Wozu ist Mathe gut?

©2015 Gerald Teschl (http://www.mat.univie.ac.at/~gerald/) und Susanne Teschl  (http://staff.technikum-wien.at/~teschl/)

Dieses Notebook ist als Ergänzung zu unserem Skriptum gedacht. Dort finden Sie auch weitere Erklärungen.

Geheimschriften knacken für Anfänger

Cäsarverschiebung

x=ToCharacterCode["DIESPINNENDIEGALLIER"]-65

y=Mod[x+3,26]

FromCharacterCode[%+65]

Mod[y + 23,26]

Caesar[x_String,k_]:=FromCharacterCode[Mod[(ToCharacterCode[x]-65)+k,26]+65];

Caesar["GALLIER",3]

Clear[x,y]

Buchstabenhäufigkeiten:

LetterFrequencies[x_String]:=Module[{xx,n,i},
xx=ToCharacterCode[x]-65;
n=Table[0,{i,26}];
Do[n[[ xx[[i]] +1]]++,{i,Length[xx]}];
n=n / Length[xx] //N
];

LetterFrequencies["INJXJWYJCYNXYSNHMYRJMWLJMJNR"]

Übersichtlichere Ausgabe dieser Routine:

{Table[FromCharacterCode[i+65],{i,0,25}],100 LetterFrequencies["INJXJWYJCYNXYSNHMYRJMWLJMJNR"]} //Transpose//TableForm

Kryptoanalyse der Cäsarverschiebung

CaesarCrack berechnet den Schlüssel fuer einen Geheimtext, der durch Caesarverschiebung eines deutschen Klartextes entstand;

Die Buchstabenhäufigkeiten wurden https://de.wikipedia.org/wiki/Buchstabenhäufigkeit entnommen.

CaesarCrack[x_String]:=Module[{lf,d0=2,i0=0},
lfg={ 6.51, 1.89,3.06,5.08, 17.40,1.66,3.01,4.76,7.55,0.27,1.21,3.33,2.53,9.78, 2.51,0.79, 0.02, 7.00,7.27,6.15,4.35,0.67,1.89, 0.03,0.04, 1.13}/100;
lf=LetterFrequencies[x];
Do[d=RotateLeft[lf,i]-lfg; d=d.d;
If[d<d0,d0=d;i0=i],
{i,0,25}];
i0
]

Beispiel:

geheimtext="PUQMZFIADFMGRPUQRDMSQZMOTPQYXQNQZPQYGZUHQDEGYGZPPQYSMZLQZDQEFUEFLIQUGZPHUQDLUS";

CaesarCrack[geheimtext]

Caesar[geheimtext,-12]

Vergleich der Buchstabenhäfigkeiten im Geheimtext mit den deutschen Buchstabenhäfigkeiten weist auf die monoalphabetische Verschlüsselung hin:

ListPlot[{Sort[LetterFrequencies[geheimtext],Greater],Sort[lfg,Greater]}]

Ebenso der Vergleich der Friedman-Charakteristik:

{((Norm[lfg])^2),((Norm[LetterFrequencies[geheimtext]])^2)}

Verschlüsselung mit Xor

(* nur für Mathematica v3.x *)
BitXor[x_Integer,y_Integer]:=Module[{xx,yy,d},xx=IntegerDigits[x,2];
    yy=IntegerDigits[y,2];
    d=Length[xx]-Length[yy];
    If[d>0,yy=Join[Table[0,{i,d}],yy],xx=Join[Table[0,{i,-d}],xx]];
    FromDigits[Mod[xx+yy,2],2]]

klartext = ToCharacterCode["Streng geheim!"]

geheimtext = BitXor[klartext, 127]

Vigenere-Verschlüsselung

Routine zur Vigenere-Verschlüsselung:

Vigenere[x_String,e_String]:= Module[{xx, ee, k, yy= {}},
xx=ToCharacterCode[x]-65;
ee=ToCharacterCode[e]-65;
k= Length[ee];
Do[
yy=Append[yy,Mod[xx[[i]] +ee[[Mod[i-1,k]+1]],26]],
{i,Length[xx]}];
FromCharacterCode[yy+65]
]

Vigenere-Verschlüsselung mit dem Schlüsselwort "VIGENERE":

Vigenere["DIESERTEXTISTGEHEIM","VIGENERE"]

Berechnung des Schlüssels zum Entschlüsseln:

FromCharacterCode[Mod[-(ToCharacterCode["VIGENERE"]-65),26]+65]

Entschlüsselung:

Vigenere["YQKWRVKISBOWGKVLZQS","FSUWNWJW"]

Koinzidenzindex

Der Koinzidenzindex (Friedman-Charakteristik) eines Texts ist die Summe der quadratischen Häufigkeiten.

FriedmanCharacteristic[geheimtext_String]:=((Norm[LetterFrequencies[geheimtext]])^2);
SetAttributes[FriedmanCharacteristic,Listable];

Für einen deutschen Text erhält man:

hd={ 6.51, 1.89,3.06,5.08, 17.40,1.66,3.01,4.76,7.55,0.27,1.21,3.33,2.53,9.78, 2.51,0.79, 0.02, 7.00,7.27,6.15,4.35,0.67,1.89, 0.03,0.04, 1.13}/100;
Kd=((Norm[hd])^2)

Die Friedman-Charakteristik eines Texts bei dem alle Buchstaben mit der gleichen Häufigkeit auftreten ist

N[1/26]

Bei einer monoalphabetischen Verschlüsselung werden die Buchstabenhäufigkeiten im Vergleich zum Klartext nur permutiert und somit haben Geheimtext und Klartext die gleiche Friedman-Charakteristik. Ein Abweichen der Friedman-Charakteristik vom Wert eines deutschen Texts zeigt, dass nicht monoalphabetisch verschlüsselt wurde.

Kryptoanalyse der Vigenere-Verschlüsselung

geheimtext= "LWAGZRUHUWVLMVRRWTOLFVZIAOQYVZTLQNHMEQMKAEIIPTAVMZFSLVUYYMFSYENDRKVIZGDIICOEXUXMAJVRJHLQKWXEZGYGUJLJLQPGWTOALPHPVNBLLUDCHORBRRMTOIEYVPTRQVLLSKXUKERHXLLBVUKMMRQAYUTVEYYMFSYENHZHYFMVUIISOIHYETFNFRSDQXLVAGCIEOECPBRLQZREKEEKHPBJNRRJHMHYJWYFIEKSKXYWMVMDZLLVUWECFDRCVSSLZYVGDRWPECXUHPVDVMLRCRYIZQTVTOSVLUWBEDFVUACVSIBMSIJUUEDBGPANGYKAJOLFMAZRULMVUNEVMAIJVNUHYWPVMKULRSOLGPFBLRKEEZHVVHQKVYIEJ";

Zerlegt man einen Geheimtext in k Teile und liegt der Koinzidenzindex aller Teile nahe am Wert eines deutschen Texts, so ist das ein Hinweis, dass die Teile monoalphabetisch verschlüsselt wurden. Dazu betrachten wir den mittlere Koinzidenzindex bei Zerlegung des Geheimtext in k Teile für von k = 1  bis 35 :

Routine, um einen Text in k Teiltexte aufzuteilen:

SplitText[x_String,k_]:=Module[{i,t},
t=Table["",{i,k}];
Do[t[[Mod[i-1,k]+1]]= t[[Mod[i-1,k]+1]] <> StringTake[x,{i}],{i,StringLength[x]}];
t
]

Routine zur Berechnung des Koinzidenzindex der Texteile:

FriedmanCharacteristic[geheimtext_String,k_]:=Mean[FriedmanCharacteristic[SplitText[geheimtext,k]]];

Anwendung auf unseren Geheimtext:

Table[FriedmanCharacteristic[geheimtext,m],{m,1,35}]

Graphische Darstellung: Verschiebung k ...x-Achse, Koinzidenzindex ...y-Achse

ListPlot[%]

Der Anstieg ergibt sich da die einzelnen Texte immer kürzer werden (ab 16 Teiltexten gibt es z.B. schon weniger als 26 Buchstaben und immer mehr Häufigkeiten müssen Null sein, was den Wert nach oben treibt).

Aufteilung des Geheimtexts in 11 Teiltexte:

SplitText[geheimtext,11]

Jeder der Teil-Geheimtexte ist mit einer Cäsarverschiebung verschlüsselt. Für jeden wird mit CaesarCrack der zugehörige Schlüssel berechnet:

CaesarCrack /@ SplitText[geheimtext,11]

Damit lautet das Schlüsselwort für die Vigenere-Verschiebung:

FromCharacterCode[%+65]

Zugehöriges Schlüsselwort für die Entschlüsselung:

FromCharacterCode[Mod[-(ToCharacterCode[%]-65),26]+65]

Entschlüsselung des Geheimtexts:

Vigenere[geheimtext,"GWSNBWJTAJX"]

Ist noch nicht ganz sinnvoll. Erster Buchstabe des Klartexts sollte wohl ein  E sein. Damit L in E verschlüsselt wird, muss  der zugehörige Schlüsselwortbuchstabe H sein, das Schlüsselwort daher HEINZERHARD:

FromCharacterCode[Mod[-(ToCharacterCode["HEINZERHARD"]-65),26]+65]

Entschlüsselung damit:

Vigenere[geheimtext,"TWSNBWJTAJX"]

Bei dem Text handelt es sich um das Gedicht "`Ritter Fips"' von Heinz Erhardt. Mit freundlicher Genehmigung des Lappan Verlags, http://www.lappan.de

Vom Lotfällen zur Bildkompression

Die diskrete Kosinustransformation

Eine Animation der DCT

DCT[j_,k_]=(Sqrt[((If[k==1,1,2])/n)])Cos[(((2j-1)(k-1)π)/(2n))];

Manipulate[ListPlot[Table[DCT[j,k],{j,n}],PlotRange->{-(Sqrt[(2.1/n)]),(Sqrt[(2.1/n)])}],{k,1,n,1},Initialization:>{n=12;}]

Eine Animation der zwei-dimensionalen DCT

Manipulate[ArrayPlot[Table[DCT[j1,k1]DCT[j2,k2],{j1,n},{j2,n}],ImageSize->250,PlotRange->{-(2/n),(2/n)},Frame->False],{k1,1,n,1},{k2,1,n,1},Initialization:>{n=12;}]

Zur Berechnung der DCT kann man in Mathematica einfach FourierDCT[ ] und für die inverse DCT FourierDCT[ ,3] verwenden.

JPEG

Unsere Bildkompression erfolgt durch wegwerfen von Koeffizienten mittels folgender Funktion:

Manipulate[MatrixForm[truncate[Table[1,{10},{10}],p]],{p,0,99},Initialization:>{truncate[data_,p_]:=
Module[{i,j},
{i,j}=Map[Max[#,1]&,Floor[Dimensions[data](Sqrt[((100-p)/100)])]];
PadRight[Take[data,i,j],Dimensions[data],0]
];}]

Wir nehmen ein einfaches Graustufenbild aus der eingebauten Bibliothek:

img=255-ExampleData[{"TestImage","Boat"},"Data"];
ArrayPlot[img,ImageSize->200,Frame->False]

Jetzt nehmen wir unser Bild, berechnen die DCT und werfen einen Teil der Koeffizienten wie oben beschrieben weg. Das resultierende Bild wird zusammen mit dem Original dargestellt.

Manipulate[Grid[{Map[Style[#,"Label"]&,{"original",StringJoin[ToString[p],"% compressed"]}],{oplot,ArrayPlot[Map[Max[0,Min[#,255]]&,FourierDCT[truncate[dct,p],3],{2}],ImageSize->400,PlotRange->{0,255},Frame->False]}}],{{p,0,"compression percentage"},0,99},Initialization:>{
img=255-ExampleData[{"TestImage","Boat"},"Data"];
oplot=ArrayPlot[img,ImageSize->400,PlotRange->{0,255},Frame->False];
dct=FourierDCT[img];
truncate[data_,p_]:=
Module[{i,j},
{i,j}=Map[Max[#,1]&,Floor[Dimensions[data](Sqrt[((100-p)/100)])]];
PadRight[Take[data,i,j],Dimensions[data],0]
];}]

AlkoMat(h). Modellierung in der Atemgasanalyse

Modellgleichungen

Lungenkompartiment

Konzentration in der Aussenluft ist 0

(Subscript[C, I])[t]=0;

eq1=(Subscript[C, A])'[t]== (((Subscript[(OverDot[V]), alv])[t]*((Subscript[C, I])[t]-(Subscript[C, A])[t])+ (Subscript[(OverDot[Q]), c])[t] ((Subscript[C, mv])[t]- (Subscript[C, a])[t]  ))/(Subscript[V, A]));

Gesetz von Henry

(Subscript[C, a])[t] = (Subscript[λ, ba]) (Subscript[C, A])[t] ;

initmv=(Subscript[C, mv])[0]/.FullSimplify[Flatten[Solve[eq1/.t-> 0/.(Subscript[C, A])'[0]->0,(Subscript[C, mv])[0]]]]

Muskelkompartiment mit Produktion und linearer metabolischer Kinetik

eq2=(Subscript[C, per])'[t]==(( (Subscript[q, per])[t](Subscript[(OverDot[Q]), c])[t]((Subscript[C, a])[t]-(Subscript[λ, bper]) (Subscript[C, per])[t])+((Subscript[pr, per])-(Subscript[met, per]) (Subscript[λ, bper])(Subscript[C, per])[t]))/(Subscript[V, per]));

initper=(Subscript[C, per])[0]/.FullSimplify[Flatten[Solve[eq2/.{t-> 0,(Subscript[C, per])'[t]->0},(Subscript[C, per])[0]]]]

Kompartiment mit Leber und restlichem Körper mit Produktion und linearer metabolischer Kinetik

eq3=(Subscript[C, rpt])'[t]== (((1-(Subscript[q, per])[t])(Subscript[(OverDot[Q]), c])[t]((Subscript[C, a])[t]-(Subscript[λ, brpt]) (Subscript[C, rpt])[t])+((Subscript[pr, rpt])-(Subscript[met, rpt])(Subscript[λ, brpt]) (Subscript[C, rpt])[t]))/(Subscript[V, rpt]));

initrpt=(Subscript[C, rpt])[0]/. FullSimplify[Flatten[Solve[eq3/.{t-> 0,(Subscript[C, rpt])'[t]->0},(Subscript[C, rpt])[0]]]]

Mittlere Konzentration im venösen bzw. arteriellen Blut

(Subscript[C, mv])[t_]:=( (1-(Subscript[q, per])[t])(Subscript[λ, brpt]) (Subscript[C, rpt])[t] +   (Subscript[q, per])[t](Subscript[λ, bper]) (Subscript[C, per])[t])

Model für Blutflußverteilung

(Subscript[q, per])[t_]:= (Subscript[q, per0]) +((Subscript[q, permax])-(Subscript[q, per0])  )(1-Exp[-τ  Max[0,(((Subscript[(OverDot[Q]), c])[t]-(Subscript[(OverDot[Q]), c])[0])/((Subscript[(OverDot[Q]), c])[0]))]])

Experimentelle Daten

isopren={142.945,142.954,142.964,142.975,142.985,142.996,143.007,143.019,143.03,143.042,143.054,143.066,143.079,143.091,143.104,143.118,143.131,143.145,143.16,143.175,143.191,143.207,143.224,143.242,143.26,143.279,143.298,143.318,143.339,143.36,143.383,143.406,143.429,143.454,143.479,143.505,143.531,143.558,143.586,143.615,143.644,143.675,143.706,143.739,143.772,143.806,143.842,143.878,143.915,143.954,143.994,144.034,144.076,144.118,144.161,144.205,144.25,144.295,144.341,144.388,144.436,144.484,144.533,144.582,144.633,144.684,144.735,144.787,144.84,144.894,144.947,145.002,145.057,145.113,145.169,145.225,145.282,145.339,145.395,145.451,145.507,145.563,145.618,145.673,145.727,145.781,145.835,145.888,145.94,145.992,146.042,146.092,146.141,146.189,146.236,146.282,146.327,146.37,146.413,146.454,146.494,146.533,146.571,146.607,146.642,146.675,146.706,146.737,146.765,146.792,146.818,146.842,146.864,146.884,146.904,146.922,146.94,146.955,146.97,146.984,146.997,147.009,147.02,147.03,147.039,147.047,147.054,147.061,147.067,147.073,147.077,147.082,147.085,147.088,147.09,147.092,147.093,147.094,147.095,147.096,147.096,147.097,147.098,147.098,147.099,147.099,147.099,147.099,147.099,147.099,147.1,147.099,147.099,147.099,147.099,147.098,147.098,147.097,147.096,147.095,147.093,147.091,147.089,147.086,147.083,147.08,147.077,147.073,147.07,147.065,147.061,147.056,147.051,147.046,147.04,147.033,147.026,147.018,147.009,146.999,146.988,146.975,146.962,146.947,146.931,146.913,146.894,146.874,146.851,146.828,146.802,146.776,146.747,146.717,146.686,146.653,146.618,146.582,146.544,146.504,146.463,146.42,146.375,146.329,146.282,146.233,146.182,146.13,146.077,146.022,145.966,145.908,145.85,145.79,145.728,145.666,145.602,145.538,145.472,145.406,145.338,145.269,145.2,145.129,145.058,144.986,144.913,144.839,144.764,144.689,144.613,144.538,144.462,144.386,144.311,144.236,144.162,144.089,144.017,143.946,143.878,143.813,143.75,143.691,143.636,143.586,143.541,143.502,143.471,143.447,143.432,143.427,143.433,143.449,143.478,143.52,143.577,143.649,143.737,143.843,143.968,144.113,144.279,144.467,144.679,144.916,145.179,145.47,145.791,146.142,146.525,146.943,147.397,147.887,148.416,148.985,149.595,150.248,150.946,151.689,152.478,153.317,154.205,155.145,156.137,157.183,158.284,159.442,160.658,161.932,163.267,164.662,166.118,167.637,169.218,170.864,172.574,174.349,176.191,178.099,180.073,182.115,184.223,186.399,188.642,190.951,193.325,195.765,198.268,200.833,203.459,206.144,208.888,211.688,214.543,217.451,220.411,223.42,226.475,229.575,232.717,235.899,239.119,242.372,245.657,248.97,252.308,255.669,259.048,262.441,265.846,269.258,272.673,276.088,279.499,282.902,286.294,289.671,293.03,296.368,299.681,302.965,306.218,309.436,312.614,315.751,318.843,321.887,324.879,327.817,330.696,333.515,336.272,338.962,341.585,344.137,346.616,349.018,351.342,353.585,355.745,357.82,359.807,361.706,363.513,365.228,366.85,368.379,369.812,371.15,372.392,373.539,374.59,375.546,376.406,377.169,377.837,378.41,378.888,379.274,379.566,379.767,379.878,379.899,379.833,379.682,379.446,379.128,378.731,378.257,377.708,377.087,376.396,375.637,374.814,373.93,372.985,371.984,370.928,369.82,368.661,367.454,366.201,364.905,363.568,362.193,360.781,359.335,357.856,356.348,354.813,353.252,351.668,350.062,348.436,346.792,345.132,343.458,341.773,340.077,338.373,336.664,334.949,333.232,331.513,329.796,328.079,326.366,324.657,322.952,321.254,319.563,317.88,316.206,314.542,312.888,311.246,309.617,308.001,306.398,304.81,303.236,301.678,300.135,298.607,297.096,295.6,294.121,292.657,291.209,289.776,288.357,286.953,285.563,284.188,282.826,281.478,280.143,278.82,277.511,276.213,274.929,273.656,272.395,271.145,269.907,268.68,267.464,266.258,265.063,263.877,262.702,261.536,260.379,259.231,258.092,256.962,255.84,254.727,253.621,252.522,251.431,250.347,249.269,248.198,247.133,246.075,245.022,243.976,242.937,241.904,240.877,239.856,238.842,237.834,236.832,235.836,234.846,233.861,232.881,231.907,230.938,229.974,229.015,228.062,227.113,226.168,225.228,224.293,223.362,222.436,221.513,220.596,219.684,218.776,217.873,216.975,216.082,215.195,214.312,213.436,212.565,211.699,210.84,209.987,209.14,208.299,207.464,206.636,205.814,204.999,204.19,203.388,202.593,201.805,201.024,200.25,199.484,198.725,197.973,197.229,196.493,195.764,195.044,194.331,193.626,192.928,192.239,191.556,190.881,190.213,189.552,188.897,188.249,187.607,186.972,186.342,185.719,185.102,184.491,183.887,183.289,182.698,182.112,181.534,180.961,180.395,179.836,179.283,178.737,178.198,177.667,177.142,176.625,176.115,175.611,175.115,174.626,174.144,173.669,173.2,172.737,172.28,171.829,171.384,170.945,170.511,170.082,169.659,169.242,168.828,168.42,168.017,167.618,167.224,166.834,166.448,166.066,165.689,165.315,164.944,164.578,164.215,163.855,163.5,163.147,162.798,162.452,162.109,161.77,161.434,161.102,160.774,160.449,160.127,159.81,159.496,159.186,158.88,158.578,158.279,157.983,157.691,157.402,157.116,156.833,156.553,156.277,156.003,155.732,155.463,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5.1899,25.206,25.2221,25.2383,25.2546,25.2709,25.2875,25.3042,25.3211,25.3383,25.3557,25.3733,25.3912,25.4092,25.4275,25.446,25.4647,25.4837,25.503,25.5226,25.5424,25.5626,25.5831,25.6041,25.6256,25.6476,25.6698,25.6924,25.7152,25.7382,25.7615,25.785,25.8087,25.8325,25.8566,25.8807,25.905,25.9293,25.9535,25.9776,26.0016,26.0253,26.0487,26.0719,26.0949,26.1176,26.1401,26.1622,26.1841,26.2057,26.2267,26.2472,26.2671,26.2865,26.3053,26.3237,26.3416,26.359,26.3759,26.3924,26.4083,26.4239,26.439,26.4537,26.4679,26.4815,26.4946,26.5071,26.519,26.5302,26.5407,26.5504,26.5594,26.5676,26.5749,26.5812,26.5866,26.5908,26.5938,26.5956,26.5963,26.5956,26.5938,26.5905,26.5859,26.5798,26.5722,26.5633,26.5529,26.5411,26.5279,26.5131,26.4968,26.4789,26.4594,26.4382,26.4155,26.3911,26.3649,26.3371,26.3076,26.2763,26.2433,26.2085,26.1718,26.1333,26.0928,26.0503,26.0055,25.9585,25.9092,25.8576,25.8039,25.7478,25.6893,25.6283,25.5646,25.4983,25.4293,25.3577,25.2833,25.2061,25.126,25.043,24.957,24.8682,24.7764,24.6817,24.5842,24.4839,24.3807,24.2746,24.1658,24.0542,23.9398,23.8228,23.7032,23.5812,23.4568,23.3301,23.2011,23.0699,22.9363,22.8006,22.6626,22.5225,22.3803,22.2359,22.0895,21.9412,21.7909,21.6389,21.4853,21.3301,21.1734,21.0154,20.8561,20.6957,20.5342,20.3718,20.2084,20.0443,19.8795,19.714,19.548,19.3815,19.2147,19.0477,18.8808,18.7142,18.5482,18.3828,18.2181,18.0541,17.8911,17.729,17.5682,17.4087,17.2506,17.0939,16.9389,16.7855,16.6339,16.4841,16.3362,16.1903,16.0465,15.9048,15.7652,15.6277,15.4924,15.3593,15.2285,15.1001,14.9741,14.8504,14.7292,14.6105,14.4942,14.3804,14.2691,14.1603,14.0542,13.9508,13.8501,13.7521,13.6568,13.5644,13.4746,13.3876,13.3032,13.2215,13.1422,13.0654,12.991,12.9189,12.8491,12.7814,12.7157,12.6522,12.5906,12.5309,12.4731,12.4172,12.3632,12.311,12.2606,12.2119,12.1648,12.1191,12.0747,12.0315,11.9893,11.9482,11.9082,11.8692,11.8312,11.7942,11.7581,11.7229,11.6885,11.6551,11.6224,11.5904,11.5592,11.5286,11.4987,11.4695,11.4408,11.4128,11.3854,11.3585,11.3321,11.3063,11.2809,11.2559,11.2313,11.207,11.183,11.1592,11.1355,11.112,11.0885,11.0651,11.0417,11.0182,10.9946,10.9711,10.9475,10.9239,10.9003,10.8768,10.8533,10.8301,10.8069,10.7841,10.7614,10.7391,10.7171,10.6955,10.6742,10.6533,10.6329,10.6128,10.5932,10.5741,10.5554,10.5372,10.5194,10.502,10.4851,10.4686,10.4526,10.437,10.4217,10.4067,10.392,10.3775,10.3631,10.3489,10.3347,10.3205,10.3063,10.2922,10.278,10.2638,10.2496,10.2354,10.2211,10.2067,10.1922,10.1776,10.163,10.1483,10.1335,10.1186,10.1037,10.0887,10.0737,10.0587,10.0437,10.0288,10.0138,9.99876,9.98371,9.96863,9.95346,9.93818,9.92275,9.90712,9.89123,9.87503,9.8585,9.84163,9.82441,9.80683,9.78884,9.77043,9.75155,9.73221,9.71239,9.69207,9.67123,9.64983,9.62783,9.60522,9.58194,9.55798,9.53334,9.50804,9.48214,9.45567,9.42869,9.40126,9.37341,9.34522,9.31672,9.288,9.2591,9.23008,9.20098,9.17186,9.14276,9.11371,9.08472,9.05582,9.02705,8.99842,8.96995,8.94167,8.9136,8.88575,8.85814,8.83075,8.80358,8.77664,8.74991,8.72341,8.69713,8.67107,8.64524,8.61964,8.5943,8.56924,8.54449,8.52009,8.49606,8.47243,8.44926,8.4266,8.40448,8.38295,8.36208,8.34182,8.32217,8.3031,8.28459,8.26666,8.24935,8.23268,8.21667,8.20133,8.18669,8.17266,8.15923,8.14644,8.1343,8.12282,8.11201,8.10186,8.09237,8.08351,8.07526,8.06761,8.06051,8.05396,8.04792,8.04236,8.03726,8.03258,8.0283,8.02439,8.02083,8.01758,8.01465,8.01199,8.00961,8.00747,8.00556,8.00386,8.00236,8.00104,7.99989};

(Subscript[C, meas])=Interpolation[Transpose[{Table[i,{i,0,Length[isopren]-1}],(1/27)isopren}]];

(Subscript[(OverDot[Q]), c])=Interpolation[Transpose[{Table[i,{i,0,Length[co]-1}],(1/60)co}]];

(Subscript[(OverDot[V]), alv])=Interpolation[Transpose[{Table[i,{i,0,Length[alv]-1}],(1/60)alv}]];

(Subscript[t, max])=3600;

Numerische Lösung der Differentialgleichung

psol=ParametricNDSolve[{eq1,eq2,eq3, (Subscript[C, A])[0]==(Subscript[C, meas])[0], 
(Subscript[C, per])[0]==initper,(Subscript[C, rpt])[0]==initrpt},{(Subscript[C, A]),(Subscript[C, per]),(Subscript[C, rpt])},{t,0,(Subscript[t, max])},{ (Subscript[met, rpt]),(Subscript[pr, rpt]),(Subscript[met, per]),(Subscript[pr, per]),(Subscript[q, per0]),(Subscript[q, permax]),(Subscript[V, rpt]),(Subscript[V, per]),(Subscript[V, A]),τ,(Subscript[λ, bper]),(Subscript[λ, brpt]),(Subscript[λ, ba])}];

Liste der optimierten Parameter (Einheiten in Liter, Nanomol, Sekunde)

parlist={metrpt->0.05,prrpt->0.34,metper->0.016,prper->1.47,q0->0.094`,qmax->0.7,vrpt->13.,vper->9.2,va->4.1,s->2.1,lbper->0.34,lbrpt->0.4,lba->0.75};

Vergleich zwischen gemessener und berechneter Isoprenkonzentration

Plot[{(Subscript[C, A])[metrpt,prrpt,metper,prper,q0,qmax,vrpt,vper,va,s,lbper,lbrpt,lba][t]/.parlist/.psol, (Subscript[C, meas])[t] },{t,0,(Subscript[t, max])},PlotRange->All,PlotLegends-> {"(Subscript[C, A])[nmol/l]","(Subscript[C, meas])[nmol/l]"}]

Simulation der Auswirkungen der einzelnen Parameter

Manipulate[
Plot[{(Subscript[C, A])[metrpt,prrpt,metper,prper,q0,qmax,vrpt,vper,va,s,lbper,lbrpt,lba][t]/.psol,(1/10)(Subscript[C, per])[metrpt,prrpt,metper,prper,q0,qmax,vrpt,vper,va,s,lbper,lbrpt,lba][t]/.psol,(Subscript[C, meas])[t],(Subscript[C, rpt])[metrpt,prrpt,metper,prper,q0,qmax,vrpt,vper,va,s,lbper,lbrpt,lba][t]/.psol}  ,{t,0,(Subscript[t, max])},PlotRange->{{0,(Subscript[t, max])},{0,20}},PlotStyle->{Red,Blue,{Red,Dashed},Cyan},PlotLegends-> {"(Subscript[C, A])","(1/10)(Subscript[C, per])","(Subscript[C, meas])","(Subscript[C, rpt])"}],
{{metrpt,0.05,(Subscript[met, rpt])},0.008,0.08},{{prrpt,0.34,(Subscript[pr, rpt])},0.03,0.5},{{ metper,0.016,(Subscript[met, per])},0.01,0.03},  {{prper,1.47,(Subscript[pr, per])},0.5,5},{{q0,0.09,(Subscript[q, per0])},0.01,0.1},{{qmax,0.7,(Subscript[q, permax])},0.1,1},{{vrpt,13,(Subscript[V, rpt])},3,30},{{vper,9.2,(Subscript[V, per])},3,20},{{va,4.1,(Subscript[V, A])},2,10}, {{s,2.1,τ},1,5},{{lbper,0.4,(Subscript[λ, per])},0.1,2},{{lbrpt,0.4,(Subscript[λ, rpt])},0.1,2},  {{lba,0.75,(Subscript[λ, ba])},0.2,2} ]