On the Solution of the Cupula Equation using Fractional Calculus.

Moustafa ElShahed
MisrUniversity for Science and Technology
Faculty of Engineering
6 October City-Giza-Egypt
e-mail:m_1240@hotmail.com

Solve[(x^4) + 2 η* (ω^((3/2)))*x + (ω^2) == 0, 
      x]

M = (1/ω) ((((9/2)))^((1/3)));

N1 = 2*((((4/3)))^((1/3)))*ω;

H = (((9 (η^2) + (Sqrt[3 (27 (η^4) - 16 )])))^((1/3)));

N1 = 2*((((4/3)))^((1/3)))*ω;

Ψ = (Sqrt[(N1/H) + (H/M)]);

λ1 = (1/2) Ψ - (1/2) (Sqrt[- (Ψ^2) - ((4  η (ω^((3/2))) )/( Ψ))]);

λ2 = (1/2) Ψ + (1/2) (Sqrt[- (Ψ^2) - ((4  η (ω^((3/2))) )/( Ψ))]);

λ3 = ((-1)/2) Ψ - (1/2) (Sqrt[- (Ψ^2) + ((4  η (ω^((3/2))) )/( Ψ))]);

λ4 = ((-1)/2) Ψ + (1/2) (Sqrt[- (Ψ^2) + ((4  η (ω^((3/2))) )/Ψ)]);

x[
          t_, η_, ω_] = (1/((λ1 - λ2) (λ1 - λ3) (λ1 - λ4))) (λ1 *(E^(t λ(1^2)))*
                Erfc[-λ1 (Sqrt[t])]) + (1/((λ2 - λ1) (λ2 - λ3) (λ2 - λ4))) (λ2 *(E^(t λ(2^2)))*
                Erfc[-λ2 (Sqrt[t])]) + (1/((λ3 - λ2) (λ3 - λ1) (λ3 - λ4))) (λ3 *(E^(t λ(3^2)))*
                Erfc[-λ3 (Sqrt[t])]) + (1/((λ4 - λ2) (λ4 - λ3) (λ4 - λ1))) (λ4*(E^(t λ(4^2)))*
                Erfc[-λ4 (Sqrt[t])]);

Plot[4  x[t, 10, 3], {t, 0, 5}];

To define the fractional derivative with mathematica, the land market paper in this field is to [Gerd Baumann 1999] as follow:

Unprotect[D];

\(D[x_\^m_. , {x_, q_}] := \(Gamma[m + 1]\/Gamma[m - q + 1]\) 
          x\^\(m - q\) /; 
        Head[q] == Real\  || \ Head[q] == Rational || 
          Head[q] == Complex\)

Protect[D];

\(∂\_{x, \(-\(1\/2\)\)}\ x\)

\(∂\_{x, 2.1}\ x\^2\)

\(∂\_{x, 11.5 + I}\ x\^4\)

. The Riemann-Liouville and Weyl calculi

\(\(\(Remove[RiemannLiouville]\)\(
\)\(
\)
    \) (*\ \(\(--\(-\ 
              treatment\)\)\ of\ \(constants\ --\)\)\(-\)\ *) \)
\(\(RiemannLiouville[f_, {x_, order_, a_:  0}] := 
        f\ RiemannLiouville[1, {x, order, a}] /; 
          FreeQ[f, x] && \ Not[SameQ[f, 1]];\)
	
 (*\ \(--\(-\ 
            remove\)\)\ constants\ from\ the\ \(argument\ --\)\ *) \)
\(\(RiemannLiouville[c_\ f_, {x_, order_, a_:  0}] := 
        c\ RiemannLiouville[f, {x, order, a}]\  /; \ 
          FreeQ[c, 
            x];\)

 (*\ \(--\(-\ \(linerity\ --\)\)\)\(-\)\ *) \)
\(\(\(RiemannLiouville[f_\  + \ g_, {x_, order_, a_:  0}] := 
      RiemannLiouville[f, {x, order, a}]\  + \ 
        RiemannLiouville[g, {x, order, a}]\)\(
\)\(	\)\(
\)
    \) (*\ \(--\(-\ \(composition\ --\)\)\)\(-\)\ *) \)
\(\(\(RiemannLiouville[\ 
        RiemannLiouville[f_, {x_, order1_, a_:  0}], {x_, order2_, a_:  0}] := 
      RiemannLiouville[f, {x, order1 + order2, a}]\  /; 
        order1 < 0\)\(
\)\(
\)
    \) (*\ \(--\(-\ \(Identity\ --\)\)\)\(-\)\ *) \)
\(\(\(RiemannLiouville[f_, {x_, 0, a_:  0}] := f\)\(
\)\(
\)\(
\)
    \) (*\(\(--\(-\ main\)\)\ \(function\ --\)\)\(-\)\ *) \)
\(\(\(RiemannLiouville[f_, {x_, order_, a_:  0}] := 
      Block[{n, int, $y1, q}, 
		If[
          NumericQ[order] && Simplify[order > 0], n = Floor[order]; \ 
          q = \ order - n, q = order; n = 0]; 
		int\  = \ 
          Integrate[\(\((x - $y1)\)\^\(\(-q\) - 1\)\) \((f /. 
                  x :> $y1)\), {$y1, a, x}, 
            GenerateConditions -> False]; 
		D[
            int\ /\ Gamma[\(-q\)], {x, n}]\  /; \ 
          FreeQ[int, $y1]
		]\)\(
\)
    \)			\)