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tomleslie

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These are replies submitted by tomleslie

Clarify...

tomleslie 13876
February 19 2019
0 0

The maximum absolute error of what???

Some observations...

tomleslie 13876
February 16 2019
0 0

@dorna01

  1. By default Maple will perform most calculations using "software floats" with 10-digit precision. You can change this to any precision level you want up to the value reurned by Maple_floats(MAX_DIGITS). This may(?) be machine-dependent, but on my machine is 38,654,705,646 digits (gulp!?). The higher you set the value of Digits, the more precise the calculation will be, but the longer it will take
  2. To save on execution time, many Maple commands will use "hardware floats" because this allows number-crunching to be done in hardware rather than software. This will generally be noticeably faster than using "software floats" but you are restricted to conventional double precision arithmetic; roughly 16 digits, and exponents between -308 and 308. Whether or not Maple is allowed use "hardware floats" depends on the value assigned to Digits, and the value of kernelopts(useHardwareFloats), By default these are set to 10 and "deduced". Since a "hardware float" is 16-digits, Maple "deduces" that under these default settings, it is acceptable to use "hardware floats" if desired. If Digits is set >16, then Maple "deduces" that using "hardware floats" does not offer sufficient precision, so won't do it.
  3. I'm mot exactly sure why you need/want 10 results. I note that each successive result is almost exactly 1/2 of the previous one. Is this coincidence, or does it arise for some "physical" reason? Without more understanding of your problem, it is difficult for me to come up with a better strategy. The for-loop is not the problem: it is the total number of function evaluations. Even something like simplifying the input equation as much as possible starts to become important. For example, if the expresssion contains multiple terms containin omega^3, it is inefficent to perform this exponentiation multiple times
  4. On my machine it takes ~3secs to execute the for loop, so 500*500*5 equations would take ~1040hours. Even on an 8-core machine like mine that's ~6days. I would consider this to be completely impractical and try to come up with an alternative method of solving the problem!!!

I should have mentioned...

tomleslie 13876
February 16 2019
0 0

Since each iteration in your for-loop depends on the results of the previous iteration, the calculation would seem to be inherently "sequential", so that I doubt if it can be parallelized :-(

Don't get it...

tomleslie 13876
February 15 2019
0 0

@Gabriel samaila 

Code runs perfectly in Maple 2016. Check the attached: note output from the kernelopts(version). Please post woksheet with the problem you are getting

> 

restart;
kernelopts(version);

`Maple 2016.2, X86 64 WINDOWS, Jan 13 2017, Build ID 1194701`

 (1)
> 

ODEs:= {diff(phi(y), y, y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(theta(y), y, y)+(diff(theta(y), y))/y)/Nb = 0,
         diff(v(y), y, y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0,
         diff(theta(y), y, y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0};

{diff(diff(phi(y), y), y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(diff(theta(y), y), y)+(diff(theta(y), y))/y)/Nb = 0, diff(diff(v(y), y), y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0, diff(diff(theta(y), y), y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0}

 (2)
> 

BCs := {phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1};

{phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1}

 (3)
> 

HaVals:=[1, 2, 5, 10]:
for j from 1 by 1 to numelems(HaVals) do
    pVals := [eta = .5, Ha = HaVals[j], Sc = .8, Nt = .1, Nb = .1, Re = 2, Ec = 0.1e-1, Pr = 10];
    sol[j]:=dsolve( `union`(eval(ODEs, pVals), eval(BCs, pVals)),
                     numeric,
                     output=listprocedure
                  );
od:
displ_opt:= titlefont=[times, bold, 20],
            legend=[seq(typeset("Ha= ", HaVals[j]),j=1..4)],
            legendstyle=[font=[times, bold, 16]],
            size=[800,600],
            color=[red, green, blue, black]:
plot( [seq( eval(theta(y), sol[j])(y),j=1..4)],
      y=0.5..1,
      title="theta(y) vs y",
      displ_opt
    );
plot( [seq( eval(phi(y), sol[j])(y),j=1..4)],
      y=0.5..1,
      title="phi(y) vs y",
      displ_opt
     );
plot( [seq( eval(v(y), sol[j])(y),j=1..4)],
      y=0.5..1,
      title="v(y) vs y",
      displ_opt
   );

 

 

 
> 

 

> 

 


 

Download odeProb4.mw

This may actually have a minimum...

tomleslie 13876
February 15 2019
0 0

Latest expression is no longer homogeneous so may well have a minimum (or possibly many minima!)

The attached does some checks on the expression to confirm the above statement, then obtains a minimum in two different ways - using NLPSolve and DirectSearch()

In both cases the value of the obtained minimum is around 1.376072*10^6, and both methods agree (more-or-less) on the required values of d[i, j] for i,j=0..2. However they disagree somewhat on the required values of c[i,j] for i,j=0..2

This suggests that the function may have multiple (more-or-less equivalent) minima. Don't have time to investigate further at the moment. May(?) get back to this later depending on how my day works out

> 

restart;
Digits:=20:
J:=1.0416666692708333334*10^8*d[0, 0]*d[0, 1]+2.0833328385416666666*10^7*d[0, 0]*d[0, 2]+4.1666666145833333334*10^7*d[0, 1]*d[0, 2]+3.4722222309027777780*10^7*d[1, 0]*d[1, 1]+6.9444427951388888895*10^6*d[1, 0]*d[1, 2]+1.3888888715277777780*10^7*d[1, 1]*d[1, 2]+2.0833333385416666666*10^7*d[2, 0]*d[2, 1]+4.1666656770833333334*10^6*d[2, 0]*d[2, 2]+8.3333332291666666665*10^6*d[2, 1]*d[2, 2]+7.4599330357142857145*10^(-16)*c[0, 1]*c[2, 2]^3+4.1666666666666666666*10^(-19)*c[0, 2]*c[2, 0]^3-3.1666666666666666666*10^(-17)*c[0, 2]*c[2, 1]^3+2.5000000000000000000*10^(-18)*c[1, 0]^3*c[2, 1]-4.1666666666666666666*10^(-19)*c[1, 0]^3*c[2, 2]+1.2500000000000000000*10^(-18)*c[1, 0]*c[2, 1]^3+2.5000000000000000000*10^(-18)*c[1, 1]^3*c[2, 0]-4.3854166666666666666*10^(-17)*c[1, 1]^3*c[2, 2]+1.2500000000000000000*10^(-18)*c[1, 1]*c[2, 0]^3-9.8100074404761904760*10^(-16)*c[1, 2]^3*c[2, 1]+4.2708333333333333334*10^(-18)*c[0, 0]*c[1, 1]^3-2.5000000000000000000*10^(-18)*c[0, 1]*c[1, 0]^3-1.0235007440476190476*10^(-15)*c[0, 1]*c[1, 2]^3+8.7500000000000000000*10^(-18)*c[0, 2]*c[1, 0]^3-1.2645833333333333334*10^(-16)*c[0, 2]*c[1, 1]^3-2.5000000000000000000*10^(-18)*c[1, 0]^3*c[1, 1]-3.7500000000000000000*10^(-18)*c[1, 0]^3*c[1, 2]+1.1562500000000000000*10^(-17)*c[1, 0]*c[1, 1]^3-1.0666666666666666666*10^(-16)*c[1, 1]^3*c[1, 2]+4.4327752976190476190*10^(-16)*c[1, 1]*c[1, 2]^3+1.2500000000000000000*10^(-17)*c[0, 2]^3*c[1, 1]-1.4592782738095238095*10^(-15)*c[0, 2]^3*c[2, 1]-6.2500000000000000000*10^(-18)*c[0, 0]^3*c[1, 1]+8.3333333333333333335*10^(-18)*c[0, 0]^3*c[1, 2]-5.7291666666666666665*10^(-18)*c[0, 1]^3*c[1, 0]+1.7708333333333333334*10^(-17)*c[0, 1]^3*c[1, 2]+2.9166666666666666666*10^(-18)*c[0, 0]^3*c[2, 2]+1.5312500000000000000*10^(-17)*c[0, 1]^3*c[2, 0]-9.6875000000000000000*10^(-18)*c[0, 1]^3*c[2, 2]+1.2500000000000000000*10^(-18)*c[0, 0]^3*c[2, 1]+6.6311284638019735570*10^6+72.544642857142857135*c[0, 0]^2*c[1, 2]*c[2, 2]+260.41666666666666666*c[0, 0]*c[0, 1]^2*c[1, 0]+104.16666666666666666*c[0, 0]*c[0, 1]^2*c[1, 2]+111.60714285714285711*c[0, 0]*c[0, 1]^2*c[2, 0]+44.642857142857142951*c[0, 0]*c[0, 1]^2*c[2, 2]+61.383928571428571495*c[1, 0]^2*c[1, 2]*c[2, 2]+57.043650793650793760*c[1, 0]*c[1, 1]^2*c[1, 2]+102.30654761904761905*c[1, 0]*c[1, 1]^2*c[2, 0]+40.922619047619047749*c[1, 0]*c[1, 1]^2*c[2, 2]+61.383928571428571495*c[1, 0]*c[1, 2]^2*c[2, 0]+17.538265306122449348*c[1, 0]*c[1, 2]^2*c[2, 2]+33.820346320346320272*c[1, 0]*c[1, 2]*c[2, 1]^2+14.494434137291279463*c[1, 0]*c[1, 2]*c[2, 2]^2+66.964285714285714350*c[0, 1]*c[1, 1]^2*c[2, 1]+52.614795918367345815*c[0, 1]*c[1, 1]*c[1, 2]^2+93.005952380952380950*c[0, 1]*c[1, 1]*c[2, 0]^2+55.803571428571428595*c[0, 1]*c[1, 1]*c[2, 1]^2+29.230442176870748487*c[0, 1]*c[1, 1]*c[2, 2]^2+35.076530612244898366*c[0, 1]*c[1, 2]^2*c[2, 1]+71.022727272727272735*c[0, 1]*c[2, 0]^2*c[2, 1]+22.321428571428571789*c[0, 1]*c[2, 1]*c[2, 2]^2+35.076530612244897394*c[0, 2]^2*c[1, 0]*c[1, 2]+72.544642857142857135*c[0, 2]^2*c[1, 0]*c[2, 0]+20.727040816326531178*c[0, 2]^2*c[1, 0]*c[2, 2]+37.999574829931971258*c[0, 2]^2*c[1, 1]*c[2, 1]+20.727040816326531178*c[0, 2]^2*c[1, 2]*c[2, 0]+156.25000000000000000*c[0, 0]^2*c[0, 2]*c[1, 2]+66.964285714285714355*c[0, 0]^2*c[0, 2]*c[2, 2]+362.72321428571428572*c[0, 0]^2*c[1, 0]*c[2, 0]+120.90773809523809524*c[0, 0]^2*c[1, 1]*c[2, 1]+71.022727272727272735*c[0, 0]*c[2, 0]*c[2, 1]^2+1.9841269841269841302*c[2, 1]*c[2, 2]*d[0, 0]+.39682539682539680895*c[2, 1]*c[2, 2]*d[0, 1]-.82199546485260771110*c[2, 1]*c[2, 2]*d[0, 2]+1.3640873015873015860*c[2, 1]*c[2, 2]*d[1, 0]+.27281746031746032180*c[2, 1]*c[2, 2]*d[1, 1]-.56512188208616780715*c[2, 1]*c[2, 2]*d[1, 2]+.86805555555555555655*c[2, 1]*c[2, 2]*d[2, 0]+.17361111111111111314*c[2, 1]*c[2, 2]*d[2, 1]-.35962301587301587203*c[2, 1]*c[2, 2]*d[2, 2]+167.41071428571428569*c[0, 1]*c[1, 0]^2*c[1, 1]+111.60714285714285711*c[0, 1]*c[1, 0]^2*c[2, 1]+1.3640873015873015908*c[1, 0]*c[2, 2]*d[2, 1]-.40922619047619048710*c[1, 0]*c[2, 2]*d[2, 2]+1.6121031746031746060*c[1, 1]*c[1, 2]*d[2, 0]+.32242063492063492331*c[1, 1]*c[1, 2]*d[2, 1]-.66787131519274374115*c[1, 1]*c[1, 2]*d[2, 2]+156.25000000000000000*c[0, 0]*c[0, 2]^2*c[1, 0]+44.642857142857142857*c[0, 0]*c[0, 2]^2*c[1, 2]+66.964285714285714355*c[0, 0]*c[0, 2]^2*c[2, 0]+19.132653061224489599*c[0, 0]*c[0, 2]^2*c[2, 2]+81.845238095238095200*c[0, 0]*c[0, 2]*c[1, 1]^2+35.076530612244897394*c[0, 0]*c[0, 2]*c[1, 2]^2+44.642857142857142951*c[0, 0]*c[0, 2]*c[2, 1]^2+19.132653061224489599*c[0, 0]*c[0, 2]*c[2, 2]^2+334.82142857142857142*c[0, 0]*c[1, 0]^2*c[2, 0]+167.41071428571428569*c[0, 0]*c[1, 0]*c[1, 1]^2+100.44642857142857130*c[0, 0]*c[1, 0]*c[1, 2]^2+279.01785714285714285*c[0, 0]*c[1, 0]*c[2, 0]^2+93.005952380952380950*c[0, 0]*c[1, 0]*c[2, 1]^2+55.803571428571428505*c[0, 0]*c[1, 0]*c[2, 2]^2+66.964285714285714415*c[0, 0]*c[1, 1]^2*c[1, 2]+111.60714285714285711*c[0, 0]*c[1, 1]^2*c[2, 0]+44.642857142857142951*c[0, 0]*c[1, 1]^2*c[2, 2]+66.964285714285714355*c[0, 0]*c[1, 2]^2*c[2, 0]+19.132653061224489599*c[0, 0]*c[1, 2]^2*c[2, 2]+37.202380952380952251*c[0, 0]*c[1, 2]*c[2, 1]^2+15.943877551020407795*c[0, 0]*c[1, 2]*c[2, 2]^2+260.41666666666666666*c[0, 0]^2*c[0, 1]*c[1, 1]+111.60714285714285711*c[0, 0]^2*c[0, 1]*c[2, 1]+1.7857142857142857295*c[0, 2]*c[2, 2]*d[2, 0]+.36848072562358279097*c[0, 2]*c[2, 2]*d[2, 1]-1.3605442176870747069*c[0, 2]*c[2, 2]*d[2, 2]+4.0302579365079365080*c[1, 0]*c[1, 1]*d[2, 0]+.80605158730158730105*c[1, 0]*c[1, 1]*d[2, 1]-3.2242063492063492051*c[1, 0]*c[1, 1]*d[2, 2]+1.6121031746031746065*c[1, 0]*c[1, 2]*d[2, 1]-.48363095238095238294*c[1, 0]*c[1, 2]*d[2, 2]+7.2544642857142857140*c[1, 0]*c[2, 0]*d[1, 0]-.80605158730158730195*c[1, 0]*c[2, 0]*d[1, 1]-4.8363095238095238092*c[1, 0]*c[2, 0]*d[1, 2]+6.1383928571428571430*c[1, 0]*c[2, 0]*d[2, 0]-.68204365079365079385*c[1, 0]*c[2, 0]*d[2, 1]-4.0922619047619047612*c[1, 0]*c[2, 0]*d[2, 2]+4.0302579365079365080*c[1, 0]*c[2, 1]*d[1, 0]+.80605158730158730105*c[1, 0]*c[2, 1]*d[1, 1]-3.2242063492063492051*c[1, 0]*c[2, 1]*d[1, 2]+3.4102182539682539680*c[1, 0]*c[2, 1]*d[2, 0]+.68204365079365079425*c[1, 0]*c[2, 1]*d[2, 1]-2.7281746031746031740*c[1, 0]*c[2, 1]*d[2, 2]+1.6121031746031746065*c[1, 0]*c[2, 2]*d[1, 1]-.48363095238095238294*c[1, 0]*c[2, 2]*d[1, 2]+42.613636363636363594*c[0, 0]*c[2, 0]*c[2, 2]^2+28.409090909090909032*c[0, 0]*c[2, 1]^2*c[2, 2]+.43402777777777777747*c[2, 0]*c[2, 1]*d[2, 1]-1.7361111111111111101*c[2, 0]*c[2, 1]*d[2, 2]+1.9841269841269841262*c[2, 0]*c[2, 2]*d[0, 1]-.59523809523809522370*c[2, 0]*c[2, 2]*d[0, 2]+1.3640873015873015908*c[2, 0]*c[2, 2]*d[1, 1]-.40922619047619048710*c[2, 0]*c[2, 2]*d[1, 2]+.86805555555555555635*c[2, 0]*c[2, 2]*d[2, 1]-.26041666666666666654*c[2, 0]*c[2, 2]*d[2, 2]+42.613636363636363594*c[0, 2]*c[2, 0]^2*c[2, 2]+28.409090909090909032*c[0, 2]*c[2, 0]*c[2, 1]^2+12.175324675324676135*c[0, 2]*c[2, 0]*c[2, 2]^2+22.321428571428571789*c[0, 2]*c[2, 1]^2*c[2, 2]+102.30654761904761905*c[1, 0]^2*c[1, 1]*c[2, 1]+89.285714285714285725*c[0, 0]*c[0, 1]*c[0, 2]*c[2, 1]+409.22619047619047622*c[0, 0]*c[0, 1]*c[1, 0]*c[1, 1]-17.577326954359643821*c[0, 0]^2*c[1, 0]+1.6121031746031746060*c[1, 1]*c[2, 2]*d[1, 0]+.32242063492063492331*c[1, 1]*c[2, 2]*d[1, 1]-.66787131519274374115*c[1, 1]*c[2, 2]*d[1, 2]+1.3640873015873015860*c[1, 1]*c[2, 2]*d[2, 0]+.27281746031746032180*c[1, 1]*c[2, 2]*d[2, 1]-.56512188208616780715*c[1, 1]*c[2, 2]*d[2, 2]+1.6121031746031746065*c[1, 2]*c[2, 0]*d[1, 1]-.48363095238095238294*c[1, 2]*c[2, 0]*d[1, 2]+1.3640873015873015908*c[1, 2]*c[2, 0]*d[2, 1]-.40922619047619048710*c[1, 2]*c[2, 0]*d[2, 2]+1.6121031746031746060*c[1, 2]*c[2, 1]*d[1, 0]+.32242063492063492331*c[1, 2]*c[2, 1]*d[1, 1]-.66787131519274374115*c[1, 2]*c[2, 1]*d[1, 2]+1.3640873015873015860*c[1, 2]*c[2, 1]*d[2, 0]+.27281746031746032180*c[1, 2]*c[2, 1]*d[2, 1]-.56512188208616780715*c[1, 2]*c[2, 1]*d[2, 2]+1.4508928571428571338*c[1, 2]*c[2, 2]*d[1, 0]+.29939058956916098696*c[1, 2]*c[2, 2]*d[1, 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1]^2-11.951943892452272855*c[0, 1]^2*c[1, 0]+9.9278273809523809525*10^(-17)*c[1, 1]*c[2, 2]^3-1.2500000000000000000*10^(-19)*c[1, 2]*c[2, 0]*d[0, 0]-1.2500000000000000000*10^(-19)*c[0, 2]*c[2, 0]*d[1, 0]-1.2500000000000000000*10^(-19)*c[0, 0]*c[1, 2]*d[2, 0]-1.8645833333333333334*10^(-16)*c[1, 1]^2*c[1, 2]*c[2, 1]+1.1979166666666666666*10^(-17)*c[1, 1]^2*c[2, 0]*c[2, 1]+2.5416666666666666666*10^(-16)*c[1, 1]^2*c[2, 1]*c[2, 2]-1.1875000000000000000*10^(-16)*c[1, 1]*c[1, 2]^2*c[2, 0]-2.8437500000000000000*10^(-16)*c[0, 1]^2*c[0, 2]*c[2, 1]+1.1041666666666666666*10^(-16)*c[0, 1]^2*c[1, 0]*c[1, 1]-2.4479166666666666666*10^(-17)*c[0, 1]^2*c[1, 0]*c[2, 1]-9.7916666666666666665*10^(-17)*c[0, 1]^2*c[1, 1]*c[1, 2]-2.4479166666666666666*10^(-17)*c[0, 1]^2*c[1, 1]*c[2, 0]+9.7916666666666666665*10^(-17)*c[0, 1]^2*c[1, 1]*c[2, 2]+9.7916666666666666665*10^(-17)*c[0, 1]^2*c[1, 2]*c[2, 1]+3.8541666666666666666*10^(-17)*c[0, 1]^2*c[2, 0]*c[2, 1]-2.8437500000000000000*10^(-16)*c[0, 1]^2*c[2, 1]*c[2, 2]-2.0104166666666666666*10^(-16)*c[0, 1]*c[0, 2]^2*c[2, 0]-4.3327752976190476190*10^(-16)*c[0, 1]*c[0, 2]^2*c[2, 2]+1.1562500000000000000*10^(-16)*c[0, 1]*c[0, 2]*c[1, 0]^2-9.7916666666666666665*10^(-17)*c[0, 1]*c[0, 2]*c[1, 1]^2+4.7672247023809523810*10^(-15)*c[0, 1]*c[0, 2]*c[1, 2]^2-1.2500000000000000000*10^(-19)*c[0, 2]*c[1, 0]*d[2, 0]+1.2500000000000000000*10^(-19)*c[0, 0]*c[0, 2]*d[2, 0]-1.2500000000000000000*10^(-19)*c[0, 0]*c[2, 2]*d[1, 0]-3.1250000000000000000*10^(-18)*c[0, 2]*c[1, 0]^2*c[2, 1]-4.1666666666666666666*10^(-18)*c[0, 2]*c[1, 0]*c[2, 0]^2-2.8437500000000000000*10^(-16)*c[0, 2]*c[1, 1]^2*c[2, 1]-6.9225074404761904760*10^(-15)*c[0, 2]*c[1, 1]*c[1, 2]^2+3.5416666666666666666*10^(-17)*c[0, 2]*c[1, 1]*c[2, 0]^2+1.8645833333333333334*10^(-16)*c[0, 2]*c[1, 1]*c[2, 1]^2+1.4667782738095238095*10^(-15)*c[0, 2]*c[1, 1]*c[2, 2]^2-4.3327752976190476190*10^(-16)*c[0, 2]*c[1, 2]^2*c[2, 1]+3.1041666666666666666*10^(-17)*c[0, 2]*c[2, 0]^2*c[2, 1]-1.4667782738095238095*10^(-15)*c[1, 1]*c[1, 2]^2*c[2, 2]-1.9270833333333333334*10^(-17)*c[1, 1]*c[1, 2]*c[2, 0]^2+2.5416666666666666666*10^(-16)*c[1, 1]*c[1, 2]*c[2, 1]^2+1.2906250000000000000*10^(-15)*c[1, 1]*c[1, 2]*c[2, 2]^2+3.5312500000000000000*10^(-16)*c[1, 2]^2*c[2, 0]*c[2, 1]+1.2906250000000000000*10^(-15)*c[1, 2]^2*c[2, 1]*c[2, 2]-15.066280246593980418*c[0, 0]*c[1, 0]*c[2, 0]:
> 

#
# Check expression to see if there are any
# terms independent of the variables by setting
# all variables to zero and evaluating J.
#
  vals:=[seq(k=0, k in indets(J, name))];
  eval(J, vals);

[c[0, 0] = 0, c[0, 1] = 0, c[0, 2] = 0, c[1, 0] = 0, c[1, 1] = 0, c[1, 2] = 0, c[2, 0] = 0, c[2, 1] = 0, c[2, 2] = 0, d[0, 0] = 0, d[0, 1] = 0, d[0, 2] = 0, d[1, 0] = 0, d[1, 1] = 0, d[1, 2] = 0, d[2, 0] = 0, d[2, 1] = 0, d[2, 2] = 0]

  

6631128.4638019735570

 (1)
> 

#
# From the above, expression is not "homogeneous"
# Where are the non-zero terms!???
#
  terms:=[seq(eval(k,vals), k in op([1..-1],J))]:
  seq( `if`(terms[j]>0, [j, terms[j]], NULL), j=1..numelems(terms));

[143, 6631128.4638019735570]

 (2)
> 

#
# Thus if all variables are 0, then J will
# evaluate to 6.6311284638019735570000000*10^6
#
# Is there some combination of variables which
# will produce less??
#
  infolevel[Optimization]:=2;
  Optimization:-NLPSolve(J, iterationlimit=1000);

2

  

NLPSolve: calling NLP solver
NLPSolve: using method=pcg
NLPSolve: number of problem variables 18
NLPSolve: trying evalf mode

attemptsolution: number of major iterations taken 608

  

[1376072.3614515957994, [c[0, 0] = -.25297616138377927973, c[0, 1] = -.39478309897588939049, c[0, 2] = 0.15211305487012686035e-3, c[1, 0] = .27642043497993973641, c[1, 1] = .68470203127474910490, c[1, 2] = .47418006681942999119, c[2, 0] = -0.55834416654511992361e-1, c[2, 1] = -.31556717438509396543, c[2, 2] = -.44709255482404861288, d[0, 0] = -0.67205030036231457236e-1, d[0, 1] = .10613609848181940555, d[0, 2] = 0.24170100044583235998e-1, d[1, 0] = -.32413315285346089311, d[1, 1] = 1.1944070031913440028, d[1, 2] = -.86054957288795628372, d[2, 0] = .28146384027544243459, d[2, 1] = -.48696268495323882506, d[2, 2] = -.17308234882015350758]]

 (3)
> 

#
# For comparison, see what DirectSearch gets
#
  sol:=DirectSearch:-Search( J );

Warning, limiting number of function evaluations (10000) is reached; set initial point equal to extremum point obtained, increase evaluationlimit option and continue search

  

[1376072.6668412643231, [c[0, 0] = 0.8804209794690149926e-1, c[0, 1] = -.20142528173472715327, c[0, 2] = .32554616141396666182, c[1, 0] = 0.3998228018922962427e-1, c[1, 1] = .51619905141111506726, c[1, 2] = -.56781364468877512347, c[2, 0] = -0.409677826757984239e-2, c[2, 1] = -.57635175842887362663, c[2, 2] = .18884411637045077739, d[0, 0] = -0.67198816955305295394e-1, d[0, 1] = .10612397156413182108, d[0, 2] = 0.24171147342192657570e-1, d[1, 0] = -.32412670830939822682, d[1, 1] = 1.1944154502284577955, d[1, 2] = -.86054710245522323027, d[2, 0] = .28147277692527091699, d[2, 1] = -.48694977124031602112, d[2, 2] = -.17310809563948536227], 10000]

 (4)
> 

 


 

Download optProb2.mw

My code was developed in Maple2018.2...

tomleslie 13876
February 15 2019
0 0

I have just checked back some earlier versions. It works in Maple 2017and later, but fails in earlier versions

In the attached I have rewritten it slightly so that it produces the same output as before. This code runs in

Maple 18      (used this version to write/test the revised code)
Maple 2015
Maple 2016
Maple 2017
Maple 2018

If your version is earlier than Maple 18, then the attached *might* work, but I can't guarantee it because I no longer have any earlier versions available

> 

restart;
kernelopts(version);

`Maple 18.02, X86 64 WINDOWS, Oct 20 2014, Build ID 991181`

 (1)
> 

ODEs:= {diff(phi(y), y, y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(theta(y), y, y)+(diff(theta(y), y))/y)/Nb = 0,
         diff(v(y), y, y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0,
         diff(theta(y), y, y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0};

{diff(diff(phi(y), y), y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(diff(theta(y), y), y)+(diff(theta(y), y))/y)/Nb = 0, diff(diff(v(y), y), y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0, diff(diff(theta(y), y), y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0}

 (2)
> 

BCs := {phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1};

{phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1}

 (3)
> 

HaVals:=[1, 2, 5, 10]:
for j from 1 by 1 to numelems(HaVals) do
    pVals := [eta = .5, Ha = HaVals[j], Sc = .8, Nt = .1, Nb = .1, Re = 2, Ec = 0.1e-1, Pr = 10];
    sol[j]:=dsolve( `union`(eval(ODEs, pVals), eval(BCs, pVals)),
                     numeric,
                     output=listprocedure
                  );
od:
displ_opt:= titlefont=[times, bold, 20],
            legend=[seq(typeset("Ha= ", HaVals[j]),j=1..4)],
            legendstyle=[font=[times, bold, 16]],
            size=[800,600],
            color=[red, green, blue, black]:
plot( [seq( eval(theta(y), sol[j])(y),j=1..4)],
      y=0.5..1,
      title="theta(y) vs y",
      displ_opt
    );
plot( [seq( eval(phi(y), sol[j])(y),j=1..4)],
      y=0.5..1,
      title="phi(y) vs y",
      displ_opt
     );
plot( [seq( eval(v(y), sol[j])(y),j=1..4)],
      y=0.5..1,
      title="v(y) vs y",
      displ_opt
   );

 

 

 
> 

 

> 

 

Download odeProb4.mw

I suggest you read my answer very carefu...

tomleslie 13876
February 15 2019
0 0

partcularly this part where I said

NB you will not been able to execute the attached until/unless you have the DirectSearch package loaded :-(

Then you should read the comments from vv, which are correct. I actually missed the fact that you original expression was

  1. homogeneous
  2. every term is of total degree 4

Hence solving expr=0 simply requires that cij=0 for all i and j.

Now exactly what are you trying to achieve?

Unless I'm missing something...

tomleslie 13876
February 14 2019
1 0

this question has already been asked (and answered) here

https://www.mapleprimes.com/questions/225962-How-To-Perform-Numerical-Integration#comment256680

Or...

tomleslie 13876
February 14 2019
0 0

with colou
 

> 

restart;
> 

ODEs:= {diff(phi(y), y, y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(theta(y), y, y)+(diff(theta(y), y))/y)/Nb = 0,
         diff(v(y), y, y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0,
         diff(theta(y), y, y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0};

{diff(diff(phi(y), y), y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(diff(theta(y), y), y)+(diff(theta(y), y))/y)/Nb = 0, diff(diff(v(y), y), y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0, diff(diff(theta(y), y), y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0}

 (1)
> 

BCs := {phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1};

{phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1}

 (2)
> 

HaVals:=[1, 2, 5, 10]:
colors:=[red, green, blue, black]:
for j from 1 by 1 to numelems(HaVals) do
    pVals := [eta = .5, Ha = HaVals[j], Sc = .8, Nt = .1, Nb = .1, Re = 2, Ec = 0.1e-1, Pr = 10];
    sol:=dsolve( `union`(eval(ODEs, pVals), eval(BCs, pVals)), numeric);
    ptheta[j]:= plots:-odeplot(sol, [y, theta(y)], y=0.5..1, color=colors[j]);
    pphi[j]:= plots:-odeplot(sol, [y, phi(y)], y=0.5..1, color=colors[j]);
    pv[j]:= plots:-odeplot(sol, [y, v(y)], y=0.5..1, color=colors[j]);
od:
displ_opt:= titlefont=[times, bold, 20],
            legend=[seq(typeset("Ha= ", HaVals[j]),j=1..4)],
            legendstyle=[font=[times, bold, 16]],
            size=[800,600]:
plots:-display( convert(ptheta, list),
                title="theta(y) vs y",
                displ_opt
              );
plots:-display( convert(pphi, list),
                title="phi(y) vs y",
                displ_opt
              );
plots:-display( convert(pv, list),
                title="v(y) vs y",
                displ_opt
              );

 

 

 
> 

``


 

Download odeProb3.mw

rs, legends and title on the plots

 

So that would be...

tomleslie 13876
February 14 2019
0 0

most easily achieved in by the method shown in the attached


 

> 

restart;
> 

ODEs:= {diff(phi(y), y, y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(theta(y), y, y)+(diff(theta(y), y))/y)/Nb = 0,
         diff(v(y), y, y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0,
         diff(theta(y), y, y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0};

{diff(diff(phi(y), y), y)-Re*Sc*v(y)*(diff(phi(y), y))+Nt*(diff(diff(theta(y), y), y)+(diff(theta(y), y))/y)/Nb = 0, diff(diff(v(y), y), y)+(diff(v(y), y))/y-(Ha^2/(1-eta)^2+1/y^2)*v(y)-Re*v(y)*(diff(v(y), y)) = 0, diff(diff(theta(y), y), y)+Ec*Pr*(diff(v(y), y)-v(y)/y)^2-Pr*Re*v(y)*(diff(theta(y), y))+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0}

 (1)
> 

BCs := {phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1};

{phi(1) = 0, phi(eta) = 1, theta(1) = 0, theta(eta) = 1, v(1) = 0, v(eta) = 1}

 (2)
> 

HaVals:=[1,2,5,10]:
for j from 1 by 1 to numelems(HaVals) do
    pVals := [eta = .5, Ha = HaVals[j], Sc = .8, Nt = .1, Nb = .1, Re = 2, Ec = 0.1e-1, Pr = 10];
    sol:=dsolve( `union`(eval(ODEs,pVals), eval(BCs, pVals)), numeric);
    ptheta[j]:=plots:-odeplot(sol, [y, theta(y)], y=0.5..1);
    pphi[j]:=plots:-odeplot(sol, [y, phi(y)], y=0.5..1);
    pv[j]:=plots:-odeplot(sol, [y, v(y)], y=0.5..1);
od:
plots:-display( convert(ptheta, list));
plots:-display( convert(pphi, list));
plots:-display( convert(pv, list));

[1, 2, 5, 10]

  

 

 

 
> 

``


 

Download odeProb2.mw

Essentially the same problem as before...

tomleslie 13876
February 13 2019
0 0

when you posted this question

https://www.mapleprimes.com/questions/226439-Solve-Figure-Draw-Problem

Try reading the answers

This was never a question!...

tomleslie 13876
February 13 2019
0 0

@sand15 

This was a 'comment' on a post where the OP was trying to sort() a set, and was surprised that sorting a set did nothing. As the response states, the original query probably should have been a "Question" not a "Post".

Someone smarter than me has transferred my comment from "Posts" to "Questions" without transferring the original query

My original response...

tomleslie 13876
February 12 2019
0 0

stated that there was little reason for doing this - other than (maybe?) formatting a "header" for some (unspecified) printed output. Unlikely, I admit,  but possible.

If I was really bothered by the "dangling comma" I'd probably fix it the easy way by using a different printf() statement for the final entry, as in

printf("\t\t"):
seq(`if`( j<5,
              printf( "op\(S\[%d\]\),    ",j),
              printf( "op\(S\[%d\]\)", j)
           ),
           j=1..5
      );

I have enough problems with printf() formats without getting 'clever'!!

My thoughts...

tomleslie 13876
February 12 2019
0 0

@anton_dys 

I have no direct info on the efficiency of MapToBasis, but if you only need this for "visualisation" parts of you code, I would guess that the "coordinate transformation" would only be a smal fraction of the actual "plotting time" - just because omplicated plots can take significant time

Code indentation: when I'm doing a "significant" Maple project I tend to write code in a standalone editor, which is reasonably "Maple-aware" so provides me with syntax-highlighting, automatic identation etc, etc. A side-effect of this usage model is that I find code (much!) easier to read/debug/maintain if it is always formatted in this way. Thus even when I'm writing "scratch" code in Maple, I tend to use the same general layout utilising shift+enter+spacebar. As a general rule I find that the increased "typing time" is only a small fraction of the associated "thinking time". This is an individual choice which works for me.

Simple?...

tomleslie 13876
February 12 2019
0 0

@anton_dys 

You appear to want to define a VectorField using spherical polar coordinates.

Then you wish to plot this VectorField with a "uniform" spacing in cartesian coordinates (rather than a "uniform" spacing in spherical coordinates).

You have to be consistent.

You can use MapToBasis() to (temporarily??) convert between coordinate systems, so for example, you could consider the command

PlotVector( MapToBasis
                    ( VectorField
                      ( <0, theta+Pi, 0>,
                         mypolar[r, theta, phi]
                      ),
                      cartesian[x,y,z]
                   ),
                   x = -5..5,
                   y = -5..5,
                   z = -5..5,
                   arrows = THICK,
                   color = black,
                   grid = [10, 10, 3],
                  view = [-5..5, -5..5, -3..3]
               ) 

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