2
$\begingroup$

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation}
\begin{equation} \frac{ds}{dt}= \lambda_b P_C \ \epsilon \ c (1-s)- \lambda_r (1-q) \ s, \nonumber \end{equation} \begin{equation} \frac{dq}{dt}= K_P (1-q) \frac{P_C}{P_Q} \ \ c - \gamma \ q, \nonumber \end{equation} I want to use asymptotic expansion on $c, s$ and $q$. And values of parameters are:

$K_F = 6.7 \times 10^{-2},$

$K_N = 6.03 \times 10^{-1}$

$K_P = 2.92 \times 10^{-2}$,

$K_D = 4.94 \times 10^{-2}$,

$\lambda_b= 0.0087$,

$I=1200$

$P_C = 3 \times 10^{11}$

$P_Q = 2.304 \times 10^{9}$

$\gamma=2.74 $

$\lambda_{b}=0.0087 $

$\lambda_{r}= 835$

$\alpha=1.14437 \times 10^{-3}$

For initial conditions:

\begin{equation} c_0(0)= c(0) = 0.25 \nonumber \end{equation} \begin{equation} s_0(0)= cs(0) = 0.02 \nonumber \nonumber \end{equation} \begin{equation} q_0(0)=q(0) = 0.98 \nonumber \nonumber \end{equation} and \begin{equation} c_i(0)= 0, \ i>0\nonumber \end{equation} \begin{equation} s_i(0)= 0, \ i>0 \nonumber \nonumber \end{equation} \begin{equation} q_i(0)=0, i>0. \nonumber \nonumber \end{equation}

=> i started with the expansions : \begin{equation} c= c_0+ \epsilon c_1 + \epsilon^2 c_2+......... \nonumber \end{equation} \begin{equation} s= s_0+ \epsilon s_1 + \epsilon^2 s_2+......... \nonumber \end{equation} \begin{equation} q= q_0+ \epsilon q_1 + \epsilon^2 q_2+......... \nonumber \end{equation} we are only interseted in up to fisrt power of $\epsilon$. so, we should get total 6 approximate differential equations to get answer for $\dfrac{dc_0}{dt}, \dfrac{ds_0}{dt}, \dfrac{dq_0}{dt}, \dfrac{dc_1}{dt}, \dfrac{ds_1}{dt}$ and $\dfrac{dq_1}{dt}$

but i think $\dfrac{dc_1}{dt}$ will disappear while expanding and equating the up to first power of $\epsilon$, do i need to go further up to $\epsilon{^2}$ because $\dfrac{dc_1}{dt}$ is very important to find and we need 6 approximate differetial equations in total. what can i do? please some one help me.

$\endgroup$
3
  • $\begingroup$ What is the variable $I$ in the first equation? $\endgroup$
    – David H
    Jul 25 '14 at 21:26
  • $\begingroup$ variable $I=1200$ $\endgroup$
    – Manjushree
    Jul 25 '14 at 21:41
  • $\begingroup$ I've added the tags differential-equations and perturbation-theory and removed the tag math-software since it didn't seem to be relevant. Please re-add it though if you feel it should be there. $\endgroup$ Jul 28 '14 at 3:09
2
+25
$\begingroup$

This is an example of a singular perturbation problem and so the type of perturbative expansion you are attempting is not likely to give you anything useful. I would recommend reading about multi-scale analysis, which can be found in most graduate books on nonlinear differential equations (Glenddings book has an introduction).
For a detailed exposition i would refer you to Kervokian and Coles book.

The system you have presented has $3$ time scales; fast time $\epsilon^{-1}t$, 'slow' time $t$, and 'really slow' time $\epsilon t$. On the fast time scale, $c$ will rapidly approach the slow manifold, ie the fixed point at $$\dot{c} =0 \implies c(q,s) = \frac{I\alpha}{(K_F+K_D+K_Ns+K_P(1−q))}.$$

You can then plug this into the equations for $\dot{s}$ and $\dot{q}$ to produce a two dimensional nonlinear equation for the motion on this slow manifold. Note that in this reduced system, there is still $\epsilon$ floating around. Since it is not a coefficient of a derivative term, the perturbative expansion attempted in the original post will work.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.