# Evaluating a double integral involving MittagLefflerE

Here is the double integral, I want to evaluate and then plot.

Clear[z, k, t, tau];

N1 = 1; (*N1 is in fact infinity*)

alpha = 1;

u[z_?NumericQ, t_?NumericQ, alpha_?NumericQ] := 2/Pi*NIntegrate[k*Sin[k*z]*
NIntegrate[Cos[t - tau]*tau^(alpha - 1)*MittagLefflerE[alpha, alpha, -k^2*tau^alpha],
{tau, 0, t}], {k, 0, N1}]


One of the integral involves MittagLefflerE, which seems to be causing the warnings.

when I try,

u[1, Pi/2, 1]


I get

NIntegrate::inumr:

and then an output.

I even tried Normal[Series[MittagLefflerE[alpha, alpha, -k^2*tau^alpha], {tau, 0, 2}]].

But when I try to plot u for different values of t, accept all those warnings, it take ages.

Plot[{u[z, Pi/2, 1], u[z, 3*Pi/2, 1], u[z, Pi, 1], u[z, 2*Pi, 1]}, {z, -1, 1}]


But so far no luck. Any suggestion?

• Why didn't you bundle it all up in a single NIntegrate[]? u[z_?NumericQ, t_?NumericQ] := With[{α = 1}, 2/π NIntegrate[k Sin[k z] Cos[t - τ] τ^(α - 1) MittagLefflerE[α, α, -k^2 τ^α], {k, 0, ∞}, {τ, 0, t}]]; Plot[Table[u[z, t], {t, π/2, 2 π, π/2}] // Evaluate, {z, -1, 1}] – J. M. is in limbo Sep 14 '17 at 7:07
• @J.M. I am trying your suggestion for the last 10 mins and its still running.... – zhk Sep 14 '17 at 7:21
• Well, I got it to plot, even tho it takes long. – J. M. is in limbo Sep 14 '17 at 7:29

Define a function of the inner integration an call it

nint1[t_?NumericQ, k_?NumericQ] :=
NIntegrate[
Cos[t - tau]*tau^(alpha - 1)*
MittagLefflerE[alpha, alpha, -k^2*tau^alpha], {tau, 0, t}]

u[z_?NumericQ, t_?NumericQ, alpha_?NumericQ] :=
2/Pi*NIntegrate[k*Sin[k*z]*nint1[t, k], {k, 0, N1}]


For me this workes very well.

• This work flawlessly for this particular case. But when I change alpha=0.8 and {z,0,10}, then it is still running for the last 2 hours. – zhk Sep 14 '17 at 11:49