Hello

Mathematica have No built-in function to support fractional calculus.

I have pointed out Wolfram Support several years ago, but nothing at all happens. It's too bad!.

Code below it's not pefect but it works in most simple cases. My code is fractional calculus of Riemann–Liouville.

   FractionalD[nu_, f_, t_, opts___] := 
   Integrate[(t - x)^(-nu - 1) (f /. t -> x), {x, 0, t}, opts, 
   GenerateConditions -> False]/Gamma[-nu]

  FractionalD[mu_?Positive, f_, t_, opts___] := 
  Module[{m = Ceiling[mu]}, 
  D[FractionalD[-(m - mu), f, t, opts], {t, m}]]

  f1[x_] := x;
  FractionalD[1/2, f1[t], t] /. t -> x (* half derivative*)
   (* (2 Sqrt[x])/Sqrt[\[Pi]] *)

  f2[x_] := (2 Sqrt[x])/Sqrt[\[Pi]];
  FractionalD[1/2, f2[t], t] /. t -> x (* half derivative*)
   (* 1 *)

 f3[x_] := x;
 FractionalD[-1/2, f3[t], t] /. t -> x (* half integral*)
 (* (4 x^(3/2))/(3 Sqrt[\[Pi]])*)


f4[x_] :=  (4 x^(3/2))/(3 Sqrt[\[Pi]]);
 FractionalD[-1/2, f4[t], t] /. t -> x (* half integral*)
 (* x^2/2*)

 f5[x_] :=  x;
 FractionalD[-1, f5[t], t] /. t -> x (* integral*)
 (* x^2/2*)

 f6[x_] := x;
 FractionalD[1, f6[t], t] /. t -> x (* derivative. Dosen't work .Use D[f6[x],x] *)
 (* 0 *)

 f7[x_] := Sin[x];
 FractionalD[1/2, f7[t], t] /. t -> x (* half integral*)
 (* Sqrt[2] (Cos[x] FresnelC[Sqrt[2/\[Pi]] Sqrt[x]] + 
    FresnelS[Sqrt[2/\[Pi]] Sqrt[x]] Sin[x])*)

PS:A bonus

  f8[x_] := x;
  n = 1/2;
  InverseLaplaceTransform[LaplaceTransform[f8[x], x, s]*s^n, s, x] (*half derivative*)
   (* (2 Sqrt[x])/Sqrt[\[Pi]]*)
  n1 = -1/2;
  InverseLaplaceTransform[
  LaplaceTransform[f8[x], x, s]*s^n1, s, x](*half integral*)
  (* (4 x^(3/2))/(3 Sqrt[\[Pi]])*)

EDITED: 4.4.2018

Improved a little code.

 FractionalD[\[Alpha]_, f_, x_, opts___] := Integrate[(x - t)^(-\[Alpha] - 1) (f /. x -> t), {t, 0, x}, opts, GenerateConditions -> False]/Gamma[-\[Alpha]]
 FractionalD[\[Alpha]_?Positive, f_, x_, opts___] :=  Module[{m = Ceiling[\[Alpha]]}, If[\[Alpha] \[Element] Integers, D[f, {x, \[Alpha]}], 
 D[FractionalD[-(m - \[Alpha]), f, x, opts], {x, m}]]]

f[x_] := x^2
FractionalD[1, f[x], x]
(* 2 x *)

f[x_] := Sin[x]
FractionalD[-1, f[x], x]
(* 1 - Cos[x]. Where 1 is a integration constant *)

Regards,MI

From this paper:

  RieszD[\[Alpha]_, f_, x_, opts___] := -1/(2*Cos[\[Alpha]*Pi/2])*1/Gamma[\[Alpha]]*(Integrate[(x - t)^(-\[Alpha] - 1) (f /. x -> t), {t, -Infinity, x}, opts, 
  GenerateConditions -> False] + Integrate[(t - x)^(-\[Alpha] - 1) (f /. x -> t), {t, x, Infinity},opts, GenerateConditions -> False])
  RieszD[\[Alpha]_?Positive, f_, x_, opts___] := Module[{m = Ceiling[\[Alpha]]}, D[RieszD[-(m - \[Alpha]), f, x, opts], {x, m}]]

  f[x_] := Exp[-x]
  RieszD[1/2, f[x], x, Assumptions -> x > 0]
  (* -(((1/2 - I/2) E^-x)/Sqrt[2]) *)

  f1[x_] := Sin[x]
  RieszD[1/2, f1[x], x]
  (* (Sqrt[\[Pi]/2] Sqrt[1/x] Sqrt[x] (Cos[x] - Sin[x]) + Sqrt[\[Pi]/2] (Cos[x] + Sin[x]))/(2 Sqrt[2 \[Pi]]) *)

I'm don't know is my code right, because I have not found any examples to check it out?