0
$\begingroup$

I compiled the following Mathematica code

m = 40;
eea = Integrate[(x - y)^(-(α - 1) - 
     1) ((Sum[(z^(s))*(Cos[((b)^(s))*(π)*(y)]), {s, 0, m}]* 
        Sum[((λ)^((d - 2)*k))*(Sin[(λ^k)*(y)]), {k, 1,
           m}]) - (Sum[(z^(s))*(Cos[((b)^(s))*(π)*(0)]), {s, 0, 
          m}]* Sum[((λ)^((d - 2)*
              k))*(Sin[(λ^k)*(0)]), {k, 1, m}])), {y, 0, x}];
sa = D[eea/ Gamma[-(α - 1)], {x, 1}]

So that others may avoid the 1 hour computation, the output of eea from Mathematica 12.0 can be acquired from the following:

eea = Uncompress[Uncompress["1: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"]];

After an hour i got an output and for the same i gave the following substitution

% /. x -> 1 /. λ -> 3 /. d -> 15/10 /. 
   z -> 1/10 /. α -> 2/10 /. b -> 35

Then for the resulting output i calculated the numerical value using the code

N[%]

So for the above numerical compilation i got the following error message

General::ovfl: Overflow occurred in computation.
General::stop: Further output of General::ovfl will be suppressed during this calculation.

Could anyone please suggest any answer to solve this issue.

Thank You

$\endgroup$
  • $\begingroup$ how long does it take to integrate? $\endgroup$ – Nasser May 29 at 13:34
  • $\begingroup$ @Nasser about an hour. Even when i set m=30 i got the same overflow error $\endgroup$ – vicky May 29 at 13:38
  • 2
    $\begingroup$ about an hour in this case, do you not think it is important to warn people about this in your post? $\endgroup$ – Nasser May 29 at 13:40
  • $\begingroup$ @vicky since the integration seems to work, albeit slowly, could you add the value of eea in your post so people don’t have to run it themselves but can still run the rest of your code to try and help you? $\endgroup$ – MarcoB May 29 at 13:41
  • 2
    $\begingroup$ After putting a fair bit more work into investigating sa via numerical means, it's very unstable. The integrand is very oscillatory, and numerical estimates of that derivative fluctuate randomly even with WorkingPrecision of 100+. I'm not sure there is a good way to answer this problem. $\endgroup$ – eyorble May 29 at 17:41
2
$\begingroup$

This is more of an extended comment unless you're willing to "believe" in an inference I propose.

The first step is to take a look at your function with m undefined:

m =.;
integrand = (x - y)^(-(α - 1) - 1) ((Sum[(z^(s))*(Cos[((b)^(s))*(π)*(y)]), {s, 0, m}]*
  Sum[((λ)^((d - 2)*k))*(Sin[(λ^k)*(y)]), {k, 1, m}]) - 
  (Sum[(z^(s))*(Cos[((b)^(s))*(π)*(0)]), {s, 0, m}]*
  Sum[((λ)^((d - 2)*k))*(Sin[(λ^k)*(0)]), {k, 1, m}]))

Formula of interest

So we can integrate the individual terms.

integral = Integrate[(x - y)^-α z^s Cos[b^s π y] λ^((-2 + d) k) Sin[y λ^k], 
  {y, 0, x}, Assumptions -> {x > 0, α < 1}]

integral of an individual term

An individual term of the derivative you want is

derivative = D[integral/Gamma[-(\[Alpha] - 1)], {x, 1}] // FullSimplify

derivative of an individual term

Now set the parameter values that you list and try values of s and k that give you the overflow error when getting a numerical value for the term (i.e., using //N). Here's one that does that.

parms = {x -> 1, λ -> 3, d -> 15/10, z -> 1/10, α -> 2/10, b -> 35};
derivative /. parms /. s -> 22 /. k -> 1 // N

overflow error overflow error overflow error Further output suppressed

That term without //N is

exact value of term

So we see the likely culprit is the $b^s$ term (how appropriate!). What to do about that?

Consider rewriting one of the hypergeometric functions as

HypergeometricPFQ[{1}, {7/5, 19/10}, -(1/4) (3 - bigBs π)^2]

We can evaluate this function for values of bigBs until we get overflow errors. We see that the above function approaches zero with larger values of bigBs. So we could set some threshold for values of bigBs. The proposal is to do so in the following manner:

threshold = 10^20;
m = 40;
Sum[derivative /. b^s -> Min[b^s, threshold] /. parms, {s, 0, m}, {k, 1, m}] // N
(* -0.30444 *)

This takes about 30 seconds.

| improve this answer | |
$\endgroup$
2
$\begingroup$

Try higher precision:

eea /. x -> 1 /. λ -> 3 /. d -> 15/10 /. 
    z -> 1/10 /. α -> 2/10 /. b -> 35.`50 // AbsoluteTiming

(*  {13.6045, -0.08742710084680679010043795135152006816}  *)
| improve this answer | |
$\endgroup$
  • $\begingroup$ +1 Can't beat a short, to-the-point, fast, accurate answer. $\endgroup$ – JimB May 30 at 3:40

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