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(%i1) kill(all)$

ode : part5,intro: ricatti
[email protected],05/02/2017

Theory Ricatti : ODE (reaction kinetics)

(dy/dx) = p(x)*y^2+q(x)*y+r(x)
Why: Consecutive Reactions and Ricatti
                     k1   k2
reaction kinetics : A → B → C
dCA/dt = - k1*CA^n          (1)
dCB/dt = k1*CA^n-k2*CB^m    (2)
dCC/dt = k2*CC^l            (3)

integer values :  n,m,l
concentration  : CA,CB,CC

(%i2) load('contrib_ode)$
assume(t>0,CA>0,CB>0)$

equation (2) n=1,m=2, by 't=0 → CA=CA0'

(%i4) eq1: CA = CA0*%e^(-k1*t);
eq2:'diff(CB,t)=k1*CA^1-k2*CB^2;
\[\tag{eq1}\label{eq1}\mathit{CA}=\mathit{CA0}\,{{\%{}e}^{-\mathit{k1}t}}\] \[\tag{eq2}\label{eq2}\frac{d}{dt}\mathit{CB}=\mathit{CA}\,\mathit{k1}-{{\mathit{CB}}^{2}}\,\mathit{k2}\]

method1:

(%i7) eq3:subst(rhs(eq1),CA,eq2);
eq4:part(eq3,2,2);
eq5: lhs(eq3)-eq4=rhs(eq3)-eq4;
\[\tag{eq3}\label{eq3}\frac{d}{dt}\mathit{CB}=\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{-\mathit{k1}t}}-{{\mathit{CB}}^{2}}\,\mathit{k2}\] \[\tag{eq4}\label{eq4}-{{\mathit{CB}}^{2}}\,\mathit{k2}\] \[\tag{eq5}\label{eq5}{{\mathit{CB}}^{2}}\,\mathit{k2}+\frac{d}{dt}\mathit{CB}=\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{-\mathit{k1}t}}\]
(%i9) eq6:contrib_ode(eq5,CB,t)[1];
method;
\[\tag{eq6}\label{eq6}[\mathit{CB}=\frac{\frac{d}{dt}\mathit{\%{}u}}{\mathit{\%{}u}\,\mathit{k2}},\mathit{\%{}u}\,\mathit{CA0}\,\mathit{k1}\,{{\mathit{k2}}^{2}}\,{{\%{}e}^{-\mathit{k1}t}}-\left( \frac{{{d}^{2}}}{d{{t}^{2}}}\mathit{\%{}u}\right) \,\mathit{k2}=0]\] \[\tag{\%{}o9}\label{o9} \mathit{riccati}\]

check : solution n=1,m=2

(%i11) eq7:part(eq6,1);
eq8:part(eq6,2);
\[\tag{eq7}\label{eq7}\mathit{CB}=\frac{\frac{d}{dt}\mathit{\%{}u}}{\mathit{\%{}u}\,\mathit{k2}}\] \[\tag{eq8}\label{eq8}\mathit{\%{}u}\,\mathit{CA0}\,\mathit{k1}\,{{\mathit{k2}}^{2}}\,{{\%{}e}^{-\mathit{k1}t}}-\left( \frac{{{d}^{2}}}{d{{t}^{2}}}\mathit{\%{}u}\right) \,\mathit{k2}=0\]

eq7 : What subst. to use → linear second order
     equation with nonconstant coefficients.

(%i12) depends([%u],[t]);
\[\tag{\%{}o12}\label{o12} [\operatorname{\%{}u}(t)]\]
(%i13) dependencies;
\[\tag{\%{}o13}\label{o13} [\operatorname{\%{}u}(t)]\]
(%i14) eq9:subst(eq7,eq5);
\[\tag{eq9}\label{eq9}\frac{{{\left( \frac{d}{dt}\mathit{\%{}u}\right) }^{2}}}{{{\mathit{\%{}u}}^{2}}\,\mathit{k2}}+\frac{d}{dt}\frac{\frac{d}{dt}\mathit{\%{}u}}{\mathit{\%{}u}\,\mathit{k2}}=\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{-\mathit{k1}t}}\]
(%i15) eq10:ev(part(eq9,1),nouns)=ev(part(eq9,2),nouns);
\[\tag{eq10}\label{eq10}\frac{\frac{{{d}^{2}}}{d{{t}^{2}}}\mathit{\%{}u}}{\mathit{\%{}u}\,\mathit{k2}}=\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{-\mathit{k1}t}}\]
(%i16) eq11:part(eq10,1,2);
\[\tag{eq11}\label{eq11}\mathit{\%{}u}\,\mathit{k2}\]
(%i17) eq12:eq10*eq11;
\[\tag{eq12}\label{eq12}\frac{{{d}^{2}}}{d{{t}^{2}}}\mathit{\%{}u}=\mathit{\%{}u}\,\mathit{CA0}\,\mathit{k1}\,\mathit{k2}\,{{\%{}e}^{-\mathit{k1}t}}\]
(%i18) dependencies;
\[\tag{\%{}o18}\label{o18} [\operatorname{\%{}u}(t)]\]
(%i19) eq13:lhs(eq12)-rhs(eq12)=0;
\[\tag{eq13}\label{eq13}\frac{{{d}^{2}}}{d{{t}^{2}}}\mathit{\%{}u}-\mathit{\%{}u}\,\mathit{CA0}\,\mathit{k1}\,\mathit{k2}\,{{\%{}e}^{-\mathit{k1}t}}=0\]
(%i20) eq14:ratsubst(lhs(eq1),rhs(eq1),eq13);
\[\tag{eq14}\label{eq14}\frac{{{d}^{2}}}{d{{t}^{2}}}\mathit{\%{}u}-\mathit{\%{}u}\,\mathit{CA}\,\mathit{k1}\,\mathit{k2}=0\]

try to solve:eq12:  d/dt(d%u/dt)=f(t)*%u

(%i23) eq14:part(eq12,1,1);
eq15:part(eq12,1,2);
eq16:part(eq12,1,3);
\[\tag{eq14}\label{eq14}\mathit{\%{}u}\] \[\tag{eq15}\label{eq15}t\] \[\tag{eq16}\label{eq16}2\]

convert : 'two' first order ODE's
Numerical Solution find with 4th Order Runge-Kutta.

(%i25) eq17:'diff(%u,eq15,eq16-1)=h;
eq18:'diff(rhs(eq17),eq15,eq16-1)=rhs(eq12);
\[\tag{eq17}\label{eq17}\frac{d}{dt}\mathit{\%{}u}=h\] \[\tag{eq18}\label{eq18}\frac{d}{dt}h=\mathit{\%{}u}\,\mathit{CA0}\,\mathit{k1}\,\mathit{k2}\,{{\%{}e}^{-\mathit{k1}t}}\]
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