I am having trouble using NIntegrate
in the following case:
$$\int_0^{100}\int_0^{100}\int_0^{0.5}\int_0^{0.5}\frac{378e^{-\frac{(x-y)^4}{10000}}e^{-625(x_3-y_3)^4}x^4}{\pi[0.02+\sqrt{((x-y)^2+(x_3-y_3)^2)^3}]}\mathrm dx\mathrm dy\mathrm dx_3\mathrm dy_3$$
When I evaluate my NIntegrate
expression I get a message saying the integration is slow at converging. It also says it suspects that the cause is either a singularity or the value of integration being zero or highly oscillatory. I really need a way to integrate this monster, Any help?
In order to obtain this formula I wrote:
expr =
E^(-((x - y)^4/10000) - 625 (x3 -
y3)^4) ((e x + f x^2 + (m x + n x^2) x3 - e y -
f y^2 - (m y + n y^2) y3) ((
945 (x - y)^2 (e x + f x^2 + (m x + n x^2) x3 - e y -
f y^2 - (m y + n y^2) y3))/(
2 π (Sqrt[((x - y)^2 + (x3 - y3)^2)^3]) + 1/50) + (
945 (x - y) (x3 - y3) (g x + h x^2 + (o x + p x^2) x3 - g y -
h y^2 - (o y + p y^2) y3))/(
2 π (Sqrt[((x - y)^2 + (x3 - y3)^2)^3]) + 1/50)) + (g x +
h x^2 + (o x + p x^2) x3 - g y - h y^2 - (o y + p y^2) y3) ((
945 (x - y) (x3 - y3) (e x + f x^2 + (m x + n x^2) x3 - e y -
f y^2 - (m y + n y^2) y3))/(
2 π (Sqrt[((x - y)^2 + (x3 - y3)^2)^3]) + 1/50) + (
945 (x3 - y3)^2 (g x + h x^2 + (o x + p x^2) x3 - g y -
h y^2 - (o y + p y^2) y3))/(
2 π (Sqrt[((x - y)^2 + (x3 - y3)^2)^3]) + 1/50)));
kist = List @@ Expand[expr]
Then in order to evaluate the total integral I need to evaluate the integral of each term of the list. I am stuck at the first term which is the one I posted.
NIntegrate[
(945 E^(-((x - y)^4/10000) - 625 (x3 - y3)^4) x^4) /
(1/50 + 2 π Sqrt[((x - y)^2 + (x3 - y3)^2)^3]),
{x, 0, 100}, {y, 0, 100}, {x3, 0, 0.5}, {y3, 0, 0.5}]
NIntegrate[]
code you used? $\endgroup$ – J. M.'s ennui♦ Mar 13 '18 at 15:46NIntegrate
is likely the incorrect tool for this. Instead, obtain an asymptotic approximation using Laplace's method to get a very accurate (analytic) result. $\endgroup$ – QuantumDot Mar 13 '18 at 16:13