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tomleslie

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These are answers submitted by tomleslie

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tomleslie 13876
May 10 2021
1 1

fsolve() can only find an answer for k__rq from 0.001 to 0.032.

Values for wi and wr for these  values of k_rq don't do anything very interesting!

Allowing for a bit of floating point rounding, I think one could justifiably say that wi=0.0, and wr=1.0 for all values of  k_rq.

See the attached.


 

> 

 

> 

restart;
> 

Ka_rq:=200:
> 

Ma_rq:=35:
> 

Bi_rq:=1:
> 

EQ1 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(0.3678181674e28*I)*R^(55/3)*Bi_rq^25*k_rq*wr^2*wi^2-(0.1438313716e28*I)*R^(61/3)*Bi_rq^23*k_rq^3*wr^2*wi^2-(0.7409924485e23*I)*R^(58/3)*Bi_rq^26*k_rq^3*wr^2*wi^2-(0.4957028881e26*I)*R^(55/3)*Bi_rq^25*k_rq^2*wr^3*wi^2+(0.2478514441e26*I)*R^(55/3)*Bi_rq^25*k_rq^2*wr*wi^4-(0.3054309160e30*I)*R^(61/3)*Bi_rq^21*k_rq^4*wr*wi^2+(0.1426033362e25*I)*R^(58/3)*Bi_rq^26*k_rq^2*wr*wi^2+(0.1648644952e39*I)*R^(43/3)*Bi_rq^11*k_rq^2*wr*wi^2-(0.1990270655e38*I)*R^(43/3)*Bi_rq^9*k_rq^4*wr*wi^2-(0.1189446090e32*I)*R^(37/3)*Bi_rq*k_rq^2*wr^3*wi^2+(0.5947230449e31*I)*R^(37/3)*Bi_rq*k_rq^2*wr*wi^4+(0.1102565637e31*I)*R^(34/3)*k_rq^6*Ka_rq*wr*wi^2+(0.1211235957e35*I)*R^(46/3)*Bi_rq^10*k_rq^2*wr^3*wi^2-(0.6056179785e34*I)*R^(46/3)*Bi_rq^10*k_rq^2*wr*wi^4+(0.2420302833e37*I)*R^(43/3)*Bi_rq^13*k_rq^2*wr^3*wi^2-(0.1210151417e37*I)*R^(43/3)*Bi_rq^13*k_rq^2*wr*wi^4+(0.5003573000e34*I)*R^(43/3)*Bi_rq^7*k_rq^2*wr^3*wi^2-(0.2501786500e34*I)*R^(43/3)*Bi_rq^7*k_rq^2*wr*wi^4-(0.6943153470e36*I)*R^(40/3)*Bi_rq^10*k_rq^2*wr^3*wi^2+(0.3471576736e36*I)*R^(40/3)*Bi_rq^10*k_rq^2*wr*wi^4-(0.3298794177e34*I)*R^(46/3)*Bi_rq^6*k_rq^4*wr*wi^2+(0.8281673789e30*I)*R^(53/3)*Bi_rq^19*k_rq*wr^4*wi^2-(0.8281673789e30*I)*R^(53/3)*Bi_rq^19*k_rq*wr^2*wi^4-(0.4815182787e31*I)*R^(56/3)*Bi_rq^20*k_rq^2*wr^3*wi^2+(0.2407591394e31*I)*R^(56/3)*Bi_rq^20*k_rq^2*wr*wi^4-(0.5644441055e31*I)*R^(53/3)*Bi_rq^23*k_rq*wr^2*wi^2-(0.6841260389e35*I)*R^(53/3)*Bi_rq^13*k_rq^4*wr*wi^2-(0.9761550588e36*I)*R^(50/3)*Bi_rq^18*k_rq^2*wr*wi^2+(0.1138535672e33*I)*R^(50/3)*Bi_rq^16*k_rq*wr^4*wi^2-(0.1000644068e33*I)*R^(38/3)*Bi_rq^2*k_rq^2*wr^3*wi^2+(0.5003220342e32*I)*R^(38/3)*Bi_rq^2*k_rq^2*wr*wi^4+(0.1772201032e36*I)*R^(35/3)*Bi_rq^5*k_rq^2*wr^3*wi^2-(0.8861005161e35*I)*R^(35/3)*Bi_rq^5*k_rq^2*wr*wi^4+(0.4708347300e37*I)*R^(35/3)*Bi_rq^3*k_rq^2*wr*wi^2+(0.1028681642e35*I)*R^(35/3)*Bi_rq*k_rq^4*wr*wi^2+(0.2990909109e31*I)*R^(35/3)*Bi_rq*k_rq*wr^4*wi^2-(0.2990909109e31*I)*R^(35/3)*Bi_rq*k_rq*wr^2*wi^4-(0.2700592497e37*I)*R^(47/3)*Bi_rq^15*k_rq^3*wr^2*wi^2-(0.1561354645e38*I)*R^(44/3)*Bi_rq^12*k_rq^3*wr^2*wi^2+(0.1704125028e35*I)*R^(44/3)*Bi_rq^8*k_rq^2*wr^3*wi^2-(0.8520625141e34*I)*R^(44/3)*Bi_rq^8*k_rq^2*wr*wi^4+(0.1052392611e37*I)*R^(41/3)*Bi_rq^11*k_rq^2*wr^3*wi^2-(0.5261963055e36*I)*R^(41/3)*Bi_rq^11*k_rq^2*wr*wi^4-(0.2804946795e34*I)*R^(41/3)*Bi_rq^5*k_rq^2*wr^3*wi^2+(0.1402473397e34*I)*R^(41/3)*Bi_rq^5*k_rq^2*wr*wi^4-(0.7504005376e38*I)*R^(41/3)*Bi_rq^11*k_rq*wr^2*wi^2+(0.9517388254e37*I)*R^(41/3)*Bi_rq^9*k_rq^3*wr^2*wi^2+(0.7275092418e35*I)*R^(50/3)*Bi_rq^12*k_rq^3*wr^2*wi^2-(0.1042304384e40*I)*R^(41/3)*k_rq^4*Bi_rq^9*wr*wi-(0.4840198688e34*I)*R^(41/3)*k_rq^3*Bi_rq^7*wr^4*wi+(0.9680397378e34*I)*R^(41/3)*k_rq^3*Bi_rq^7*wr^2*wi^3-(0.3577640806e39*I)*R^(34/3)*Bi_rq^10*Ma_rq*k_rq^3*wi+(0.1072077667e36*I)*R^(40/3)*Bi_rq^8*Ma_rq*k_rq^5*wi+(0.5978506002e39*I)*R^(37/3)*Bi_rq^13*Ma_rq*k_rq^3*wi+(0.2794281398e40*I)*R^(29/3)*k_rq^4*Bi_rq^7*Ka_rq*wr+(0.2999593162e37*I)*R^(41/3)*k_rq^4*Bi_rq^15*Ka_rq*wr^3-(0.6924487677e36*I)*R^(44/3)*Bi_rq^14*k_rq^2*wr*wi^4-(0.6332983016e27*I)*R^(59/3)*Bi_rq^25*k_rq^4*wr*wi^2+(0.1240126012e23*I)*R^(56/3)*Bi_rq^26*k_rq^2*wr^3*wi^2-(0.6200630059e22*I)*R^(56/3)*Bi_rq^26*k_rq^2*wr*wi^4+(0.3276634201e38*I)*R^(47/3)*Bi_rq^13*k_rq^4*wr*wi^2+(0.2810613986e37*I)*R^(44/3)*Bi_rq^18*k_rq^2*wr*wi^2-(0.3244702138e35*I)*R^(46/3)*Bi_rq^8*k_rq^3*wr^2*wi^2-(0.1519841088e39*I)*R^(43/3)*Bi_rq^13*k_rq*wr^2*wi^2-(0.6670161982e37*I)*R^(43/3)*Bi_rq^11*k_rq^3*wr^2*wi^2+(0.2445979622e36*I)*R^(49/3)*Bi_rq^17*k_rq^3*wr^2*wi^2+(0.4095329528e33*I)*R^(52/3)*Bi_rq^22*k_rq*wr^2*wi^2-(0.2094996789e34*I)*R^(52/3)*Bi_rq^20*k_rq^3*wr^2*wi^2+(0.1183173628e35*I)*R^(50/3)*Bi_rq^14*k_rq^2*wr*wi^4-(0.1531465443e35*I)*R^(47/3)*Bi_rq^11*k_rq^2*wr^3*wi^2+(0.7657327213e34*I)*R^(47/3)*Bi_rq^11*k_rq^2*wr*wi^4-(0.4526477388e35*I)*R^(47/3)*Bi_rq^9*k_rq^3*wr^2*wi^2-(0.8140924838e38*I)*R^(44/3)*Bi_rq^14*k_rq*wr^2*wi^2+(0.1384897535e37*I)*R^(44/3)*Bi_rq^14*k_rq^2*wr^3*wi^2-(0.8605707758e31*I)*R^(32/3)*k_rq^3*Bi_rq^5*Ma_rq*wi^4+(0.8907839488e36*I)*R^(29/3)*k_rq^3*Bi_rq^6*Ma_rq*wr^2-(0.8907839488e36*I)*R^(29/3)*k_rq^3*Bi_rq^6*Ma_rq*wi^2-(0.2590568235e32*I)*R^(35/3)*k_rq^5*Bi_rq^4*Ma_rq*wr^2+(0.2590568235e32*I)*R^(35/3)*k_rq^5*Bi_rq^4*Ma_rq*wi^2-(0.4975751558e36*I)*R^(38/3)*Bi_rq^8*Ma_rq*k_rq^6*wr-(0.8980966636e39*I)*R^(35/3)*Bi_rq^13*Ma_rq*k_rq^4*wr-(0.3011260462e35*I)*R^(23/3)*Bi_rq*Ma_rq*k_rq^4*wr-(0.1773488496e38*I)*R^(26/3)*Bi_rq^4*Ma_rq*k_rq^4*wr+(0.1030536314e36*I)*R^(35/3)*Bi_rq^5*Ma_rq*k_rq^6*wr+(0.4544820399e39*I)*R^(32/3)*Bi_rq^10*Ma_rq*k_rq^4*wr+(0.2410317190e29*I)*R^(44/3)*Bi_rq^24*Ma_rq*k_rq^2*wr+(0.1357681408e37*I)*R^(44/3)*Bi_rq^16*k_rq^4*Ma_rq*wr-(0.3905673713e35*I)*R^(41/3)*Bi_rq^17*k_rq^2*Ma_rq*wr^3-(0.4372771287e33*I)*R^(47/3)*Bi_rq^11*k_rq^6*Ma_rq*wr-(0.5529015844e32*I)*R^(41/3)*Bi_rq^13*k_rq^2*Ma_rq*wr^5+(0.2798413468e38*I)*R^(41/3)*Bi_rq^13*k_rq^4*Ma_rq*wr+(0.2515465823e37*I)*R^(38/3)*Bi_rq^14*k_rq^2*Ma_rq*wr^3+(0.3668029463e33*I)*R^(44/3)*Bi_rq^8*k_rq^6*Ma_rq*wr+(0.2493817675e32*I)*R^(38/3)*Bi_rq^10*k_rq^2*Ma_rq*wr^5-(0.1138535672e33*I)*R^(50/3)*Bi_rq^16*k_rq*wr^2*wi^4-(0.3866391118e35*I)*R^(47/3)*Bi_rq^17*k_rq^2*wr^3*wi^2+(0.1933195559e35*I)*R^(47/3)*Bi_rq^17*k_rq^2*wr*wi^4-(0.1116339219e34*I)*R^(56/3)*Bi_rq^16*k_rq^4*wr*wi^2+(0.5454758823e27*I)*R^(59/3)*Bi_rq^23*k_rq^2*wr^3*wi^2-(0.2727379411e27*I)*R^(59/3)*Bi_rq^23*k_rq^2*wr*wi^4+(0.2796076392e30*I)*R^(59/3)*Bi_rq^21*k_rq^3*wr^2*wi^2-(0.7432957153e24*I)*R^(56/3)*Bi_rq^26*k_rq*wr^2*wi^2-(0.1392147783e34*I)*R^(26/3)*k_rq^4*Ka_rq*wr*wi^2+(0.7719364488e34*I)*R^(53/3)*Bi_rq^19*k_rq^4*wr*wi^2-(0.7307114813e32*I)*R^(50/3)*Bi_rq^24*k_rq^2*wr*wi^2+(0.2171382652e29*I)*R^(53/3)*Bi_rq^23*k_rq^2*wr^3*wi^2-(0.1085691326e29*I)*R^(53/3)*Bi_rq^23*k_rq^2*wr*wi^4-(0.2381200684e35*I)*R^(50/3)*Bi_rq^20*k_rq*wr^2*wi^2+(0.5511887834e35*I)*R^(50/3)*Bi_rq^18*k_rq^3*wr^2*wi^2+(0.5466899311e33*I)*R^(53/3)*Bi_rq^17*k_rq^2*wr^3*wi^2-(0.2733449655e33*I)*R^(53/3)*Bi_rq^17*k_rq^2*wr*wi^4+(0.3280242435e33*I)*R^(50/3)*Bi_rq^20*k_rq^2*wr^3*wi^2-(0.1640121217e33*I)*R^(50/3)*Bi_rq^20*k_rq^2*wr*wi^4-(0.1775037760e21*I)*R^(62/3)*Bi_rq^26*k_rq^2*wr^3*wi^2+(0.8875188801e20*I)*R^(62/3)*Bi_rq^26*k_rq^2*wr*wi^4+(0.2365256825e32*I)*R^(53/3)*Bi_rq^21*k_rq^3*wr^2*wi^2-(0.1311950441e39*I)*R^(41/3)*Bi_rq^15*k_rq^2*wr*wi^2-(0.4359116962e36*I)*R^(32/3)*Bi_rq^2*k_rq*wr^2*wi^2+(0.3620302634e34*I)*R^(44/3)*Bi_rq^6*k_rq^3*wr^2*wi^2+(0.2430171380e31*I)*R^(52/3)*Bi_rq^22*k_rq^2*wr*wi^4-(0.6444555337e29*I)*R^(55/3)*Bi_rq^23*k_rq^3*wr^2*wi^2-(0.5919818141e34*I)*R^(43/3)*Bi_rq^9*k_rq*wr^4*wi^2+(0.5919818141e34*I)*R^(43/3)*Bi_rq^9*k_rq*wr^2*wi^4-(0.1411773145e40*I)*R^(40/3)*Bi_rq^14*k_rq^2*wr*wi^2+(0.5353011462e34*I)*R^(43/3)*Bi_rq^5*k_rq^3*wr^2*wi^2+(0.3962260413e38*I)*R^(40/3)*Bi_rq^10*k_rq*wr^2*wi^2+(0.7010758092e37*I)*R^(40/3)*Bi_rq^8*k_rq^3*wr^2*wi^2+(0.3810867117e30*I)*R^(58/3)*Bi_rq^24*k_rq^4*wr*wi^2+(0.1013880267e27*I)*R^(58/3)*Bi_rq^24*k_rq*wr^4*wi^2-(0.1013880267e27*I)*R^(58/3)*Bi_rq^24*k_rq*wr^2*wi^4+(0.3090464401e31*I)*R^(44/3)*k_rq^3*Bi_rq^17*Ma_rq*wr^4+(0.3090464401e31*I)*R^(44/3)*k_rq^3*Bi_rq^17*Ma_rq*wi^4+(0.2409859012e37*I)*R^(34/3)*Bi_rq^6*Ma_rq*k_rq^4*wr+(0.6422743130e26*I)*R^(50/3)*k_rq^3*Bi_rq^23*Ma_rq*wr^4+(0.6422743130e26*I)*R^(50/3)*k_rq^3*Bi_rq^23*Ma_rq*wi^4+(0.6282696054e29*I)*R^(50/3)*k_rq^4*Bi_rq^21*Ma_rq*wr^3+(0.4974871558e31*I)*R^(50/3)*k_rq^5*Bi_rq^19*Ma_rq*wr^2-(0.4974871558e31*I)*R^(50/3)*k_rq^5*Bi_rq^19*Ma_rq*wi^2-(0.5108948639e33*I)*R^(52/3)*Bi_rq^16*k_rq^2*wr^3*wi^2+(0.2554474320e33*I)*R^(52/3)*Bi_rq^16*k_rq^2*wr*wi^4-(0.1024854691e36*I)*R^(49/3)*Bi_rq^19*k_rq*wr^2*wi^2+(0.5471168249e36*I)*R^(38/3)*Bi_rq^8*k_rq^2*wr*wi^4-(0.3071104923e33*I)*R^(56/3)*Bi_rq^18*k_rq^3*wr^2*wi^2-(0.2834801641e34*I)*R^(53/3)*Bi_rq^21*k_rq^2*wr*wi^2+(0.2645208732e20*I)*R^(58/3)*k_rq^5*Bi_rq^26*Ka_rq*wr^4+(0.2645208732e20*I)*R^(58/3)*k_rq^5*Bi_rq^26*Ka_rq*wi^4-(0.1584521075e21*I)*R^(64/3)*k_rq^8*Bi_rq^26*Ka_rq*wr-(0.7973059801e24*I)*R^(61/3)*k_rq^7*Bi_rq^25*Ka_rq*wr^2+(0.1901829253e33*I)*R^(40/3)*Bi_rq^2*k_rq^3*wr^2*wi^2+(0.2326581318e38*I)*R^(37/3)*Bi_rq^7*k_rq*wr^2*wi^2-(0.1138495431e37*I)*R^(37/3)*Bi_rq^5*k_rq^3*wr^2*wi^2-(0.1517985840e34*I)*R^(40/3)*Bi_rq^4*k_rq^2*wr^3*wi^2+(0.7589929204e33*I)*R^(40/3)*Bi_rq^4*k_rq^2*wr*wi^4-(0.3351785666e36*I)*R^(37/3)*Bi_rq^7*k_rq^2*wr^3*wi^2+(0.1675892832e36*I)*R^(37/3)*Bi_rq^7*k_rq^2*wr*wi^4-(0.1395003596e35*I)*R^(55/3)*Bi_rq^15*k_rq^4*wr*wi^2+(0.3532944233e32*I)*R^(49/3)*k_rq^5*Bi_rq^18*Ma_rq*wr^2-(0.3532944233e32*I)*R^(49/3)*k_rq^5*Bi_rq^18*Ma_rq*wi^2-(0.2216223222e31*I)*R^(46/3)*k_rq^3*Bi_rq^23*Ma_rq*wr^2+(0.2216223222e31*I)*R^(46/3)*k_rq^3*Bi_rq^23*Ma_rq*wi^2+(0.3980838386e33*I)*R^(49/3)*k_rq^6*Bi_rq^16*Ma_rq*wr-(0.1512416173e34*I)*R^(46/3)*k_rq^5*Bi_rq^15*Ma_rq*wr^2+(0.1512416173e34*I)*R^(46/3)*k_rq^5*Bi_rq^15*Ma_rq*wi^2+(0.2568285424e34*I)*R^(43/3)*k_rq^3*Bi_rq^20*Ma_rq*wr^2-(0.2568285424e34*I)*R^(43/3)*k_rq^3*Bi_rq^20*Ma_rq*wi^2-(0.6883676006e31*I)*R^(35/3)*k_rq^6*Bi_rq*Ka_rq*wr^3-(0.7143412266e37*I)*R^(32/3)*k_rq^5*Bi_rq^4*Ka_rq*wr^2+(0.7143412266e37*I)*R^(32/3)*k_rq^5*Bi_rq^4*Ka_rq*wi^2-(0.3227860693e33*I)*R^(50/3)*k_rq^6*Bi_rq^16*Ka_rq*wr^3-(0.1135525381e36*I)*R^(47/3)*k_rq^5*Bi_rq^19*Ka_rq*wr^2+(0.1135525381e36*I)*R^(47/3)*k_rq^5*Bi_rq^19*Ka_rq*wi^2+(0.8013536717e31*I)*R^(47/3)*k_rq^4*Bi_rq^17*Ka_rq*wr^5+(0.9071126625e35*I)*R^(44/3)*k_rq^4*Bi_rq^22*Ka_rq*wr-(0.5945792349e33*I)*R^(53/3)*k_rq^7*Bi_rq^17*Ka_rq*wr^2+(0.5945792349e33*I)*R^(53/3)*k_rq^7*Bi_rq^17*Ka_rq*wi^2+(0.5627052842e35*I)*R^(50/3)*k_rq^6*Bi_rq^20*Ka_rq*wr-(0.2271122295e32*I)*R^(50/3)*k_rq^5*Bi_rq^18*Ka_rq*wr^4-(0.2271122295e32*I)*R^(50/3)*k_rq^5*Bi_rq^18*Ka_rq*wi^4-(0.2906616546e33*I)*R^(47/3)*k_rq^4*Bi_rq^21*Ka_rq*wr^3-(0.5437087544e37*I)*R^(23/3)*k_rq^4*Bi_rq*Ka_rq*wr+(0.2696132285e35*I)*R^(50/3)*k_rq^7*Bi_rq^14*Ka_rq*wr^2-(0.2696132285e35*I)*R^(50/3)*k_rq^7*Bi_rq^14*Ka_rq*wi^2-(0.6800412066e37*I)*R^(47/3)*k_rq^6*Bi_rq^17*Ka_rq*wr+(0.1119953967e34*I)*R^(47/3)*k_rq^5*Bi_rq^15*Ka_rq*wr^4+(0.1119953967e34*I)*R^(47/3)*k_rq^5*Bi_rq^15*Ka_rq*wi^4+(0.1616440423e35*I)*R^(32/3)*Bi_rq^6*Ma_rq*k_rq^4*wr^3-(0.2304975081e35*I)*R^(32/3)*Bi_rq^4*Ma_rq*k_rq^5*wr^2+(0.2413033647e30*I)*R^(35/3)*k_rq^6*Bi_rq^2*Ma_rq*wr+(0.1070056635e33*I)*R^(35/3)*k_rq^4*Bi_rq^6*Ma_rq*wr^3+(0.1071169013e33*I)*R^(38/3)*k_rq^6*Bi_rq^5*Ma_rq*wr-(0.2256160154e33*I)*R^(38/3)*k_rq^5*Bi_rq^7*Ma_rq*wr^2+(0.2256160154e33*I)*R^(38/3)*k_rq^5*Bi_rq^7*Ma_rq*wi^2+(0.5607698669e37*I)*R^(35/3)*k_rq^3*Bi_rq^12*Ma_rq*wr^2-(0.5607698669e37*I)*R^(35/3)*k_rq^3*Bi_rq^12*Ma_rq*wi^2+(0.3599508276e28*I)*R^(47/3)*k_rq^3*Bi_rq^24*Ma_rq*wr^2-(0.3599508276e28*I)*R^(47/3)*k_rq^3*Bi_rq^24*Ma_rq*wi^2-(0.2533958845e37*I)*R^(41/3)*k_rq^4*Bi_rq^16*Ma_rq*wr-(0.2398180353e33*I)*R^(41/3)*k_rq^3*Bi_rq^14*Ma_rq*wr^4-(0.2398180353e33*I)*R^(41/3)*k_rq^3*Bi_rq^14*Ma_rq*wi^4-(0.2301822448e30*I)*R^(59/3)*k_rq^8*Bi_rq^21*Ka_rq*wr+(0.3629959334e29*I)*R^(56/3)*k_rq^6*Bi_rq^22*Ka_rq*wr^3+(0.3964543912e28*I)*R^(53/3)*k_rq^5*Bi_rq^25*Ka_rq*wr^2-(0.3964543912e28*I)*R^(53/3)*k_rq^5*Bi_rq^25*Ka_rq*wi^2+(0.2801810054e25*I)*R^(53/3)*k_rq^4*Bi_rq^23*Ka_rq*wr^5+(0.1031963441e28*I)*R^(53/3)*k_rq^5*Bi_rq^22*Ma_rq*wr^2-(0.1031963441e28*I)*R^(53/3)*k_rq^5*Bi_rq^22*Ma_rq*wi^2-(0.3120308867e34*I)*R^(44/3)*k_rq^5*Bi_rq^13*Ma_rq*wr^2+(0.3120308867e34*I)*R^(44/3)*k_rq^5*Bi_rq^13*Ma_rq*wi^2-(0.1228004437e36*I)*R^(41/3)*k_rq^3*Bi_rq^18*Ma_rq*wr^2+(0.1228004437e36*I)*R^(41/3)*k_rq^3*Bi_rq^18*Ma_rq*wi^2+(0.6829077716e33*I)*R^(44/3)*k_rq^4*Bi_rq^15*Ma_rq*wr^3-(0.6238216011e33*I)*R^(38/3)*k_rq^4*Bi_rq^9*Ma_rq*wr^3+(0.2862119713e37*I)*R^(47/3)*Bi_rq^17*k_rq*wr^2*wi^2-(0.2366347257e35*I)*R^(50/3)*Bi_rq^14*k_rq^2*wr^3*wi^2+(0.3775774935e35*I)*R^(47/3)*k_rq^3*Bi_rq^13*wr^4*wi-(0.7551549867e35*I)*R^(47/3)*k_rq^3*Bi_rq^13*wr^2*wi^3-(0.1788080260e36*I)*R^(47/3)*k_rq^5*Bi_rq^9*wr^2*wi-(0.4037263014e39*I)*R^(44/3)*k_rq^3*Bi_rq^14*wr^2*wi+(0.6009233886e35*I)*R^(53/3)*k_rq^5*Bi_rq^15*wr^2*wi-(0.1180134552e36*I)*R^(50/3)*k_rq^3*Bi_rq^20*wr^2*wi+(0.1408901356e32*I)*R^(50/3)*Bi_rq^18*k_rq^2*wr^5*wi-(0.4696337855e32*I)*R^(50/3)*Bi_rq^18*k_rq^2*wr^3*wi^3+(0.1408901356e32*I)*R^(50/3)*Bi_rq^18*k_rq^2*wr*wi^5+(0.3020873476e31*I)*R^(53/3)*Bi_rq^19*k_rq^3*wr^4*wi-(0.6041746950e31*I)*R^(53/3)*Bi_rq^19*k_rq^3*wr^2*wi^3+(0.2099907059e34*I)*R^(53/3)*Bi_rq^17*k_rq^4*wr^3*wi-(0.2099907059e34*I)*R^(53/3)*Bi_rq^17*k_rq^4*wr*wi^3-(0.3179766260e33*I)*R^(50/3)*Bi_rq^22*k_rq^2*wr^3*wi+(0.3179766260e33*I)*R^(50/3)*Bi_rq^22*k_rq^2*wr*wi^3-(0.6774213178e38*I)*R^(28/3)*Bi_rq^6*Ma_rq*k_rq^4*wr+(0.5204700843e37*I)*R^(38/3)*k_rq^3*Bi_rq^15*Ma_rq*wr^2-(0.5204700843e37*I)*R^(38/3)*k_rq^3*Bi_rq^15*Ma_rq*wi^2-(0.2998300581e33*I)*R^(41/3)*k_rq^6*Bi_rq^8*Ma_rq*wr-(0.2632683443e38*I)*R^(38/3)*k_rq^4*Bi_rq^13*Ma_rq*wr+(0.8014422036e32*I)*R^(38/3)*k_rq^3*Bi_rq^11*Ma_rq*wr^4+(0.8014422036e32*I)*R^(38/3)*k_rq^3*Bi_rq^11*Ma_rq*wi^4-(0.1845460446e37*I)*R^(32/3)*k_rq^4*Bi_rq^7*Ma_rq*wr-(0.8605707758e31*I)*R^(32/3)*k_rq^3*Bi_rq^5*Ma_rq*wr^4+(0.6463891808e33*I)*R^(49/3)*Bi_rq^21*k_rq^4*Ma_rq*wr+(0.6494000869e28*I)*R^(52/3)*Bi_rq^24*k_rq^4*Ma_rq*wr+(0.2163652842e33*I)*R^(46/3)*Bi_rq^10*Ma_rq*k_rq^6*wr-(0.7180216485e33*I)*R^(49/3)*Bi_rq^13*Ma_rq*k_rq^6*wr+(0.2475935920e33*I)*R^(49/3)*Bi_rq^15*Ma_rq*k_rq^5*wr^2-(0.2475935920e33*I)*R^(49/3)*Bi_rq^15*Ma_rq*k_rq^5*wi^2+(0.8517902052e35*I)*R^(46/3)*Bi_rq^16*Ma_rq*k_rq^6*wr+(0.1131317333e32*I)*R^(37/3)*Bi_rq^3*Ma_rq*k_rq^5*wr^2-(0.1131317333e32*I)*R^(37/3)*Bi_rq^3*Ma_rq*k_rq^5*wi^2-(0.1918822575e34*I)*R^(46/3)*Bi_rq^20*k_rq^3*Ma_rq*wr^2+(0.1918822575e34*I)*R^(46/3)*Bi_rq^20*k_rq^3*Ma_rq*wi^2-(0.1470903330e39*I)*R^(47/3)*k_rq^2*Bi_rq^17*wr*wi+(0.6157424459e37*I)*R^(50/3)*k_rq^3*Bi_rq^14*wr^2*wi-(0.1156694956e33*I)*R^(38/3)*k_rq^9*Bi_rq^6*Ka_rq*Ma_rq-(0.8459479221e35*I)*R^(35/3)*k_rq^7*Bi_rq^11*Ka_rq*Ma_rq-(0.7502666969e31*I)*R^(35/3)*k_rq^9*Bi_rq^3*Ka_rq*Ma_rq-(0.1093337152e38*I)*R^(32/3)*k_rq^7*Bi_rq^8*Ka_rq*Ma_rq-(0.3488465717e33*I)*R^(47/3)*k_rq^9*Bi_rq^15*Ka_rq*Ma_rq+(0.2867974221e24*I)*R^(56/3)*k_rq^9*Bi_rq^24*Ka_rq*Ma_rq-(0.9181218654e31*I)*R^(47/3)*k_rq^7*Bi_rq^23*Ka_rq*Ma_rq+(0.2490101299e35*I)*R^(26/3)*k_rq^7*Bi_rq^2*Ka_rq*Ma_rq+(0.2321839922e37*I)*R^(29/3)*k_rq^7*Bi_rq^5*Ka_rq*Ma_rq+(0.5970257926e31*I)*R^(50/3)*k_rq^9*Bi_rq^18*Ka_rq*Ma_rq-(0.1213822601e34*I)*R^(44/3)*k_rq^9*Bi_rq^12*Ka_rq*Ma_rq-(0.5932727868e36*I)*R^(41/3)*k_rq^7*Bi_rq^17*Ka_rq*Ma_rq+(0.7982739506e34*I)*R^(44/3)*k_rq^7*Bi_rq^20*Ka_rq*Ma_rq+(0.8599746173e33*I)*R^(41/3)*k_rq^9*Bi_rq^9*Ka_rq*Ma_rq+(0.3433817662e38*I)*R^(38/3)*k_rq^7*Bi_rq^14*Ka_rq*Ma_rq+(0.1375886663e29*I)*R^(53/3)*k_rq^9*Bi_rq^21*Ka_rq*Ma_rq-(0.1483269582e31*I)*R^(52/3)*Bi_rq^22*Ma_rq*k_rq^6*wr-(0.9228305468e23*I)*R^(55/3)*Bi_rq^25*Ma_rq*k_rq^6*wr+(0.1535707641e36*I)*R^(40/3)*Bi_rq^14*Ma_rq*k_rq^4*wr^3-(0.3286027145e34*I)*R^(43/3)*Bi_rq^17*Ma_rq*k_rq^4*wr^3-(0.1031591138e34*I)*R^(55/3)*Bi_rq^17*k_rq^3*wr^2*wi^2+(0.2845646843e33*I)*R^(58/3)*Bi_rq^18*k_rq^4*wr*wi^2+(0.1108512641e32*I)*R^(55/3)*Bi_rq^23*k_rq^2*wr*wi^2+(0.2347964944e29*I)*R^(55/3)*Bi_rq^21*k_rq*wr^4*wi^2-(0.2347964944e29*I)*R^(55/3)*Bi_rq^21*k_rq*wr^2*wi^4+(0.2543172812e37*I)*R^(46/3)*Bi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q*wr^2*wi^3-0.2153645786e36*R^(28/3)*k_rq^4*Bi_rq^2*Ka_rq*wr^2*wi-0.2625039913e32*R^(37/3)*k_rq^7*Bi_rq*Ka_rq*wr*wi-0.5883311735e32*R^(34/3)*k_rq^5*Bi_rq^2*Ka_rq*wr^3*wi+0.5883311735e32*R^(34/3)*k_rq^5*Bi_rq^2*Ka_rq*wr*wi^3-0.5646683253e37*R^(31/3)*k_rq^4*Bi_rq^5*Ka_rq*wr^2*wi-0.1000959436e35*R^(40/3)*k_rq^5*Bi_rq^8*Ka_rq*wr*wi^3-0.3761999885e38*R^(37/3)*k_rq^4*Bi_rq^11*Ka_rq*wr^2*wi+0.1285811343e35*R^(31/3)*Bi_rq^7*Ma_rq*k_rq^2*wr^2*wi+0.2276503560e37*R^(31/3)*Bi_rq^5*Ma_rq*k_rq^3*wr*wi+0.2486418157e33*R^(37/3)*Bi_rq^5*Ma_rq*k_rq^4*wr^2*wi-0.1018264908e38*R^(34/3)*Bi_rq^8*Ma_rq*k_rq^3*wr*wi+0.2100354850e32*R^(34/3)*Bi_rq^6*Ma_rq*k_rq^2*wr^4*wi-0.4200709701e32*R^(34/3)*Bi_rq^6*Ma_rq*k_rq^2*wr^2*wi^3+0.2503692353e38*R^(40/3)*Bi_rq^14*Ma_rq*k_rq^3*wr*wi+0.2346875167e33*R^(40/3)*Bi_rq^12*Ma_rq*k_rq^2*wr^4*wi-0.4693750334e33*R^(40/3)*Bi_rq^12*Ma_rq*k_rq^2*wr^2*wi^3-0.3110388219e34*R^(46/3)*Bi_rq^12*Ma_rq*k_rq^5*wr*wi-0.1775461680e34*R^(43/3)*Bi_rq^13*Ma_rq*k_rq^3*wr^3*wi+0.1775461680e34*R^(43/3)*Bi_rq^13*Ma_rq*k_rq^3*wr*wi^3-0.5235091884e36*R^(40/3)*Bi_rq^16*Ma_rq*k_rq^2*wr^2*wi-0.2675071170e36*R^(28/3)*Bi_rq^4*Ma_rq*k_rq^2*wr^2*wi-0.4452640878e32*R^(34/3)*Bi_rq^4*Ma_rq*k_rq^3*wr^3*wi+0.4452640878e32*R^(34/3)*Bi_rq^4*Ma_rq*k_rq^3*wr*wi^3+0.1037819215e37*R^(37/3)*Bi_rq^11*Ma_rq*k_rq^3*wr*wi-0.1630963708e33*R^(37/3)*Bi_rq^9*Ma_rq*k_rq^2*wr^4*wi+0.3261927415e33*R^(37/3)*Bi_rq^9*Ma_rq*k_rq^2*wr^2*wi^3-0.2507905835e33*R^(40/3)*Bi_rq^6*Ma_rq*k_rq^5*wr*wi+0.6701308724e32*R^(37/3)*Bi_rq^7*Ma_rq*k_rq^3*wr^3*wi-0.6701308724e32*R^(37/3)*Bi_rq^7*Ma_rq*k_rq^3*wr*wi^3+0.5883077136e37*R^(34/3)*Bi_rq^10*Ma_rq*k_rq^2*wr^2*wi+0.2053105778e34*R^(43/3)*Bi_rq^9*Ma_rq*k_rq^5*wr*wi+0.7928743687e33*R^(40/3)*Bi_rq^10*Ma_rq*k_rq^3*wr^3*wi-0.7928743687e33*R^(40/3)*Bi_rq^10*Ma_rq*k_rq^3*wr*wi^3-0.1037848581e38*R^(37/3)*Bi_rq^13*Ma_rq*k_rq^2*wr^2*wi+0.2660434902e23*R^(56/3)*k_rq^5*Bi_rq^26*Ka_rq*wr*wi^2+0.2709430263e23*R^(50/3)*k_rq^3*Bi_rq^25*Ma_rq*wr*wi^2+0.1892766758e36*R^(35/3)*k_rq^3*Bi_rq^10*Ma_rq*wr*wi^2-0.2883115261e35*R^(41/3)*k_rq^3*Bi_rq^16*Ma_rq*wr*wi^2-0.2098433684e35*R^(32/3)*k_rq^3*Bi_rq^7*Ma_rq*wr*wi^2-0.2565061840e26*R^(50/3)*k_rq^2*Bi_rq^24*Ma_rq*wr^2*wi^2-0.2035915109e23*R^(53/3)*k_rq^3*Bi_rq^25*Ma_rq*wr*wi^2+0.6672568890e36*R^(43/3)*Bi_rq^15*Ma_rq*k_rq^5*wr*wi+0.3788768229e32*R^(37/3)*k_rq^8*Bi_rq^7*Ka_rq*Ma_rq*wi-0.1045004695e37*R^(29/3)*k_rq^4*Bi_rq^3*Ka_rq*wr^2*wi-0.2211451947e33*R^(38/3)*k_rq^7*Bi_rq^2*Ka_rq*wr*wi-0.2890816040e33*R^(35/3)*k_rq^5*Bi_rq^3*Ka_rq*wr^3*wi+0.2890816040e33*R^(35/3)*k_rq^5*Bi_rq^3*Ka_rq*wr*wi^3-0.3194361480e37*R^(32/3)*k_rq^4*Bi_rq^6*Ka_rq*wr^2*wi+0.2640453232e34*R^(38/3)*k_rq^6*Bi_rq^4*Ka_rq*wr^2*wi-0.5170703568e38*R^(35/3)*k_rq^5*Bi_rq^7*Ka_rq*wr*wi+0.2019761612e33*R^(35/3)*k_rq^4*Bi_rq^5*Ka_rq*wr^4*wi-0.4039523225e33*R^(35/3)*k_rq^4*Bi_rq^5*Ka_rq*wr^2*wi^3-0.2388622396e27*R^(56/3)*k_rq^5*Bi_rq^24*Ka_rq*wr^3*wi+0.2388622396e27*R^(56/3)*k_rq^5*Bi_rq^24*Ka_rq*wr*wi^3+0.2535900152e31*R^(53/3)*k_rq^6*Bi_rq^19*Ka_rq*wr^2*wi-0.9002337419e33*R^(50/3)*k_rq^5*Bi_rq^22*Ka_rq*wr*wi+0.3433409899e30*R^(50/3)*k_rq^4*Bi_rq^20*Ka_rq*wr^4*wi-0.6866819797e30*R^(50/3)*k_rq^4*Bi_rq^20*Ka_rq*wr^2*wi^3-0.1084535748e32*R^(56/3)*k_rq^7*Bi_rq^20*Ka_rq*wr*wi-0.6067520056e29*R^(53/3)*k_rq^5*Bi_rq^21*Ka_rq*wr^3*wi+0.6067520056e29*R^(53/3)*k_rq^5*Bi_rq^21*Ka_rq*wr*wi^3-0.1176876099e31*R^(50/3)*k_rq^4*Bi_rq^24*Ka_rq*wr^2*wi-0.9527996880e35*R^(40/3)*Bi_rq^14*Ma_rq*k_rq^2*wr^2*wi^2+0.3643405546e36*R^(40/3)*Bi_rq^12*Ma_rq*k_rq^3*wr*wi^2+0.1371685656e34*R^(40/3)*k_rq^4*Bi_rq^11*Ma_rq*wr^2*wi+0.4625443504e33*R^(37/3)*k_rq^5*Bi_rq^6*Ma_rq*wr*wi-0.6364981070e32*R^(49/3)*Bi_rq^17*Ma_rq*k_rq^4*wr^2*wi-0.1357270192e31*R^(46/3)*Bi_rq^18*Ma_rq*k_rq^2*wr^4*wi+0.2714540384e31*R^(46/3)*Bi_rq^18*Ma_rq*k_rq^2*wr^2*wi^3+0.2735224364e35*R^(28/3)*Bi_rq^2*Ma_rq*k_rq^3*wr*wi-0.4924939815e33*R^(25/3)*Bi_rq*Ma_rq*k_rq^2*wr^2*wi+0.3226548980e33*R^(49/3)*k_rq^5*Bi_rq^17*Ka_rq*wr^3*wi-0.3226548980e33*R^(49/3)*k_rq^5*Bi_rq^17*Ka_rq*wr*wi^3-0.1202661533e35*R^(46/3)*k_rq^4*Bi_rq^20*Ka_rq*wr^2*wi-0.2783596833e33*R^(52/3)*k_rq^6*Bi_rq^18*Ka_rq*wr^2*wi-0.3273275048e34*R^(49/3)*k_rq^5*Bi_rq^21*Ka_rq*wr*wi+0.3043607331e30*R^(49/3)*k_rq^4*Bi_rq^19*Ka_rq*wr^4*wi-0.6087214662e30*R^(49/3)*k_rq^4*Bi_rq^19*Ka_rq*wr^2*wi^3-0.4963909289e23*R^(55/3)*k_rq^4*Bi_rq^25*Ka_rq*wr^4*wi+0.9927818576e23*R^(55/3)*k_rq^4*Bi_rq^25*Ka_rq*wr^2*wi^3+0.1749367823e36*R^(34/3)*Bi_rq^6*Ma_rq*k_rq^5*wr*wi-0.3419188348e35*R^(31/3)*Bi_rq^5*Ma_rq*k_rq^4*wr^2*wi+0.4120879344e33*R^(37/3)*k_rq^4*Bi_rq^8*Ma_rq*wr^2*wi+0.1043190142e34*R^(40/3)*k_rq^3*Bi_rq^13*Ma_rq*wr^3*wi-0.1043190142e34*R^(40/3)*k_rq^3*Bi_rq^13*Ma_rq*wr*wi^3+0.8872238174e33*R^(43/3)*Bi_rq^11*Ma_rq*k_rq^4*wr^2*wi-0.4129683649e35*R^(43/3)*k_rq^6*Bi_rq^9*Ka_rq*wr^2*wi+0.3556543721e39*R^(40/3)*k_rq^5*Bi_rq^12*Ka_rq*wr*wi-0.8528054640e33*R^(40/3)*k_rq^4*Bi_rq^10*Ka_rq*wr^4*wi+0.1705610928e34*R^(40/3)*k_rq^4*Bi_rq^10*Ka_rq*wr^2*wi^3+0.1411704486e30*R^(58/3)*k_rq^7*Bi_rq^22*Ka_rq*wr*wi+0.2528007384e27*R^(55/3)*k_rq^5*Bi_rq^23*Ka_rq*wr^3*wi-0.2528007384e27*R^(55/3)*k_rq^5*Bi_rq^23*Ka_rq*wr*wi^3+0.7263784375e27*R^(58/3)*k_rq^6*Bi_rq^24*Ka_rq*wr^2*wi+0.1368186868e35*R^(49/3)*k_rq^6*Bi_rq^15*Ka_rq*wr^2*wi-0.9200691040e36*R^(46/3)*k_rq^5*Bi_rq^18*Ka_rq*wr*wi-0.8434015658e30*R^(49/3)*k_rq^4*Bi_rq^20*Ma_rq*wr^2*wi-0.1360907621e31*R^(46/3)*k_rq^3*Bi_rq^19*Ma_rq*wr^3*wi+0.1360907621e31*R^(46/3)*k_rq^3*Bi_rq^19*Ma_rq*wr*wi^3-0.1196222138e21*R^(52/3)*k_rq^3*Bi_rq^25*Ma_rq*wr^3*wi+0.1196222138e21*R^(52/3)*k_rq^3*Bi_rq^25*Ma_rq*wr*wi^3+0.1150664248e28*R^(52/3)*k_rq^4*Bi_rq^23*Ma_rq*wr^2*wi+0.2998305288e25*R^(55/3)*k_rq^5*Bi_rq^24*Ma_rq*wr*wi+0.1094862510e35*R^(43/3)*k_rq^7*Bi_rq^7*Ka_rq*wr*wi+0.1000959436e35*R^(40/3)*k_rq^5*Bi_rq^8*Ka_rq*wr^3*wi):
> 

 

> 

EQ2 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k_rq^6*Bi_rq^12*Ma_rq*wr+0.6679692799e50*R^14*k_rq^5*Bi_rq^14*Ma_rq*wr^2-0.6679692799e50*R^14*k_rq^5*Bi_rq^14*Ma_rq*wi^2-0.1378466399e49*R^15*k_rq^5*Bi_rq^17*Ma_rq*wr^2+0.1378466399e49*R^15*k_rq^5*Bi_rq^17*Ma_rq*wi^2+0.1103799190e47*R^17*Bi_rq^17*k_rq*wr^4*wi^2-0.1103799190e47*R^17*Bi_rq^17*k_rq*wr^2*wi^4-0.1130840571e49*R^16*k_rq^2*Bi_rq^22*wr*wi^2+0.3963891402e51*R^15*k_rq^2*Bi_rq^19*wr*wi^2-0.9119586161e43*R^17*k_rq^2*Bi_rq^25*wr*wi^2:
> 

 

> 

#------------------------------------------------------------------------------------------------------------------------------------------------
> 

eqns:={EQ1,EQ2}:
> 

#--------------------------------------------------------------------------------------------------------------------------------------------#635
> 

#n:=147/100:B:=0:Delta:=1:We:=1:#use avoid{a,b,c,d,...}
> 

R:=10:
 S:=Array():
 S(1, 1..3):=Array( ['k_rq', 'wi', 'wr']):
 j:=1:
 for i from 1 by 1 to 1000 do
      k_rq:=0.001*i:
      ans:=fsolve(eqns, {wi, wr},{wi=0..0.2,wr=0..10}):
      if type(ans,set)
      then S(j+1, 1..3):=Array([k_rq, rhs~(ans)[]]) ;
           j:=j+1;
      fi;
     # v(i):=(subs(S,(wi)));
     # w(i):=(subs(S,(wr)));
 end do:
> 

#
# Check the of values of k_rq for which fsolve()
# actually produces an answer
#
  interface(rtablesize=upperbound(S)[1]):
  S;

[10, 10]

  

Array(%id = 36893488148708661236)

 (1)
> 

plot( convert(S[2..,1], list), convert(S[2..,2], list));

 

NULL


 

Download fsol.mw

I'm not sure...

tomleslie 13876
May 09 2021
0 2

I kmow what the "phase plane" between two functions is!!!

Maybe you just want to plot F(r) versus m(r) for given values of the parameters 'a' and 'l' ?? This is done in the attached.

> 

  restart;
  myProc:= proc( a, l )
                 local R, X, m, F;
                 m:= unapply( (l^2*r^2 + r^4 + a*l^2)/(2*l^2*r), r);
                 X := m(r)*l^2*(sqrt((27*l^2*m(r)^4 - 16*a^2*l^2 + 64*a^3)/(-l^2 + 4*a)) + 4*a)*(-l^2 + 4*a)^2;
                 R:= X^(1/3)/(-l^2 + 4*a) - 3*l^2*m(r)^2/X^(1/3):
                 F:=unapply( 736*R^4/sqrt((-1380*Pi*R*m(r)^3 + 368*R^4 - 1587*m(r)^2*R^2 + 1280*m(r)^2*a)^2),r):
                 return m(r), F(r);
           end proc:

  plot([myProc(1,5), r=0..2], color=red, labels=['m(r)', 'F(r)']);

 
> 

 


 

Download phplt.mw

Aaaah - the joys of 2D math input...

tomleslie 13876
May 08 2021
2 1

You appear to believe that two different methods of entering a vatriable with a "subscript" are identical, So for example you think that 'K__3' and K[3] are the same. To be fair, if you choose to use 2D math input then they will "appear" the same (as K3) in your input , but they are two completely differnt entities: K[3] is the third entry in the indexable quantity 'K' (so K is a vector, list,,table -something for whihc the concept of a third element makes sense!).On the other hand K__3, just signifies the name 'K' with an inert subscript '3' - no indexable quantity is implied/involved.

Unfortunately when you choose to use 2D math input, you can't *see* the difference between these two entirely different entities, either in your input or Maple's output.

There are mutiple places across multiple variables where you have used name[sub] and name__sub interchangeably - whihc of course you cannot do. In the attached I (think I) have made your naming consistent.

In the final few execution groups of your original worksheet you use the variables D[B}, D[C}, etc and u[B], u[C] etc - none of which are defined anywhere (with or without units) - so I took a wild guess about what you intended (and I'm probably wrong!). I replaced all D[B}, D[C}, etc with L[B}, L{C} etc and u[B], u[C] with y[B], y[C} etc.

At least the attached now runs with no errors, but with the number of corrections I have made the possibility that I have guessed correctly every time what you want is pretty small!


 

ASME
B31.3-2002 - Flexibility Analysis

This document considers the requirements for flexibility analysis of pipe work in accordance with standard ASME B31.3 2002 Edition.

 

 

Temperature Reduction for Swagelok fittings and tube.

 

Based on 316L Stainless steel fittings and tube, from Data Sheet MS-01-107 rev S and shown below a linear interpolation is used to determine the factor for operation at 100 and 130 centigrade.

with(Interpolation)

with(plots)

Temperature values from table. temp := [93, 204, 315, 426, 537]

Factor values from fable. fact := [1, .96, .85, .79, .76]

Graph of data points. pointplot(temp, fact)

 

facto := LinearInterpolation(temp, fact)

_m679396416

(1.1)

Factor for operation at 100 centigrade.facto(100) = HFloat(0.9974774774774775)NULL

Factor for operation at 130 centigrade. facto(130) = HFloat(0.9866666666666667)NULL

Swagelok compatability

 

From the B31.3 standard, Table 326.1 Component Standards, Swagelok fittings are not listed directly but it is considered that they are covered under the entry for Unified inch screw threads (UN and UNR thread form) with designation ASME B1.1. Stainless steel pipe is covered directly with designation ASME B36.19M.

Therefore the use of these parts is acceptable under the standard.

Specific Tube Sizes.

 

The initial specification suggests a DN15 tube size. This equates to 5/8", however the range of fittings in this size is limited. A preferred size would be to either go down to 1/2" or up to 3/4" tube. These will give an actual bore of ~10mm and ~15mm respectivly. In terms of the pressure and temperature specification the following information is provided from Swagelok data sheet MS-01-181.

1/2" x 0.049" wall pressure rating at 37C. TP05__37 := 3700*Unit('psi') = 3700*Units:-Unit(` psi `)NULL

3/4" x 0.065" wall pressure rating at 37C. TP75__37 := 3300*Unit('psi') = 3300*Units:-Unit(` psi `)NULL

Hence pressure ratings, converted to bar at 130C are:

TP05__130 := convert(TP05__37*facto(130), units, bar) = HFloat(251.70460624926636)*Units:-Unit(bar)NULL

TP75__130 := convert(TP75__37*facto(130), units, bar) = HFloat(224.49329746556185)*Units:-Unit(bar)NULL

Hence the tubing used in the design is acceptable.

Weld Fitting Specifications

 

There are several types of welded fittings as "Tee" "Reduceing Tee" and "Reducing Unions" and they have various pressure ratings.

A 3/4" tube butt weld "Tee" Tee75__37 := 2400*Unit('psi') = 2400*Units:-Unit(` psi `)NULL

The weld fittings have pressure rating at 130C of. Tee75__130 := convert(Tee75__37*facto(130), units, bar) = HFloat(163.2678527022268)*Units:-Unit(bar)NULL

Hence the weld fittings used in the design are acceptable.

NULL

Tube wall calculation

 

with(Units[Standard])

Determine the minimum wall thickness based on the design code.

Design pressure. P__D := 150*Unit('bar') = 150*Units:-Unit(bar)NULL

Outside diameter of nominal 3/4" tube. D__75 := 19.05*Unit('mm') = 19.05*Units:-Unit(mm)NULL

Quality factor. E := 1 = 1 for seamless stainless steel tube.

Allowable stress for 316L tube at 166C (300F). From B31.1 Table A1. S__allow := 115.14*Unit('MPa') = 115.14*Units:-Unit(MPa)NULL

Y__coeff := .4 = .4NULL

Miniumum wall thickness. t__min := P__D*D__75/(2*(E*S__allow+P__D*Y__coeff)) = 0.1179420505e-2*Units:-Unit(m)NULL

Actual wall thickness. W__75 := 1.65*Unit('mm') = 1.65*Units:-Unit(mm)NULL

if W__75 >= t__min then print(PASS) else print(FAIL) end if = PASSNULLNULL

NULL``

Conversions

 

From Appendix A

allowable stress for TP316L at 100C (212F).convert(16.7, units, ksi, MPa) = 115.1424468NULL

allowable stress for TP316L at room temperature.convert(25.0, units, ksi, MPa) = 172.3689323NULL

convert(18, units, m^3/h, L/minute) = 300NULL

convert(3300.0, units, psi, bar) = 227.5269907NULL

.58*% = 131.9656546NULL

convert(70.0, units, degF, degC) = 38.88888889NULL

convert(300.0, units, degF, degC) = 166.6666667NULL

convert(.117, units, Unit('dm'^3), Unit('L')) = .117NULL

with(ThermophysicalData:-Chemicals)

with(ThermophysicalData)

Property(MolarMass, He, useunits)

4.0026020*Units:-Unit(g/mol)

(3.1)

Property(density, Helium, temperature = 300*Unit(kelvin), pressure = 101325*Unit(Pa), source = CoolProp)

.1625153176*Units:-Unit(kg/m^3)

(3.2)

Helium flow rate. flow[he] := 300*Unit('L')/Unit('min')NULL

Determine molar flow rate. molflow[he] := .1625153176*flow[he]*Units[Unit](kg/m^3)/(4.0026020*Units[Unit](g/mol)) = .2030120876*Units:-Unit(mol/s)NULL

Determine mass flow rate. massflow[he] := 4.0026020*molflow[he]*Units[Unit](g/mol) = 0.8125765879e-3*Units:-Unit(kg/s)NULL

Property(MolarMass, CO2, useunits)

44.0095000*Units:-Unit(g/mol)

(3.3)

Property(density, CO2, temperature = 300*Unit(kelvin), pressure = 101325*Unit(Pa), source = CoolProp)

1.796636097*Units:-Unit(kg/m^3)

(3.4)

Property(MolarMass, Air, useunits)

28.9651159*Units:-Unit(g/mol)

(3.5)

Property(density, Air, temperature = 300*Unit(kelvin), pressure = 101325*Unit(Pa), source = CoolProp)

1.176995588*Units:-Unit(kg/m^3)

(3.6)

Determine the boiling point of water at 6 bar, the maxiumum pressure delivered by the cd90e unit.

boiling := Property(temperature, water, pressure = 6*Unit(bar), Q = 0)

431.9764773*Units:-Unit(K)

(3.7)

convert(boiling, temperature, Celsius)

158.8264773*Units:-Unit(`°C`)

(3.8)

evalf(convert(30, units, Unit('ft'^3/'min'), liters/s))

14.15842330

(3.9)

``

Input Parameters

 

Allowable stress for calculations. S[A] := evalf(f*(1.25*S[c]+.25*S[h]))NULL = 219.82275*Units:-Unit(MPa)NULL

Allowable stress at minimum temperature. S[c] := 172.37*Unit('MPa')

Allowable stress at maximum temperature. S[h] := 115.14*Unit('MPa')

Stress reduction factor for N=7000 to 14000 cycles. f := .9

From Table C6 of standard, modulus of elasticity for stainless steel at room temperature.convert(28.3*10^6, units, psi, MPa) = 195121.6314NULL

Named as E[A] := 195121*Unit('MPa') = 195121*Units:-Unit(MPa)NULL

alpha[stst] := 16*10^(-6)*(Unit('m')*(1/(Unit('m')*Unit('K')))) = (1/62500)*Units:-Unit(1/K)NULL

Flexibility Stress

 

with(Units[Standard])

text##

S[E] := sqrt(S[b]^2+4*S[t]^2) =  = (S[b]^2+4*S[t]^2)^(1/2)NULL

Allowable stress factor for analysis requirement. K[allow] := 208000*S[A]/E[A] = 234.3321939 

Pipe diameter for 1/2" tube. D__A := 12.7*Unit('mm') = 12.7*Units:-Unit(mm)NULL

Pipe diameter for 3/4" tube. D__B := 19.05*Unit('mm') = 19.05*Units:-Unit(mm)NULL

Change in temperature. Delta[T] := 130*Unit('K') = 130*Units:-Unit(K)NULL

Section A1

Combined length of section A1.L[A1] := 110*Unit('mm')+119*Unit('mm') = 229*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[A1] := evalf(alpha[stst]*Delta[T]*L[A1]) = .4763200000*Units:-Unit(mm)NULL

U__A1 := 163*Unit('mm') = 163*Units:-Unit(mm)NULL

K[A1] := D__A*y[A1]/(L[A1]-U__A1)^2 = 0.1388719927e-2NULL

if K[A1] <= K[allow] then print(PASS) else print(FAIL) end if = PASSNULL

Section A2

Combined length of section A2.L[A2] := 610*Unit('mm')+119*Unit('mm') = 729*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[A2] := evalf(alpha[stst]*Delta[T]*L[A2]) = 1.516320000*Units:-Unit(mm)NULL

U__A2 := 622*Unit('mm') = 622*Units:-Unit(mm)NULL

K[A2] := D__A*y[A2]/(L[A2]-U__A2)^2 = 0.1682004018e-2NULL

if K[A2] <= K[allow] then print(PASS) else print(FAIL) end if = PASSNULL

Section A3

Combined length of section A3.L[A3] := 723*Unit('mm')+288*Unit('mm') = 1011*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[A3] := evalf(alpha[stst]*Delta[T]*L[A3]) = 2.102880000*Units:-Unit(mm)NULL

U__A3 := 778*Unit('mm') = 778*Units:-Unit(mm)NULL

K[A3] := D__A*y[A3]/(L[A3]-U__A3)^2 = 0.4919334672e-3NULL

if K[A3] <= K[allow] then print(PASS) else print(FAIL) end if = PASSNULL

 

unwith(Units[Standard])

Length of section B. L[B] := 526*Unit('mm')NULL = 526*Units:-Unit(mm) 

Length of section C. L[C] := 1331*Unit('mm') = 1331*Units:-Unit(mm) 

Length of section D. L[D] := 416*Unit('mm') = 416*Units:-Unit(mm) 

Length of section E. L[E] := 526*Unit('mm') = 526*Units:-Unit(mm) 

Length of section F. L[F] := 762*Unit('mm') = 762*Units:-Unit(mm) 

Length of section G. L[G] := 287*Unit('mm') = 287*Units:-Unit(mm) 

Length of section H. L[H] := 440*Unit('mm') = 440*Units:-Unit(mm) 

Length of section I. L[I] := 400*Unit('mm') = 400*Units:-Unit(mm)NULL

 

Length increase due to thermal expansion. y[B] := evalf(alpha[stst]*Delta[T]*L[B]) = 1.094080000*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[C] := evalf(alpha[stst]*Delta[T]*L[C]) = 2.768480000*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[D] := evalf(alpha[stst]*Delta[T]*L[D]) = .8652800000*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[E] := evalf(alpha[stst]*Delta[T]*L[E]) = 1.094080000*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[F] := evalf(alpha[stst]*Delta[T]*L[F]) = 1.584960000*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[G] := evalf(alpha[stst]*Delta[T]*L[G]) = .5969600000*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[H] := evalf(alpha[stst]*Delta[T]*L[H]) = .9152000000*Units:-Unit(mm)NULL

Length increase due to thermal expansion. y[I] := evalf(alpha[stst]*Delta[T]*L[I]) = .8320000000*Units:-Unit(mm)NULL

NULL

K[B] := L[B]*y[B]/(L[B]-y[B])^2 = 0.2088679871e-2

> 

 

K[C] := L[C]*y[C]/(L[C]-y[C])^2 = 0.2088679871e-2NULL

K[D] := L[D]*y[D]/(L[D]-y[D])^2 = 0.2088679872e-2NULL

K[E] := L[E]*y[E]/(L[E]-y[E])^2 = 0.2088679871e-2NULL

K[F] := L[F]*y[F]/(L[F]-y[F])^2 = 0.2088679872e-2NULL

K[G] := L[G]*y[G]/(L[G]-y[G])^2 = 0.2088679872e-2NULL

K[H] := L[H]*y[H]/(L[H]-y[H])^2 = 0.2088679872e-2NULL

K[I] := L[I]*y[I]/(L[I]-y[I])^2 = 0.2088679872e-2NULL

NULL

> 

 

> 

L[B]

NULL

``


 

Download 2DMathAgain.mw

 

From Maple help...

tomleslie 13876
May 08 2021
3 0

emphasis added

Curl, Divergence and Gradient check their arguments (for consistency) before sending the task to Nabla. So, if "A_" is a vector, then Gradient(A_) will interrupt the computation with an error message, as well as Divergence(A) and Curl(A) when  "A" is a scalar (not a vector). All these differential operations are realized just w.r.t the geometrical coordinates  "{phi, r, rho, theta, x, y, z}"  Therefore, if  "A"does not depend on these global geometrical coordinates, these commands (as well as Nabla) return 0.

 

 

 

 

 

 

Looking in the wrong place!!...

tomleslie 13876
May 08 2021
2 2

The expression 'lieu' only has solutions for x-values in the range -infinity..-10, and 10..infinity, so the command

Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green):

with a=5 and b=3, has no solutions. In the attached, I changed the range of 'x' to -20..20 and the required data appears

> 

restart: with(geometry): with(plots): `local`(O):
_EnvHorizontalName := x: _EnvVerticalName := y:
corde := a*x/cos(theta)-b*y/sin(theta) = a^2-b^2:
isolate(corde, a/cos(theta)):
eq1 := (a^2-b^2)*X/a^2 = a/cos(theta):
c := solve(eq1, cos(theta)):
eq2 := (a^2-b^2)*Y/b^2 = -b/sin(theta):
s := solve(eq2, sin(theta)):
lieu := simplify(expand((a^2-b^2)^2*X^2*Y^2*(c^2+s^2 = 1))):
allvalues(eliminate({eq1, eq2}, theta))[1][2]:
ell := x^2/a^2+y^2/b^2 = 1:
P := [a*cos(theta), b*sin(theta)]:
tgP := x*P[1]/a^2+y*P[2]/b^2 = 1:

sol := solve({corde, ell}, {x, y}):
tgP1 := simplify(x*rhs(sol[2][1])/a^2+y*rhs(sol[2][2])/b^2 = 1):
 

Error, (in isolate) a*x/cos(theta)-b*y/sin(theta) = a^2-b^2 does not contain a/cos(theta)

  
> 

#Drawing in a case
a := 5: b := 3: theta := (1/6)*Pi:

line(l1, corde): conic(co, ell):
Pole(P1, l1, co): coordinates(P1):
a := 5: b := 3: theta := (1/6)*Pi:
Ell := implicitplot(ell, x = -a .. a, y = -b .. b, color = red):
Cor := implicitplot(corde, x = -a-1 .. a, y = -b-1 .. b, color = blue):
TgP := implicitplot(tgP, x = 0 .. 10, y = -5 .. 10, color = magenta):
TgP1 := implicitplot(tgP1, x = -5 .. 10, y = -5 .. 10, color = magenta):
> 

lieu := subs(X = x, Y = y, lieu):
subs(x = 125*sqrt(3)*(1/24), y = -27/8, lieu):
#Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green):
Lieu := implicitplot(lieu, x = -20 .. 20, y = -b .. b, color = green):
dr := draw(P1):
display([Ell, Cor, Lieu, TgP, TgP1, dr], axes = normal, view = [-20 .. 20, -20 .. 20], scaling = constrained);

 
> 

seq( fsolve(eval(lieu, y=j)), j=-3..3,0.1);

-9.449111825, 9.449111825, -9.606365709, 9.606365709, -9.790295969, 9.790295969, -10.00800961, 10.00800961, -10.26939539, 10.26939539, -10.58865856, 10.58865856, -10.98702101, 10.98702101, -11.49779264, 11.49779264, -12.17672717, 12.17672717, -13.12565630, 13.12565630, -14.55556274, 14.55556274, -17.00075502, 17.00075502, -22.45066275, 22.45066275, -64.54227854, 64.54227854, 0., 0., -64.54227854, 64.54227854, -22.45066275, 22.45066275, -17.00075502, 17.00075502, -14.55556274, 14.55556274, -13.12565630, 13.12565630, -12.17672717, 12.17672717, -11.49779264, 11.49779264, -10.98702101, 10.98702101, -10.58865856, 10.58865856, -10.26939539, 10.26939539, -10.00800961, 10.00800961, -9.790295969, 9.790295969, -9.606365709, 9.606365709, -9.449111825, 9.449111825

 (1)
> 

 


 

Download fsdol.mw

 

 

 

 

 

It would be a lot easier...

tomleslie 13876
May 07 2021
2 3

if you actually supplieed the relevant equations (The big green up-arrow in the MApleprimes toolbar is your friend!)

Maybe you can use a spacecurve(), as in the attached, which plots x, y, z, values satisfying the pair of equations, which I just "made up"

> 

eqns:=[2*x+3*cos(y)+z^2=0, sin(x)+y+2*z=1];

[2*x+3*cos(y)+z^2 = 0, sin(x)+y+2*z = 1]

 (1)
> 

plots:-spacecurve( [ 'eval(x,fsolve(eval(eqns, z=Z)))',
                     'eval(y,fsolve(eval(eqns, z=Z)))',
                     Z
                   ],
                   Z=-5...5,
                   color=red,
                   thickness=4,
                   labels=[x,y,z]
                 );

 
> 

 


 

Download scurve.mw

 

Something like...

tomleslie 13876
May 07 2021
2 1

the attached, maybe?

> 

  restart;
  A:= nu^(2/3)*B/(g^(1/3)*(sin(beta))^(1/3)*h);
  ans:= simplify
        ( subs
          ( isolate
            ( R=h^3*g*sin(beta)/(3*nu^2),
              g
            ),
            A
          ),
          symbolic
        );

nu^(2/3)*B/(g^(1/3)*sin(beta)^(1/3)*h)

  

(1/3)*B*3^(2/3)/R^(1/3)

 (1)
> 

 


 

Download maybe.mw

 

Use a piecewise() definition...

tomleslie 13876
May 07 2021
1 0

as in the attached

> 

  restart;
  f:=x-> piecewise( x <= 0,  x^2 + x,
                    x < 3*Pi, sin(x),
                    x^2 - 6*x*Pi + 9*Pi^2 - x + 3*Pi
                  );
#
# Differentiate 'f'
#
  df:= diff(f(x), x);
#
# or, as an applicable function
#
  df:= unapply( diff(f(x), x),x);
#
# Plot the function and its derivative
#
  plot( [f, df], -10..20, color=[ red,blue]);
  

f := proc (x) options operator, arrow; piecewise(x <= 0, x^2+x, x < 3*Pi, sin(x), x^2-6*x*Pi+9*Pi^2-x+3*Pi) end proc

  

df := piecewise(x <= 0, 2*x+1, x <= 3*Pi, cos(x), 3*Pi < x, -6*Pi+2*x-1)

  

proc (x) options operator, arrow; piecewise(x <= 0, 2*x+1, x <= 3*Pi, cos(x), 3*Pi < x, -6*Pi+2*x-1) end proc

  

 
> 

 

> 

 


 

Download pw.mw

After fixing a couple of typos...

tomleslie 13876
May 05 2021
1 0

I disccovered that within the procedure Fig(), the solve() command produces no solutions. Hence the subsequent attempt to assign the first and thiird of these solutions using

P := [subs(sol[1], x), subs(sol[1], y)];
Q := [subs(sol[3], x), subs(sol[3], y)];

If there are no solutions then in the above statements sol[1], and sol[3], will produce Error, (in Fig) invalid subscript selector

See the attached. For most of the frames Fig/solve returns no solutions - for a few frames, 4 solutions are produced.


 

> 

restart: with(geometry): with(plots): local O:
_EnvHorizontalName := x: _EnvVerticalName := y:
ell := x^2/a^2+y^2/b^2 = 1:
a := 5: b := 3: alpha := (1/6)*Pi: p := sqrt(a^2*b^2/(a^2+b^2)):
PQ := x*cos(alpha)+y*sin(alpha)-p: drPQ := solve(PQ, y):
OPQ := x^2/a^2+y^2/b^2-((x*cos(alpha)+y*sin(alpha))/p)^2:
sol := solve({OPQ, ell}, {x, y}, explicit): P := [subs(sol[1], x), subs(sol[1], y)]: Q := [subs(sol[3], x), subs(sol[3], y)]:
O := [0, 0]:
Ell := implicitplot(ell, x = -a .. a, y = -b .. b, color = red):
DrOPQ := implicitplot(OPQ, x = -a .. a, y = -b .. b, color = magenta, numpoints = 5000):
DrPQ := plot(drPQ, x = -6 .. 6, color = green):
line(OP, 2*x-y): line(OQ, -(1/2)*x-y):

Points := pointplot([O[],P[],Q[]], symbol = solidcircle, color = red, symbolsize = 10):

T := textplot([[O[], "O"],[P[],"P"],[Q[],"Q"]], font = [times, 15], align = {below, right}):
cir := x^2+y^2 = p^2:
Cir := implicitplot(cir, x = -a .. a, y = -b .. b, color = black):
display([Ell, Cir, DrPQ, DrOPQ, Points, T], view = [-6 .. 6, -4 .. 6], axes = normal, scaling = constrained):
> 

Fig := proc (k)
             local alpha, PQ, drPQ, DrPQ, OPQ, DrOPQ, sol, P, Q, Points, T;
             global a, b, p, ell, Ell, Cir;
             alpha := k;
             PQ := x*cos(alpha)+y*sin(alpha)+p;
             drPQ := solve(PQ, y);
             OPQ := x^2/a^2+y^2/b^2-(x*cos(alpha)+y*sin(alpha))^2/p^2;
             sol:=solve({ell, OPQ}, {x, y}, explicit);
             return numelems([sol]);
          (*
             P := [subs(sol[1], x), subs(sol[1], y)];
             Q := [subs(sol[3], x), subs(sol[3], y)];
             Points := pointplot([P[], Q[]], symbol = solidcircle, color = red, symbolsize = 10);
             T := textplot([[P[], "P"], [Q[], "Q"]], font = [times, 15], align = {below, right});
             DrPQ := plot(drPQ, x = -6 .. 6, color = green);
             DrOPQ := implicitplot(OPQ, x = -a .. a, y = -b .. b, color = magenta, numpoints = 5000);
             display([Ell, Cir, DrPQ, DrOPQ, Points, T], view = [-a .. a, -b .. b], axes = normal, scaling = constrained);
          *)
       end proc:
#
# Check whethe the solve() command within the Fig() procedure
# actually produces solutions
#
  nframes:=100:
  seq( Fig(eval(2*Pi*i/nframes,i=k)), k=0..nframes);
  

4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4

 (1)
> 

 


 

Download noSol.mw

 

You seem to have several problems...

tomleslie 13876
May 05 2021
1 0

but I am only going to address the first one. (I'm assuming that subsequent errors are just a propagation of the first error)

You define a sequence of boundary condiitions as

ics := Y(i) = 1.8, D(Y)(i) = 0, X(1) = 0.4, D(X)(i) = 0, H(i) = 0.7;

in which 'i' is a random number in the range 0.60658..1. Thus the boundary value solution from the dsolve() command will only exist in the range of the independent variable 'z' defined by i..1.

However in a subsequnt referencce to this solution,

kk := [subs(solution, Y(z))];
          if    op(kk(0))>.7

The reference kk(0) in the above will attempt to evaluate the solution at z=0, when the solution only exists for i<=z<=1. Hence you get an error message  similar to

  Error, (in anonymous procedure) solution is only defined in the range .76226371411128628..1.

Given that you have defined one of the boundaries for the independent variable 'z' to be in the range 0.60658..1, with the other one at 1, you will never be able to compute the solution at z=0, which the  statement op(kk(0))>.7 requires

 

 

 

 

With care!!...

tomleslie 13876
May 04 2021
5 3

the pdsolve/numeric command does not return any derivative values, so these have to be computed from the returned (numerical) value of the dependent variable.

This *can* be achieved using the D() operator, but this should be used with care. Essentially one will be computing a numerical derivative, and this calculation is often numerically "unstable" - so be warned! The relevant graphs in the attached worksheet show signs of these numeical problems

The attached shows how to produce various calculations/graphs (indluding derivative values) from the solution of your PDE. The final execution group will generate a matrix of values which are then exported to the EXCEL file   pdeDat.xlsx. In order to produce your own EXCEL file you will have to change the pathname "C:/Users/TomLeslie/Desktop/pdeDat.xlsx" in the ExcelTools:-Export()  command to something appropriate for your machine

> 

  restart:
> 

  ra:=2: b1:=1.41: na:=0.7: we:=0.5: eta[1]:=4*0.1: d:=0.5/1:
  xi:=0.1: m:=na: ea:=0.5: pr:=21: gr:=0.1: R:=0.9323556933:
> 

  PDE1:=ra*(diff(f(x,t),t))=+b1*(1+ea*cos(t))+(1/(R^2))*((diff(f(x,t),x,x))+(1/x)*diff(f(x,t),x));
  IBC:= {D[1](f)(0,t)=0,f(1,t)=0,f(x,0)=0};

2*(diff(f(x, t), t)) = 1.41+.705*cos(t)+1.150367877*(diff(diff(f(x, t), x), x))+1.150367877*(diff(f(x, t), x))/x

  

{f(1, t) = 0, f(x, 0) = 0, (D[1](f))(0, t) = 0}

 (1)
> 

#
# Solve the PDE then use the returned methods 'plot', 'plot3d',
# and 'animate' to produce various plots
#
  sol :=  pdsolve({PDE1}, IBC, numeric) :
  sol:-plot(f(x, t), t = 1.2, linestyle = "solid", title = "Velocity Profile", labels = ["r", "f"]);
  sol:-plot3d(f(x, t), x=0..1, t=0..2, style=surface, color=cyan );
  sol:-animate( f(x,t), x=0..1, t=0..2);

 

 

 
> 

#
# Use the 'value' method to facilitate computation of
# assorted numerical values
#
# Check which quantities are "immediately available"
#
  sol:-value(f(x,t), output=listprocedure);
#
# Aow get some numerical values from this basic method
#
  sol:-value(f(x,t))(0.5,0.5);
#
# For ease of use, one can assign the 'f(x,t)' procedure to,
# the name 'fN' and then compute the values, as in
#
  fN:=eval( f(x,t), sol:-value(f(x,t), output=listprocedure)):
  fN(0.5,0.5);

[x = proc () option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; evalf(args[1]) end proc, t = proc () option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; evalf(args[2]) end proc, f(x, t) = proc () local tv, xv, solnproc, stype, ndsol, vals; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; Digits := trunc(evalhf(Digits)); solnproc := proc (tv, xv) local INFO, errest, nd, dvars, dary, daryt, daryx, vals, msg, i, j; option `Copyright (c) 2001 by Waterloo Maple Inc. All rights reserved.`; table( [( "soln_procedures" ) = array( 1 .. 1, [( 1 ) = (36893488148122041876) ] ) ] ) INFO := table( [( "banded" ) = true, ( "solspace" ) = Vector(21, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = 1.0}, datatype = float[8]), ( "intspace" ) = Matrix(21, 1, {(1, 1) = .0, (2, 1) = .0, (3, 1) = .0, (4, 1) = .0, (5, 1) = .0, (6, 1) = .0, (7, 1) = .0, (8, 1) = .0, (9, 1) = .0, (10, 1) = .0, (11, 1) = .0, (12, 1) = .0, (13, 1) = .0, (14, 1) = .0, (15, 1) = .0, (16, 1) = .0, (17, 1) = .0, (18, 1) = .0, (19, 1) = .0, (20, 1) = .0, (21, 1) = .0}, datatype = float[8], order = C_order), ( "dependson" ) = [{1}], ( "solmat_ne" ) = 0, ( "eqnords" ) = [[2, 1]], ( "t0" ) = 0, ( "stages" ) = 1, ( "startup_only" ) = false, ( "solmat_i2" ) = 0, ( "periodic" ) = false, ( "autonomous" ) = true, ( "adjusted" ) = false, ( "pts", x ) = [0, 1], ( "method" ) = theta, ( "maxords" ) = [2, 1], ( "timeidx" ) = 2, ( "eqndep" ) = [1], ( "inputargs" ) = [{2*(diff(f(x, t), t)) = 1.41+.705*cos(t)+1.150367877*(diff(diff(f(x, t), x), x))+1.150367877*(diff(f(x, t), x))/x}, {f(1, t) = 0, f(x, 0) = 0, (D[1](f))(0, t) = 0}], ( "solmat_i1" ) = 0, ( "extrabcs" ) = [0], ( "matrixhf" ) = true, ( "PDEs" ) = [2*(diff(f(x, t), t))-141/100-(141/200)*cos(t)-(1150367877/1000000000)*(diff(diff(f(x, t), x), x))-(1150367877/1000000000)*(diff(f(x, t), x))/x], ( "soltimes" ) = Vector(3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8]), ( "solvec3" ) = Vector(21, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0}, datatype = float[8]), ( "explicit" ) = false, ( "rightwidth" ) = 0, ( "norigdepvars" ) = 1, ( "minspcpoints" ) = 4, ( "totalwidth" ) = 7, ( "solmat_v" ) = Vector(147, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0, (44) = .0, (45) = .0, (46) = .0, (47) = .0, (48) = .0, (49) = .0, (50) = .0, (51) = .0, (52) = .0, (53) = .0, (54) = .0, (55) = .0, (56) = .0, (57) = .0, (58) = .0, (59) = .0, (60) = .0, (61) = .0, (62) = .0, (63) = .0, (64) = .0, (65) = .0, (66) = .0, (67) = .0, (68) = .0, (69) = .0, (70) = .0, (71) = .0, (72) = .0, (73) = .0, (74) = .0, (75) = .0, (76) = .0, (77) = .0, (78) = .0, (79) = .0, (80) = .0, (81) = .0, (82) = .0, (83) = .0, (84) = .0, (85) = .0, (86) = .0, (87) = .0, (88) = .0, (89) = .0, (90) = .0, (91) = .0, (92) = .0, (93) = .0, (94) = .0, (95) = .0, (96) = .0, (97) = .0, (98) = .0, (99) = .0, (100) = .0, (101) = .0, (102) = .0, (103) = .0, (104) = .0, (105) = .0, (106) = .0, (107) = .0, (108) = .0, (109) = .0, (110) = .0, (111) = .0, (112) = .0, (113) = .0, (114) = .0, (115) = .0, (116) = .0, (117) = .0, (118) = .0, (119) = .0, (120) = .0, (121) = .0, (122) = .0, (123) = .0, (124) = .0, (125) = .0, (126) = .0, (127) = .0, (128) = .0, (129) = .0, (130) = .0, (131) = .0, (132) = .0, (133) = .0, (134) = .0, (135) = .0, (136) = .0, (137) = .0, (138) = .0, (139) = .0, (140) = .0, (141) = .0, (142) = .0, (143) = .0, (144) = .0, (145) = .0, (146) = .0, (147) = .0}, datatype = float[8], order = C_order, attributes = [source_rtable = (Matrix(21, 7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (8, 7) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (9, 7) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (10, 7) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (11, 7) = .0, (12, 1) = .0, (12, 2) = .0, (12, 3) = .0, (12, 4) = .0, (12, 5) = .0, (12, 6) = .0, (12, 7) = .0, (13, 1) = .0, (13, 2) = .0, (13, 3) = .0, (13, 4) = .0, (13, 5) = .0, (13, 6) = .0, (13, 7) = .0, (14, 1) = .0, (14, 2) = .0, (14, 3) = .0, (14, 4) = .0, (14, 5) = .0, (14, 6) = .0, (14, 7) = .0, (15, 1) = .0, (15, 2) = .0, (15, 3) = .0, (15, 4) = .0, (15, 5) = .0, (15, 6) = .0, (15, 7) = .0, (16, 1) = .0, (16, 2) = .0, (16, 3) = .0, (16, 4) = .0, (16, 5) = .0, (16, 6) = .0, (16, 7) = .0, (17, 1) = .0, (17, 2) = .0, (17, 3) = .0, (17, 4) = .0, (17, 5) = .0, (17, 6) = .0, (17, 7) = .0, (18, 1) = .0, (18, 2) = .0, (18, 3) = .0, (18, 4) = .0, (18, 5) = .0, (18, 6) = .0, (18, 7) = .0, (19, 1) = .0, (19, 2) = .0, (19, 3) = .0, (19, 4) = .0, (19, 5) = .0, (19, 6) = .0, (19, 7) = .0, (20, 1) = .0, (20, 2) = .0, (20, 3) = .0, (20, 4) = .0, (20, 5) = .0, (20, 6) = .0, (20, 7) = .0, (21, 1) = .0, (21, 2) = .0, (21, 3) = .0, (21, 4) = .0, (21, 5) = .0, (21, 6) = .0, (21, 7) = .0}, datatype = float[8], order = C_order))]), ( "timeadaptive" ) = false, ( "allocspace" ) = 21, ( "leftwidth" ) = 1, ( "vectorhf" ) = true, ( "solmat_is" ) = 0, ( "depords" ) = [[2, 1]], ( "timei" ) = 3, ( "linear" ) = true, ( "erroraccum" ) = true, ( "spacestep" ) = 0.500000000000000e-1, ( "solvec2" ) = Vector(21, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0}, datatype = float[8]), ( "solmatrix" ) = Matrix(21, 7, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (8, 7) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (9, 7) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (10, 7) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (11, 7) = .0, (12, 1) = .0, (12, 2) = .0, (12, 3) = .0, (12, 4) = .0, (12, 5) = .0, (12, 6) = .0, (12, 7) = .0, (13, 1) = .0, (13, 2) = .0, (13, 3) = .0, (13, 4) = .0, (13, 5) = .0, (13, 6) = .0, (13, 7) = .0, (14, 1) = .0, (14, 2) = .0, (14, 3) = .0, (14, 4) = .0, (14, 5) = .0, (14, 6) = .0, (14, 7) = .0, (15, 1) = .0, (15, 2) = .0, (15, 3) = .0, (15, 4) = .0, (15, 5) = .0, (15, 6) = .0, (15, 7) = .0, (16, 1) = .0, (16, 2) = .0, (16, 3) = .0, (16, 4) = .0, (16, 5) = .0, (16, 6) = .0, (16, 7) = .0, (17, 1) = .0, (17, 2) = .0, (17, 3) = .0, (17, 4) = .0, (17, 5) = .0, (17, 6) = .0, (17, 7) = .0, (18, 1) = .0, (18, 2) = .0, (18, 3) = .0, (18, 4) = .0, (18, 5) = .0, (18, 6) = .0, (18, 7) = .0, (19, 1) = .0, (19, 2) = .0, (19, 3) = .0, (19, 4) = .0, (19, 5) = .0, (19, 6) = .0, (19, 7) = .0, (20, 1) = .0, (20, 2) = .0, (20, 3) = .0, (20, 4) = .0, (20, 5) = .0, (20, 6) = .0, (20, 7) = .0, (21, 1) = .0, (21, 2) = .0, (21, 3) = .0, (21, 4) = .0, (21, 5) = .0, (21, 6) = .0, (21, 7) = .0}, datatype = float[8], order = C_order), ( "solvec5" ) = 0, ( "spacepts" ) = 21, ( "indepvars" ) = [x, t], ( "BCS", 1 ) = {[[1, 0, 1], b[1, 0, 1]], [[1, 1, 0], b[1, 1, 0]]}, ( "errorest" ) = false, ( "multidep" ) = [false, false], ( "timevar" ) = t, ( "solvec1" ) = Vector(21, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0}, datatype = float[8]), ( "fdepvars" ) = [f(x, t)], ( "spacevar" ) = x, ( "theta" ) = 1/2, ( "matrixproc" ) = proc (v, vp, vpp, t, x, k, h, n, mat) local _s1, _s2, _s3, xi; _s1 := -1150367877*h; _s2 := 4000000000*h^2; _s3 := (1/1000000000)*(2000000000*h^2+1150367877*k)/(k*h^2); mat[3] := -(3/2)/h; mat[4] := 2/h; mat[5] := -(1/2)/h; mat[7*n-4] := 1; for xi from 2 to n-1 do mat[7*xi-4] := _s3; mat[7*xi-5] := -(_s1+2300735754*x[xi])/(_s2*x[xi]); mat[7*xi-3] := (_s1-2300735754*x[xi])/(_s2*x[xi]) end do end proc, ( "depshift" ) = [1], ( "ICS" ) = [0], ( "bandwidth" ) = [1, 3], ( "depeqn" ) = [1], ( "timestep" ) = 0.500000000000000e-1, ( "vectorproc" ) = proc (v, vp, vpp, t, x, k, h, n, vec) local _s1, _s2, _s3, _s4, _s5, _s6, _s7, _s8, xi; _s1 := cos(t+(1/2)*k); _s4 := 2300735754*k; _s5 := 8000000000*h^2; _s6 := 1150367877*h*k; _s7 := 4000000000*k*h^2; _s8 := 2820000000*k*h^2*(_s1+2); vec[1] := 0; vec[n] := 0; for xi from 2 to n-1 do _s2 := -vp[xi-1]+vp[xi+1]; _s3 := vp[xi-1]-2*vp[xi]+vp[xi+1]; vec[xi] := (_s3*_s4*x[xi]+_s5*vp[xi]*x[xi]+_s2*_s6+_s8*x[xi])/(_s7*x[xi]) end do end proc, ( "solvec4" ) = 0, ( "solution" ) = Array(1..3, 1..21, 1..1, {(1, 1, 1) = .0, (1, 2, 1) = .0, (1, 3, 1) = .0, (1, 4, 1) = .0, (1, 5, 1) = .0, (1, 6, 1) = .0, (1, 7, 1) = .0, (1, 8, 1) = .0, (1, 9, 1) = .0, (1, 10, 1) = .0, (1, 11, 1) = .0, (1, 12, 1) = .0, (1, 13, 1) = .0, (1, 14, 1) = .0, (1, 15, 1) = .0, (1, 16, 1) = .0, (1, 17, 1) = .0, (1, 18, 1) = .0, (1, 19, 1) = .0, (1, 20, 1) = .0, (1, 21, 1) = .0, (2, 1, 1) = .0, (2, 2, 1) = .0, (2, 3, 1) = .0, (2, 4, 1) = .0, (2, 5, 1) = .0, (2, 6, 1) = .0, (2, 7, 1) = .0, (2, 8, 1) = .0, (2, 9, 1) = .0, (2, 10, 1) = .0, (2, 11, 1) = .0, (2, 12, 1) = .0, (2, 13, 1) = .0, (2, 14, 1) = .0, (2, 15, 1) = .0, (2, 16, 1) = .0, (2, 17, 1) = .0, (2, 18, 1) = .0, (2, 19, 1) = .0, (2, 20, 1) = .0, (2, 21, 1) = .0, (3, 1, 1) = .0, (3, 2, 1) = .0, (3, 3, 1) = .0, (3, 4, 1) = .0, (3, 5, 1) = .0, (3, 6, 1) = .0, (3, 7, 1) = .0, (3, 8, 1) = .0, (3, 9, 1) = .0, (3, 10, 1) = .0, (3, 11, 1) = .0, (3, 12, 1) = .0, (3, 13, 1) = .0, (3, 14, 1) = .0, (3, 15, 1) = .0, (3, 16, 1) = .0, (3, 17, 1) = .0, (3, 18, 1) = .0, (3, 19, 1) = .0, (3, 20, 1) = .0, (3, 21, 1) = .0}, datatype = float[8], order = C_order), ( "initialized" ) = false, ( "IBC" ) = b, ( "mixed" ) = false, ( "depvars" ) = [f], ( "depdords" ) = [[[2, 1]]], ( "spaceadaptive" ) = false, ( "spaceidx" ) = 1 ] ); if xv = "left" then return INFO["solspace"][1] elif xv = "right" then return INFO["solspace"][INFO["spacepts"]] elif tv = "start" then return INFO["t0"] elif not (type(tv, 'numeric') and type(xv, 'numeric')) then error "non-numeric input" end if; if xv < INFO["solspace"][1] or INFO["solspace"][INFO["spacepts"]] < xv then error "requested %1 value must be in the range %2..%3", INFO["spacevar"], INFO["solspace"][1], INFO["solspace"][INFO["spacepts"]] end if; dary := Vector(3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8]); daryt := 0; daryx := 0; dvars := [proc (t, x, u) u[1] end proc]; errest := false; nd := nops(INFO["depvars"]); if dary[nd+1] <> tv then try `pdsolve/numeric/evolve_solution`(INFO, tv) catch: msg := StringTools:-FormatMessage(lastexception[2 .. -1]); if tv < INFO["t0"] then error cat("unable to compute solution for %1<%2: ", msg), INFO["timevar"], INFO["failtime"] else error cat("unable to compute solution for %1>%2: ", msg), INFO["timevar"], INFO["failtime"] end if end try end if; if dary[nd+1] <> tv or dary[nd+2] <> xv then `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["solspace"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, dary); if errest then `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["err_t"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, daryt); `pdsolve/interp2dto0d`(3, INFO["soltimes"], INFO["spacepts"], INFO["err_x"], nops(INFO["depvars"]), INFO["solution"], true, tv, xv, daryx) end if end if; dary[nd+1] := tv; dary[nd+2] := xv; if dvars = [] then [seq(dary[i], i = 1 .. INFO["norigdepvars"])] else vals := NULL; for i to nops(dvars) do j := eval(dvars[i]); try if errest then vals := vals, evalhf(j(tv, xv, dary, daryt, daryx)) else vals := vals, evalhf(j(tv, xv, dary)) end if catch: userinfo(5, `pdsolve/numeric`, `evalhf failure`); try if errest then vals := vals, j(tv, xv, dary, daryt, daryx) else vals := vals, j(tv, xv, dary) end if catch: vals := vals, undefined end try end try end do; [vals] end if end proc; stype := "2nd"; if nargs = 1 then if args[1] = "left" then return solnproc(0, "left") elif args[1] = "right" then return solnproc(0, "right") elif args[1] = "start" then return solnproc("start", 0) else error "too few arguments to solution procedure" end if elif nargs = 2 then if stype = "1st" then tv := evalf(args[1]); xv := evalf(args[2]) else tv := evalf(args[2]); xv := evalf(args[1]) end if; if not (type(tv, 'numeric') and type(xv, 'numeric')) then if procname <> unknown then return ('procname')(args[1 .. nargs]) else ndsol := pointto(solnproc("soln_procedures")[1]); return ('ndsol')(args[1 .. nargs]) end if end if else error "incorrect arguments to solution procedure" end if; vals := solnproc(tv, xv); vals[1] end proc]

  

[x = .5, t = .5, f(x, t) = .274730832428660143]

  

.274730832428660143

 (2)
> 

#
# The solution module returned by pdsolve() does not
# contain any derivatives, so these have to be computed
# explicitly. The simplest method is to use the 'D'
# operator.
#
# Differentiation wrt 'x' evaluated at x=0.2, t=1.2
#
  D[1](fN)(0.2, 1.2);
#
# Differentiation twice wrt 'x' and evaluate at x=0.2,
# t=1.2
#
  D[1,1](fN)(0.2, 1.2);
#
# Differentiation wrt 't' evaluated at x=0.2, t=1.2
#
  D[2](fN)(0.2, 1.2);
#
# Plot the first and second derivatives of f(x,t) wrt 'x' for t=1.2
# Note the "glitch" in the second derivative
#
  plot( [ D[1](fN)(x, 1.2),
          D[1, 1](fN)(x, 1.2)
        ],
        x=0..1,
        color=[red, blue]
      );
#
# Plot the first and second derivatives of f(x,t) wrt 't' for x=0.5
#
  plot( [ D[2](fN)(0.5, t),
          D[2, 2](fN)(0.5, t)
        ],
        t=0..2,
        color=[red, blue],
        axes=boxed
      );
  

-.1590019759

  

-1.149981230

  

-0.8026099339e-1

  

 

 
> 

  M:= Matrix([ [ "x", "f(x,t)", "diff(f(x,t),x)", "diff(f(x,t),x,x)"],
                  seq( [j, fN(j, 1.2), D[1](fN)(j,1.2), D[1,1](fN)(j,1.2)], j=0.1..0.9, 0.1)
               ]
            );
  ExcelTools:-Export( M, "C:/Users/TomLeslie/Desktop/pdeDat.xlsx")

Matrix(10, 4, {(1, 1) = "x", (1, 2) = "f(x,t)", (1, 3) = "diff(f(x,t),x)", (1, 4) = "diff(f(x,t),x,x)", (2, 1) = .1, (2, 2) = .386450395099301292, (2, 3) = -0.7957793238e-1, (2, 4) = -1.096653890, (3, 1) = .2, (3, 2) = .374519447545877127, (3, 3) = -.1590019759, (3, 4) = -1.149981230, (4, 1) = .3, (4, 2) = .354660645957600662, (4, 3) = -.2381063559, (4, 4) = -1.188912300, (5, 1) = .4, (5, 2) = .326914868358544664, (5, 3) = -.3167095051, (5, 4) = -1.254515940, (6, 1) = .5, (6, 2) = .291342358954074454, (6, 3) = -.3946031424, (6, 4) = -1.361681320, (7, 1) = .6, (7, 2) = .248026334799754833, (7, 3) = -.4715328601, (7, 4) = -1.217418660, (8, 1) = .7, (8, 2) = .197076684864131463, (8, 3) = -.5472420907, (8, 4) = -1.897191300, (9, 1) = .8, (9, 2) = .138628396303586865, (9, 3) = -.6213846287, (9, 4) = -1.192826120, (10, 1) = .9, (10, 2) = 0.728807948292487240e-1, (10, 3) = -.6937862569, (10, 4) = -50.76329468})

 (3)
> 

 

Download pdeTut.mw

Hmmm...

tomleslie 13876
April 30 2021
1 1

If I "make up" some initial conditions (you didn't provide any!) then I can obtain a numeric solution. However having tried a few different combinations of initial conditions, the numerical solution produces a singuarity for quite small values of the independent variable 't' - generally in the range 2e-03..3e-03.

Several possibilities

  1. The singularity *may* disappear when given "correct" initial conditions, staher than thos I have guessed
  2. There may be a typo somewhere either in your list of parameters or the definition of the ODE
  3. The singularity may be real

See the attached worksheet for typical output for one set of initial conditions I tried

(For future reference it is generally a better idea to upload worksheets using the big green up-arrow in the Mapleprimes toolbar)

> 

  restart;
  Parameterliste__5:=[ A=4071.5*10^(-6),
                     a__ks=-20.94481*10^(-3),
                     a__kos=14.063837*10^(-3),
                     a__sss=0,
                     l__k=100*10^(-3),
                     m__k=11.02736,
                     m__ko=0.7237868,
                     m__p=1.437419,
                     m__ss=15,
                     u=1
                  ];
  ode:= diff(phi(t),t)^2*cos(phi(t))*a__ks*m__k
        +
        cos(phi(t))*diff(phi(t),t)^2*a__sss*m__ss
        +
        diff(phi(t),t)^2*cos(phi(t))*l__k*m__ko
        +
        diff(phi(t),t)^2*cos(phi(t))*l__k*m__p
        +
        diff(diff(phi(t),t),t)*sin(phi(t))*a__ks*m__k
        +
        diff(diff(phi(t),t),t)*sin(phi(t))*a__sss*m__ss
        +
        diff(diff(phi(t),t),t)*sin(phi(t))*l__k*m__ko
        +
        diff(diff(phi(t),t),t)*sin(phi(t))*l__k*m__p
        +
        5000000*A*exp(-4/961*(100*Pi*trunc(1/4*(phi(t)+Pi)/Pi)-25*phi(t)+9)^2)*u+200000*u*A = 0;

[A = 0.4071500000e-2, a__ks = -0.2094481000e-1, a__kos = 0.1406383700e-1, a__sss = 0, l__k = 1/10, m__k = 11.02736, m__ko = .7237868, m__p = 1.437419, m__ss = 15, u = 1]

  

(diff(phi(t), t))^2*cos(phi(t))*a__ks*m__k+cos(phi(t))*(diff(phi(t), t))^2*a__sss*m__ss+(diff(phi(t), t))^2*cos(phi(t))*l__k*m__ko+(diff(phi(t), t))^2*cos(phi(t))*l__k*m__p+(diff(diff(phi(t), t), t))*sin(phi(t))*a__ks*m__k+(diff(diff(phi(t), t), t))*sin(phi(t))*a__sss*m__ss+(diff(diff(phi(t), t), t))*sin(phi(t))*l__k*m__ko+(diff(diff(phi(t), t), t))*sin(phi(t))*l__k*m__p+5000000*A*exp(-(4/961)*(100*Pi*trunc((1/4)*(phi(t)+Pi)/Pi)-25*phi(t)+9)^2)*u+200000*u*A = 0

 (1)
> 

#
# Evaluate the ode given the lit of parameters
#
  sys:= eval( ode, Parameterliste__5);
#
# Make up a couple of initial conditions (because
# the OP hasn't supplied any
#
  ics:= phi(0)=1, D(phi)(0)=0;
#
# Solve system
#
  sol:=dsolve( [sys, ics], numeric);
#
# Plot the solution.
#
# This will complain of a singularity somewhere
# around 3.5e-03 - exact value varies a little
# depending on the the choice of initial conditions.
#
  plots:-odeplot( sol, [t, phi(t)], t=0..0.35e-02);

-0.1484538000e-1*(diff(phi(t), t))^2*cos(phi(t))+814.3000000-0.1484538000e-1*(diff(diff(phi(t), t), t))*sin(phi(t))+20357.50000*exp(-(4/961)*(100*Pi*trunc((1/4)*(phi(t)+Pi)/Pi)-25*phi(t)+9)^2) = 0

  

phi(0) = 1, (D(phi))(0) = 0

  

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..65, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0, (64) = -1, (65) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.8054878798607347e-8, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..2, {(1) = 1.0, (2) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..2, {(1) = 1.0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = 626658.5608605637}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = phi(t), Y[2] = diff(phi(t),t)]`; YP[2] := -67.361024103121644579*(-814.3000000+0.1484538000e-1*Y[2]^2*cos(Y[1])-20357.50000*exp(-(4/961)*(314.159265358979*trunc(0.795774715459478e-1*Y[1]+.250000000000000)-25*Y[1]+9)^2))/sin(Y[1]); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = phi(t), Y[2] = diff(phi(t),t)]`; YP[2] := -67.361024103121644579*(-814.3000000+0.1484538000e-1*Y[2]^2*cos(Y[1])-20357.50000*exp(-(4/961)*(314.159265358979*trunc(0.795774715459478e-1*Y[1]+.250000000000000)-25*Y[1]+9)^2))/sin(Y[1]); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0) ] )) ] ); _y0 := Array(0..2, {(1) = 0., (2) = 1.}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2) ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch: end try else try procname(procname("right")) catch: end try end if end if end if; return elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch: end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch: end try end if end if end if; return elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [t, phi(t), diff(phi(t), t)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

  

Warning, cannot evaluate the solution further right of .34712926e-2, probably a singularity

  

 
> 

 

Download odeProb.mw

How I do it...

tomleslie 13876
April 28 2021
3 2
  1. I use Notepad++ (with a Maple customization file) to generated .mpl files
  2. I've never figured out a way to fire up Maple from with Notepad++
  3. So all I do is wrtie the file in Notepad++ then use a "read" statement within an open Maple instance
  4. If I'm being really fussy I may include a maplemint() call

So, for example if I want to check out the 'mpl' file heur1.mpl, stored in D:/Users/TomLeslie/myMaple/SuDoku/SudokuModule/heur1.mpl which I create using Notepad++, then I would just have a Maple worksheet with the following commands

read  "D:/Users/TomLeslie/myMaple/SuDoku/SudokuModule/heur1.mpl":
maplemint(heur1);

 

Having tried some experiments...

tomleslie 13876
April 28 2021
0 0

I don't think that you are going to save much time without a compromise in the quality of the animation. The code you provide works on a grid of 201 x 201 and produces 200 frames in the animation. This means that the argument() function is being computed 200*201*201=40,240,200 times 40 million is quite a lot of "arctangents" to evaluate!

  1. The most obvious way to speed things up is to reduce the grid used for plotting (49 x 49 is the default), as well as the number of frames. In the attached I have used a 49 x 49 grid and 100 frames. You can adjust these two values easily. This saves a lot of time. Only you can tell if the "quality of the resulting image is "good enough"
  2. I have also simplified the code being executed somewhat. This eliminates some function calls. Also, since one knows that the variables 'x', 'y', and 'a' are all floats, one can write the arctan() function explicitly bypassing some of the argument-checking done by the argument() function () This saved a little time, but probably wasn't worth the effort.
  restart;
  with(plots):
  nFrames:=100:
  gr:=[49, 49]:
  animate
  ( densityplot,
    [ arctan
      ( x*(a*x^2 + a*y^2 - a - 200*y - 100),
        a*y^3 + (a - 100)*y^2 + (a*x^2 + a - 100)*y - 100 + (a + 100)*x^2
      ),
      x=-4 .. 4,
      y=-4 .. 4,
      axes = boxed,
      view = [-2 .. 2, -2 .. 2],
      restricttoranges,
      colorstyle = HUE,
      style = surface,
      grid = gr
    ],
    a = 1 .. 200,
    frames=nFrames
);

which produces

Download phplt.mw

 

 

 

From the Maple help...

tomleslie 13876
April 28 2021
2 0

at ?Procedure Parameter Declarations

one finds

The End of Parameters Marker, $
• Usually, if there are more arguments in a function call than needed to match the called procedure's parameters, the remaining arguments are passed anyway, and are available via the special variables _rest and _passed. This allows procedures to be written to accept an arbitrary number of arguments of no particular type.
• If the parameter declaration sequence ends with the special parameter, $, such extra arguments are not allowed. An exception is raised if any arguments remain after all the parameters have been matched to arguments or default values. For example:
 

 

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