I have been using Mathematica 9 to solve a double integration (Equation 1) but without any success.
I was told that the value of \alpha will be 0.2826 when the following data is used for the constants:
I uses NIntegrate but still can’t obtain 0.2828 for the \alpha value. I am not sure if i have the code wrong and hope someone in the community can help.
It will be a good idea if you publish the notebook showing what you have tried so far.
Eqn 1 is separated into two sections and notebooks are attached.
Can anyone please comment have i made any mistakes in the attached notebooks?
Hi,
one of the problems in both notebooks is that you have not declared A in the equation for lct[t]. The numerical integration will not work like that.
Mathematica says that the integral does not converge sufficiently well, and gives three suspected reasons for it:
Numerical integration converging too slowly; suspect one of the
following: singularity, value of the integration is 0, highly
oscillatory integrand, or WorkingPrecision too small.
The integrand appears to be very nasty indeed.
If I change your 3.1416 to Pi
ClearAll["Global`*"]
lmean = 0.8814;
lc = 0.5613;
Af = 1.2;
u = 0.1;
a = 1.0813;
b = 1.2;
p = 1;
q = 2.5294;
lct[t_] := lc*(1 - Af*Tan[t])/(Exp[u*t])
f[l_] := a*b*l^(b - 1)*Exp[-a*l^b]
g[t_] := ((Sin[t])^(2*p - 1)*(Cos[t])^(2*q - 1))/
Chop[(Integrate[(Sin[w])^(2*p - 1)*(Cos[w])^(2*q - 1), {w, 0,
Pi/2}])]
NIntegrate[
f[l]*g[t]*(l/lmean)*(1 -
Af*Tan[t])*(1 - (lc*(1 - Af*Tan[t])/(2*l*Exp[u*t]))), {t, 0,
Pi/2}, {l, 0, lct[t]}]
I get
During evaluation of In[184]:= NIntegrate::zeroregion: Integration
region
{{1.,0.99999999999999415148564447489073961419065634827050978249802712372},{0.25,0.5}}
cannot be further subdivided at the specified working precision.
NIntegrate assumes zero integral there and on any further indivisible
regions. >>During evaluation of In[184]:= 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. >>During evaluation of In[184]:= NIntegrate::ncvb: NIntegrate failed to
converge to prescribed accuracy after 18 recursive bisections in t
near {t,l} =
{0.99999999998256007485943732904080818950806429498343042006069936178,0.499962}.
NIntegrate obtained
-6.7530887552778382916336050578428251258689827762888800560827789037310^893804534628964+5.4691121715425751471977471775999760599853203693952910350123540463610^893804534628964
I and
-6.7530887552778382916336050578428251258689827762888800560827789037310^893804534628964+5.4691121715425751471977471775999760599853203693952910350123540463610^893804534628964
I for the integral and error estimates. >>Out[184]= -6.75308875527783810^893804534628964 +
5.46911217154257510^893804534628964 I
In fact, if you set the Precision to infinity
lmean = SetPrecision[0.8814, Infinity];
lc = SetPrecision[0.5613, Infinity];
Af = 6/5;
u = 1/10;
a = SetPrecision[1.0813, Infinity];
b = SetPrecision[1.2, Infinity];
p = 1;
q = SetPrecision[2.5294, Infinity];
lct[t_] := lc*(1 - Af*Tan[t])/(Exp[u*t])
f[l_] := a*b*l^(b - 1)*Exp[-a*l^b]
g[t_] := ((Sin[t])^(2*p - 1)*(Cos[t])^(2*q -
1))/(Integrate[(Sin[w])^(2*p - 1)*(Cos[w])^(2*q - 1), {w, 0,
Pi/2}])
NIntegrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], {t, 0, Pi/2}, {l, 0, lct[t]}]
You get - at least in MMA10- an error message. If you click on “show all” you get a rather detailed output that shows what is going on.
Cheers,
Marco
Thanks Marco for your comment.
Not sure if the integrand can be solved or not.
Could you possibly use the exact value of the indefinite integral to help trying to find your solution
In[1]:= lmean = 8814/10^4;
lc = 5613/10^4;
Af = 12/10;
u = 1/10;
a = 10813/10^4;
b = 12/10;
p = 1;
q = 25294/10^4;
lct[t_] := lc*(1 - A*Tan[t])/(Exp[u*t]);
f[l_] := a*b*l^(b - 1)*Exp[-a*l^b];
g[t_] := ((Sin[t])^(2*p - 1)*(Cos[t])^(2*q - 1))/(Integrate[(Sin[w])^(2*p - 1)*(Cos[w])^(2*q-1), {w,0,Pi/2}]);
FullSimplify[Integrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], l], Assumptions -> 0 <= t < 2 Pi && 0 <= l]
Out[12]= -(1/2407281094647)126470000 E^(-((10813 l^(6/5))/10000) + t/10) l^(4/5) Cos[t]^(10147/2500)
(150000 + 97317 l^(6/5) + 100000 E^((10813 l^(6/5))/10000)ExpIntegralE[1/3, (10813 l^(6/5))/10000]) Sin[t]
In[13]:= FullSimplify[Integrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], t, l],
Assumptions -> 0 <= t < 2 Pi && 0 <= l]
Out[13]= -(1/(Cos[t]^(2353/2500)))(2470117187500/892485605516233971387177524702941121272167 -
(1788364843750 I)/68652738885864151645167501900226240097859) E^(-((10813 l^(6/5))/10000) +
(1/10 - I) t) (1 + E^(2 I t)) l^(4/5) (150000 + 97317 l^(6/5) + 100000 E^((10813 l^(6/5))/10000)
ExpIntegralE[1/3, (10813 l^(6/5))/10000]) (1243209104158084056000 E^(I t)
Hypergeometric2F1[2647/5000 - I/20, 1, 7353/5000 - I/20, -E^(2 I t)] - (250 + 2353 I)
(51882557652852180070 Cos[t] + 26149933547065465035 Cos[3 t] + 5233573221864243007 Cos[5 t] -
250 (19936133122065382 Sin[t] + 7069109 (485993763 Sin[3 t] + 58539109 Sin[5 t]))))
If I include A=1 then I get a ConditionalExpression depending on (1 - Tan[t])^(2/5)>0 for the exact definite integral for l. For a few other values of A I get a string of “PolynomialGCD::lrgexp: Exponent is out of bounds for function PolynomialGCD.” and no result.