# Unable to solve a double integration

GROUPS:
 kok wong 1 Vote 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.
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3 months ago
6 Replies
 Jose Calderon 3 Votes It will be a good idea if you publish the notebook showing what you have tried so far.
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3 months ago
 kok wong 2 Votes Eqn 1 is separated into two sections and notebooks are attached. Attachments:
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3 months ago
 Can anyone please comment have i made any mistakes in the attached notebooks?
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3 months ago
 Marco Thiel 2 Votes 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
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3 months ago
 Thanks Marco for your comment. Not sure if the integrand can be solved or not.
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3 months ago
 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.
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3 months ago