Solving rate equations with RK method. Don't understand how.

In summary, the authors of the article used the Runge-Kutta method to solve a system of rate equations for a three-level system in order to understand the effects of pulsed laser light transmitted through a nonlinear absorbing sample. The differential equations were first decoupled and then integrated over time, length, and along the radial direction before being solved numerically. The RK method was applied by starting with an initial position and stepping through with a chosen step size to find the corresponding path of the system.
  • #1
ale_yoman
2
0
I can't understand how authors of the article solve the system of rate equations with Runge-Kutta method.

Pulsed laser light is transmited through nonlinear absorbing sample. So, here is the system of rate equations which governs the populations of the levels in the three-level system.

[itex] \frac{dN_{0}}{dt} = -\frac{\sigma_{0} I N_{0}}{h \omega} - \frac {\beta I^{2} } {2 h \omega} - \frac{\gamma I^{3} }{3 h \omega} + \frac{N_{1}}{\tau_{s1}} [/itex]

[itex] \frac{dN_{1}}{dt} = \frac{\sigma_{0} I N_{0}}{h \omega} - \frac{\sigma_{1} I N_{1}}{h \omega} - \frac{N_{1}}{\tau_{s1}} +\frac{N_{2}}{\tau_{sn}} [/itex]

[itex] \frac{dN_{2}}{dt} = \frac{\sigma_{1} I N_{1}}{h \omega} + \frac {\beta I^{2} } {2 h \omega} + \frac{\gamma I^{3} }{3 h \omega} - \frac{N_{2}}{\tau_{sn}} [/itex]

Intensity transmitted through the sample is given by
[itex] \frac{dI}{dz} = -\sigma_{0} I N_{0} - \sigma_{1} I N_{1} - \beta I^{2} - \gamma I^{3}[/itex]

where [itex] I(r,z,t) = I_{00} \left( \frac{w_{0}^{2}}{w^{2}(z)} \right) \ \exp \left( - \frac{t^{2}}{\tau_{p}^{2}} \right) \ \exp \left( - \frac{2r^{2}}{w^{2}(z)} \right)[/itex]

Here [itex] \sigma_{i} [/itex] is the cross-section of the corresponding state, [itex] \beta[/itex] and [itex] \gamma [/itex] - two- and three-photon absoption coefficients, [itex] w(z) [/itex] - radius of the laser (assumed to be Gaussian) beam, [itex]N_{i}(t)[/itex] - population of the i-th state, [itex] \tau_{i}[/itex] - life-time of the corresponding state.

Here we come to the point where my incomprehension is rising. It is stated in the article that the differential equations were first decoupled and then integrated over time, length, and along the radial direction before solving them numerically with Runge-Kutta fourth order method. Limits of integration of [itex]r, t, z[/itex] are [itex](0,\infty), (-\infty, \infty)[/itex] and [itex](0,L)[/itex],respectively, where L is the sample length.

So, i wonder how i should do this "de-coupling"?

I thought that the RK method is aplicable for the systems where the derivatives are taken with respect to the same variable. But in this exaple the state population is the function of time, and in the equation for the intensity there is the coordinate derivative. May be I'm wrong?
 
Physics news on Phys.org
  • #2
Hi ale_yoman! Welcome to PF! :smile:

To apply RK you need to uncouple your equations, so they become of the form:
[tex]y' = f(t, y)[/tex]

Or more specifically:
[tex]\frac d {dt} \begin{pmatrix}y_1 \\ y_2 \\ ... \\ y_n \end{pmatrix}
= \vec f(t, y_1, y_2, ..., y_n)[/tex]

Can you see how this matches your set of equations?

Next the actual application of RK is to start with an initial position y, and step t up with a step size, find the corresponding path that y takes.
 

1. What is the RK method for solving rate equations?

The RK (Runge-Kutta) method is a numerical method used to solve differential equations, including rate equations. It involves breaking down the equation into smaller time intervals and using a series of calculations to approximate the solution at each interval.

2. How does the RK method work?

The RK method works by using a weighted average of several intermediate values, calculated using the slope of the differential equation at different points within each time interval. This allows for a more accurate approximation of the solution compared to simpler methods, such as Euler's method.

3. Can the RK method be used for any type of rate equation?

Yes, the RK method can be used for any type of rate equation, as long as it can be expressed as a system of first-order differential equations. This includes linear, nonlinear, and even systems of equations.

4. What are the benefits of using the RK method compared to other methods?

The RK method is known for its high accuracy and stability, making it a popular choice for solving differential equations. It also allows for greater flexibility in choosing the step size, which can improve the efficiency of the calculation.

5. Are there any limitations to using the RK method?

One limitation of the RK method is that it can be computationally expensive, especially for higher-order equations. Additionally, it may be more challenging to implement compared to simpler methods, such as Euler's method.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
979
  • Differential Equations
Replies
3
Views
1K
  • Thermodynamics
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
930
  • Differential Equations
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
7
Views
2K
  • Quantum Physics
Replies
9
Views
711
  • Differential Equations
Replies
1
Views
609
  • Differential Equations
Replies
7
Views
2K
  • Differential Equations
Replies
1
Views
699
Back
Top