Asymptotic expansion on 3 nonlinear ordinary differential equations

[ra_forever8]

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
$\frac{dc_0}{dt}, \frac{ds_0}{dt}, \frac{dq_0}{dt}, \frac{dc_1}{dt}, \frac{ds_1}{dt}$ and $\frac{dq_1}{dt}$but i think $\frac{dc_1}{dt}$ will disappear while expanding and equating the up to first power of $\epsilon$, do i need to go further up to $\epsilon{^2}$ because $\frac{dc_1}{dt}$ is very important to find and we need 6 approximate differetial equations in total. what can i do? please some one help me.

 

[jvicens]

Thank you for sharing your question on the 3 nonlinear differential equations. It seems like you are on the right track with your approach to using asymptotic expansion to solve these equations. I understand your concern about $\frac{dc_1}{dt}$ disappearing during the expansion process, but rest assured that it will not be lost. Let me explain why.

When we use asymptotic expansion, we are essentially breaking down a complex equation into simpler ones by assuming that one of the variables is much smaller (in this case, $\epsilon$) compared to the others. This allows us to ignore higher order terms and focus on the leading order terms. However, this does not mean that the higher order terms are not important or do not exist. They are still there, but their contribution to the overall solution is relatively small compared to the leading order terms.

In your case, when you expand the equations up to first order in $\epsilon$, you will indeed get 6 approximate differential equations. However, these equations will also contain terms that are of higher order in $\epsilon$, including $\frac{dc_1}{dt}$. These terms may not contribute significantly to the overall solution, but they are still present and cannot be ignored.

So, to answer your question, you do not need to go further up to second order in $\epsilon$ to find $\frac{dc_1}{dt}$. It will still be present in the first order equations, and you can solve for it by equating the coefficients of $\epsilon$ on both sides of the equations.

I hope this helps clarify your doubts. Good luck with your problem!