CODEWITHSUNDEEP
maple
Nasser
Nasser

Reputation: 13173

how to evaluate integral that contains HeunT in it?

I am trying to plot solution obtained from ODE solution which has in it an integral from 0..x. The integral contains HeunT special function. Maple is very slow in producing the plot, and traced it down to it not being able or slow in evaluating this integral. It just says evaluating at the bottom of the window.

Here is the part of the solution which contains the integral, copied as is from notebook

r := Int(exp((2/3)*(-(9/49)/Pi^4)^(1/2)*_z1*
      (3*Pi^2+3*Pi*_z1+_z1^2))/HeunT(0, 0, 0, (1/3)*3^(2/3)*
       (-(9/49)/Pi^4)^(1/6)*2^(1/3)*(_z1+Pi))^2, _z1 = 0 .. x)

To plot the solution, this integral needs to be evaluated for different x values. But Maple can't evaluate this integral for any x it seems. For example

evalf(subs(x=Pi,r));

Maple just keeps saying evaluating.... Is there a way to evaluate this integral, which is part of the the full ODE solution (given below if needed), so I can actually plot the solution of the ODE?

Here is the complete code below, starting from the ODE itself. I am basically just trying to plot the solution for a boundary value sturm-Liouville ODE

restart;
lam:=n->(9*n^2/(49*Pi^4)):
ode:=diff(y(x),x$2)+lam*(x+Pi)^4*y(x)=0:
sol:=dsolve({ode,y(0)=0},y(x)):
sol:=subs({lam=lam(1),_C2=sqrt(6/(7*Pi^3))},sol);

The solution is, which is what I am trying to plot for x=0..Pi 

sol := y(x) = (1/7)*42^(1/2)*exp(-(1/3)*(-(9/49)/Pi^4)^(1/2)*x*
 (3*Pi^2+3*Pi*x+x^2))*HeunT(0, 0, 0, (1/3)*3^(2/3)*
 (-(9/49)/Pi^4)^(1/6)*2^(1/3)*(x+Pi))*(Int(exp((2/3)*
 (-(9/49)/Pi^4)^(1/2)*_z1*(3*Pi^2+3*Pi*_z1+_z1^2))/
 HeunT(0, 0, 0, (1/3)*3^(2/3)*(-(9/49)/Pi^4)^(1/6)*2^(1/3)*
 (_z1+Pi))^2, _z1 = 0 .. x))/Pi^(3/2)

And you can see the integral mentioned at the top, is inside the solution.

Maple 2016, windows 7

Upvotes: 0

Views: 74

Answers (1)

zhk
zhk

Reputation: 331

plot(Re(convert(simplify(series(rhs(sol), x, 3), size),polynom)),x=0..Pi)

enter image description here

Upvotes: 1

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