Error to converge NIntegrate with FindRoot

Question

I have the following error at FindRoot using a function with NIntegrate:

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in y near {y} = {10.9130359139484499045371956829929066187800934528695506742224097252}. NIntegrate obtained 2.2676305277968822`*^-14 and 9.641852330564264`*^-13 for the integral and error estimates.

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in y near {y} = {10.9130359139484499045371956829929066187800934528695506742224097252}. NIntegrate obtained 2.2676305277968822`*^-14 and 9.641852330564264`*^-13 for the integral and error estimates.

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in y near {y} = {10.1455}. NIntegrate obtained -3.85109*10^-16 and 1.5965726624074823`*^-16 for the integral and error estimates.

General::stop: Further output of NIntegrate::ncvb will be suppressed during this calculation.

My code:

b = -35/100;
d = 9;
u[y_] := (Tanh[a*(y - d)] + Tanh[a*d])/(1 + Tanh[a*d]) + 
   b*Sqrt[3]*y/d*Exp[-1.5*(y/d)^2 + 0.5];
f[a_?NumericQ] := NIntegrate[u[y] (1 - u[y]), {y, 0, 80}];
a = Rationalize[FindRoot[f[a] == 1, {a, 14/100}][[1, 2]], 0];

g[ydd_?NumericQ] := NIntegrate[u[y], {y, 0, ydd}];
FindRoot[g[ydd], {ydd, 10}]

What is wrong?

Answer 1

"NIntegrate::slwcon : Numerical integration converging too slowly; suspect one of the following : singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small."

So use a higher WorkingPrecision

Clear[a]

b = -35/100;
d = 9;

u[y_] := (Tanh[a*(y - d)] + Tanh[a*d])/(1 + Tanh[a*d]) + 
   b*Sqrt[3]*y/d*Exp[-3/2*(y/d)^2 + 1/2];

f[a_?NumericQ] := NIntegrate[u[y] (1 - u[y]), {y, 0, 80}];

a = Rationalize[a /. FindRoot[f[a] == 1, {a, 14/100}], 0];

g[ydd_?NumericQ] := 
  NIntegrate[u[y], {y, 0, Rationalize[ydd, 0]}, WorkingPrecision -> 21];

FindRoot[g[ydd], {ydd, 10}, WorkingPrecision -> 20]

(* {ydd -> 10.913036109675691159} *)

Answer 2

Set an explicit value for AccuracyGoal within NIntegrate:

g[ydd_?NumericQ] := NIntegrate[u[y], {y, 0, ydd}, AccuracyGoal -> 10]

FindRoot[g[ydd], {ydd, 10}]
(* Out: {ydd -> 10.913} *)

Even AccuracyGoal -> Automatic would work in this case. I suspect the difficulty may arise because your integral value approaches zero. The above returns the same result as your code, but without warnings.

A plot supports this results:

Plot[
  NIntegrate[u[y], {y, 0, ydd}], {ydd, 10.91, 10.915},
  AxesOrigin -> {Automatic, 0},
  Epilog -> {Red, PointSize[0.02], Point[{10.913036109679442, 0}]}
]