b[0](x, t) + b[1](x, t) p + b[2](x, t) p  + b[3](x, t) p 

      / 4\
   + O\p /