# Regular perturbation nonlinear problem

• MHB
• Carla1985
In summary: I did the nondimensionalisation. Basically the system is a type of drug-receptor binding system. After nondimensionalising I have 3 parameters: $\alpha$, $\beta$ and $\gamma$. We know that $\alpha$ is $O(1)$ but have limit information about $beta$ and $gamma$. So we are trying to use asymptotic to explore these. To do this we fix one of the parameters to also be $O(1)$ and set the other to be either small ($\epsilon$) or large ($1/\epsilon$). I have done the cases of \beta=\ep Carla1985 Hi all, I have this (nondimensionalised) system of ODEs that I am trying to analyse: \begin{align} \frac{dr}{dt}= &\ - \left(\alpha+\frac{\epsilon}{2}\right)r + \left(1-\frac{\epsilon}{2}\right)\alpha p - \alpha^2\beta r p + \frac{\epsilon}{2} \\ \frac{dp}{dt}= &\ \left(1-\frac{\epsilon}{2\alpha}\right)r - \left(1+\frac{\epsilon}{2}\right)p - \alpha\beta r p + \frac{\epsilon}{2\alpha} \end{align} with initial conditionsr(0)=1, p(0)=0$and$\epsilon$is a small parameter. After substituting in the asymptotic approximations$r\approx r_0+\epsilon r_1+...$,$p\approx p_0+\epsilon p_1+...I get a leading order problem as \begin{align} \frac{dr_0}{dt}= &\ - \alpha r_0 + \alpha p_0 -\alpha^2\beta r_0 p_0 \\ \frac{dp_0}{dt}= &\ r_0 - p_0 -\alpha\beta r_0 p_0 \end{align} Clearly, this is still of no help as it's still nonlinear. So my question is, is there any way of getting past this? The only other piece of information I have is that at equilibrium the solution for both will beO(\epsilon^{1/2})$. I did attempt to rescale for this but I then get a singular perturbation problem in which the limit of the inner solution is infinity so I could not do the matching (I will explain in more detail what I did for this if needed). If anyone could suggest a way in which I can solve this problem I would be very grateful. Thank you! Carla What are $$\displaystyle \alpha$$ and $$\displaystyle \beta$$? What we know about them? The non-linearity has nothing to do with $$\epsilon$$ but rather with $$\alpha$$ and $$\beta$$.$\alpha$and$\beta$are grouped parameters from how I did the nondimensionalisation. Basically the system is a type of drug-receptor binding system. After nondimensionalising I have 3 parameters:$\alpha$,$\beta$and$\gamma$. We know that$\alpha$is$O(1)$but have limit information about$beta$and$gamma$. So we are trying to use asymptotic to explore these. To do this we fix one of the parameters to also be$O(1)$and set the other to be either small ($\epsilon$) or large ($1/\epsilon$). I have done the cases of$\beta=\epsilon$,$\beta=1/\epsilon$and$\gamma=1/\epsilon$without too many issues. This one that I'm having trouble with is$\gamma=\epsilon$. I understand the answer may well be that in this situation there is nothing I can do further but I thought I would ask in case there was something else I could do with it that I wasn't aware of. Either way I appreciate the help. Thanks Carla In that case, I'd simply use either numerical methods to find out the behaviour of the solution, or use some sort of series solution to approximate the solution. Carla1985 said:$\alpha$and$\beta$are grouped parameters from how I did the nondimensionalisation. Basically the system is a type of drug-receptor binding system. After nondimensionalising I have 3 parameters:$\alpha$,$\beta$and$\gamma$. We know that$\alpha$is$O(1)$but have limit information about$beta$and$gamma$. So we are trying to use asymptotic to explore these. To do this we fix one of the parameters to also be$O(1)$and set the other to be either small ($\epsilon$) or large ($1/\epsilon$). I have done the cases of$\beta=\epsilon$,$\beta=1/\epsilon$and$\gamma=1/\epsilon$without too many issues. This one that I'm having trouble with is$\gamma=\epsilon$. I understand the answer may well be that in this situation there is nothing I can do further but I thought I would ask in case there was something else I could do with it that I wasn't aware of. Either way I appreciate the help. Thanks Carla In your ODEs in your OP I don't see any mention of a$\gamma\$ there, where should it appear?

## 1. What is a regular perturbation nonlinear problem?

A regular perturbation nonlinear problem is a mathematical problem that involves finding solutions to nonlinear equations using a series expansion technique. It is commonly used in physics and engineering to simplify complex nonlinear problems into a series of linear problems, making them easier to solve.

## 2. What is the importance of regular perturbation in scientific research?

Regular perturbation is important in scientific research because it allows us to approximate solutions to complex nonlinear problems that cannot be solved analytically. It also provides insights into the behavior of nonlinear systems and can be used to validate numerical simulations.

## 3. What are the limitations of regular perturbation in solving nonlinear problems?

Regular perturbation has limitations in solving nonlinear problems when the perturbation parameter is too large, resulting in inaccurate solutions. It also does not work well for problems with multiple scales, where there are several parameters of different orders of magnitude.

## 4. What are some common techniques used in regular perturbation?

Some common techniques used in regular perturbation include the method of multiple scales, the Lindstedt-Poincaré method, and the method of strained coordinates. These techniques involve systematically expanding the solution in a series of terms and solving for them iteratively.

## 5. How does regular perturbation differ from singular perturbation?

Regular perturbation and singular perturbation are both methods used to solve nonlinear problems, but they differ in their approach. Regular perturbation assumes that the perturbation parameter is small, while singular perturbation allows for the parameter to be large. This results in different series expansions and solutions for the two methods.

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