hello agian i have the following problem. i need to fit data using the following model:

ode_sub := diff(S(t), t) = -k1*S(t)-S(t)/T1_s;

ode_P1 := diff(P1(t), t) = 2*k1*S(t)-k2*(P1(t)-P2(t)/keq)-P1(t)/T1_p1;

ode_P2 := diff(P2(t), t) = -k2*(-keq*P1(t)+P2(t))/keq-k4*P2(t)-P2(t)/T1_p2;

ode_P2e := diff(P2_e(t), t) = k4*P2(t)-P2_e(t)/T1_p2_e;

ode_system := ode_sub, ode_P1, ode_P2, ode_P2e

known paramters: s0 := 10000; k2 := 1000; T1_s := 14; T1_p2_e := 35; T1_p2 := T1_p1

initial conditions: init := S(0) = s0, P1(0) = 0, P2(0) = 0, P2_e(0) = 0

using the following data the fitting is fine:

T := [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100]

data_s := [10000.00001377746, 7880.718836217512, 6210.572917625112, 4894.377814704496, 3857.121616806618, 3039.689036293312, 2395.493468824973, 1887.821030424934, 1487.738721378254, 1172.445015238064, 923.970970533911, 728.1555148262845, 573.8389058920359, 452.2263410725434, 356.3868133709972, 280.8584641247961, 221.3366863178145, 174.4291648589732, 137.4627967029932, 108.3305246342751, 85.37230370803576, 67.2794974831867, 53.0210755540487, 41.78436556408739, 32.92915589156605, 25.95049906181449, 20.45092590987601, 16.11678944747619, 12.70115104646253, 10.009439492356, 7.888058666357939, 6.216464754698855, 4.899036441885205, 3.860793948052506, 3.042584031379331, 2.397778731385364, 1.889601342133927, 1.489145259940784, 1.173591417634974, .9248694992255094, .7288443588090404, .5743879613705721, .4526024635132348, .3567687570029392, .2810508404011898, .2215650452813882, .174603181767829, .1376243372528356, .1084232889842753, 0.8542884707952822e-1, 0.6738282660463157e-1]

data_p1 := [0.1194335334401124e-4, 244.8746046039949, 374.8721199398692, 430.5392805383767, 439.6598364813143, 421.0353424914179, 387.1842556288343, 346.2646897222593, 303.4377508746471, 261.8283447091155, 223.1996547160051, 188.4213144493491, 157.7924449350029, 131.2622073983344, 108.5771635112278, 89.37951009190863, 73.26979150957087, 59.84578653950572, 48.72563358658898, 39.56010490461378, 32.03855466968536, 25.88922670933322, 20.87834763772145, 16.80708274458702, 13.50774122768974, 10.84014148654258, 8.687656394505874, 6.954093898245485, 5.560224107929433, 4.441209458524726, 3.544128529596104, 2.825755811619965, 2.251247757181308, 1.792233305651086, 1.425861347838012, 1.133566009019768, .90081361320016, .7153496336919163, .5678861241754847, .4505952916932289, .3572989037753538, .2832489239941939, .2244289868248577, .1778450590752305, .1408633578784151, .1114667192753896, 0.8826814044702111e-1, 0.6979315954603076e-1, 0.5526502783606788e-1, 0.4370354298880999e-1, 0.3456334307662573e-1]

data_p2_p2e := [-0.1821397630630296e-4, 1000.40572909871, 1568.064904416198, 1848.900129881268, 1944.147939710225, 1923.352973299286, 1833.705314342611, 1706.726235937363, 1563.036042115902, 1415.741121363331, 1272.825952816517, 1138.833091575137, 1016.03557293539, 905.2470623752856, 806.3754051707843, 718.7979384094563, 641.6091822001032, 573.7825966275556, 514.2718125966452, 462.0710416682647, 416.2491499591012, 375.9656328260581, 340.4748518743264, 309.124348579787, 281.3484612079911, 256.6599441255977, 234.6416876403397, 214.9378461256197, 197.2453333305823, 181.3059798685686, 166.90059218783, 153.8422174795751, 141.9717783871606, 131.1529759058513, 121.2692959115991, 112.2199678247825, 103.9187616370328, 96.29007054510998, 89.26877137303566, 82.79743967408479, 76.82586273439793, 71.30944943617081, 66.2085909515396, 61.48838150744805, 57.11714763242225, 53.06666224006544, 49.31130738219119, 45.82807853990728, 42.59597194910467, 39.59575450632008, 36.81013335261527]

the fitting is done the following way:

P1fu, P2fu, P2e_fu, Sfu := op(subs(res, [P1(t), P2(t), P2_e(t), S(t)]))

making residuals:

Q := proc (T1_p1, k1, keq, k4) local P1v, P2v, P2e_v, Sv, resid; option remember;

res(parameters = [T1_p1, k1, keq, k4]);

try P1v := `~`[P1fu](T); P2v := `~`[P2fu](T); P2e_v := `~`[P2e_fu](T); Sv := `~`[Sfu](T); resid := [P1v-data_p1, P2v+P2e_v-data_p2_p2e, Sv-data_s]; return [seq(seq(resid[i][j], i = 1 .. 3), j = 1 .. nops(T))] catch: return [1000000$3*nops(T)] end try end proc;

q := [seq(subs(_nn = n, proc (T1_p1, k1, keq, k4) Q(args)[_nn] end proc), n = 1 .. 3*nops(T))];

finding inital point for the LSsolve:

L := fsolve(q[2 .. 5], [10, 0.2e-1, 4, 4])

fitting the data with the intial point:

solLS := Optimization:-LSSolve(q, initialpoint = L)

this is all good however, when i used the following data it did not turn out so well (using the same approch as above):

data_s := [96304.74567, 77385.03700, 62621.83067, 51239.94333, 42663.82367, 35084.74100, 28480.28367, 23066.01467, 18774.73700, 15179.13700, 12278.50767, 9937.652000, 8046.848333, 6521.242000, 5287.811667, 4277.779000, 3466.518333, 2835.467000, 2297.796333, 1861.249667, 1529.654000, 1235.353000, 999.6626667, 826.2343333, 667.9480000, 559.9230000, 449.2790000, 376.4860000, 289.1203333, 236.1483333]

data_p1 := [0.86e-1, 3.904, 26.975, 31.719, 41.067, 46.779, 52.115, 43.101, 44.344, 41.094, 36.523, 27.742, 26.543, 28.062, 22.178, 21.303, 14.951, 17.871, 11.422, 12.051, 9.232, 6.817, 6.1, .717, 1.215, 6.146, .772, .375, 2.595, .518]

data_p2_p2e := [-3.024, 22.238, 61.731, 103.816, 132.695, 159.069, 167.302, 160.188, 158.398, 152.943, 146.745, 135.22, 132.145, 120.413, 107.864, 95.339, 90.775, 81.828, 71.065, 70.475, 62.872, 49.955, 40.858, 42.938, 41.311, 35.583, 31.573, 29.841, 29.558, 21.762]

the known parameters is the case are:

s0 := 96304.74567; k2 := 10^5; T1_s := 14; T1_p2_e := 35; T1_p2 := T1_p1

additionally the following fuction affects the solution:

K:=t->cos((1/180)*beta*Pi)^(t/Tr)

i included this by doing this:

P1fu_K := proc (t) options operator, arrow; P1fu(t)*K(t) end proc;

P2fu_K := proc (t) options operator, arrow; P2fu(t)*K(t) end proc;

P2e_fu_K := proc (t) options operator, arrow;

P2e_fu(t)*K(t) end proc;

Sfu_K := proc (t) options operator, arrow; Sfu(t)*K(t) end proc

resulting in the following residuals:

Q := proc (T1_p1, k1, keq, k4) local P1v, P2v, P2e_v, Sv, resid; option remember;

res(parameters = [T1_p1, k1, keq, k4]);

try P1v := `~`[P1fu_K](T); P2v := `~`[P2fu_K](T); P2e_v := `~`[P2e_fu_K](T); Sv := `~`[Sfu_K](T); resid := [P1v-data_p1, P2v+P2e_v-data_p2_p2e, Sv-data_s]; return [seq(seq(resid[i][j], i = 1 .. 3), j = 1 .. nops(T))] catch: return [1000000$3*nops(T)] end try end proc;

q := [seq(subs(_nn = n, proc (T1_p1, k1, keq, k4) Q(args)[_nn] end proc), n = 1 .. 3*nops(T))];

i think my problem is that the inital point in this case is not known. all i know is that all the fitted parameters should be positive and that k1<1, k4>10, keq>1 and T1_p1>100 (more i do not know) - is there a way to determin the inital point without guessing?

i also know the results, which should be close to these values:

k1=0.000438, k4=0.0385, keq=2.7385 and T1_p1=36.8 the output fit should look something like this

where the red curve is (PP2(t)+PP2_e(t))*K(t)

the blue cure is PP1(t)*K(t)

anyone able to help - i've tried for 2 days now. it might be that ode_P1 := diff(P1(t), t) = 2*k1*S(t)-k2*(P1(t)-P2(t)/keq)-P1(t)/T1_p1 should be changed into ode_P1 := diff(P1(t), t) = k1*S(t)-k2*(P1(t)-P2(t)/keq)-P1(t)/T1_p1;

i've tried this but it didnt seem to do much

anyone able to help?:)

NB stiff=true can be used within the dsolve to speed up the process if needed:)