Numerically solving coupled DEs?

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In summary, the conversation discusses finding a steady-state solution for N_b and N_w using Matlab. The set of coupled differential equations involves constants \tau_b, \tau_c, and D. The recommended method is to use the RK4 algorithm, specifically running two simultaneously with the current values of N_b and N_w. There is also a mention of removing dimensions and using RK2 or Euler's method with small steps. The specific form of P_b and \tau_w is not mentioned.
  • #1
gyver
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Hi folks!

I'm trying to (numerically) find a steady-state solution for [tex]N_b[/tex] and [tex]N_w[/tex] in the following set of coupled DEs using the software package Matlab:

[tex]
\left{
\begin{array}{l}
\frac{\delta N_b}{\delta t} = P_b(N_b) - N_b \cdot \left( \frac{1}{\tau_b} - \frac{1}{\tau_c}D \right)\\
\frac{\delta N_w}{\delta t} = \frac{N_b}{\tau_c} - \frac{N_w}{\tau_w(N_w)} - P_w(N_w)
\end{array}
\right.
[/tex]

where [tex]\tau_b[/tex], [tex]\tau_c[/tex] and [tex]D[/tex] are constants. Which way would be the right one to go?
 
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  • #2
Remove the dimensions and write a program to do RK4 unless this is just for a class in which case RK2 or Euler's with really small step is easier.
 
  • #3
I'm don't know MatLab but I'd run two Runge Kutta 4th order (RK4) algorithms simultaneously, using the current values for Nb and Nw in each formula.
 
  • #4
What form do [tex]P_b[/tex] and [tex]\tau_w[/tex] take?
 

FAQ: Numerically solving coupled DEs?

How do I know which numerical method to use for solving coupled differential equations?

There are several numerical methods available for solving coupled differential equations, such as Euler's method, Runge-Kutta methods, and multistep methods. The most suitable method depends on the specific characteristics of the system, such as stability, accuracy, and computational efficiency. It is recommended to consult with a numerical analyst or refer to literature to determine the most appropriate method for your particular problem.

Can I use the same numerical method for both initial value problems and boundary value problems?

Yes, in many cases, the same numerical method can be used for both initial value problems (IVPs) and boundary value problems (BVPs). However, some methods are better suited for IVPs, while others are more efficient for BVPs. It is important to understand the differences between IVPs and BVPs and choose a method accordingly.

How does the step size affect the accuracy of the numerical solution?

The step size, also known as the time or spatial increment, determines how frequently the numerical method is applied to the differential equations. A smaller step size generally results in a more accurate solution, but it also increases the computational time. It is important to choose an appropriate step size that balances accuracy and efficiency.

What is the stability criterion for numerical methods for solving coupled differential equations?

Stability is an important consideration when choosing a numerical method for solving coupled differential equations. The stability criterion depends on the specific method and the characteristics of the system. Generally, a method is considered stable if the solution does not grow uncontrollably as the number of steps increases. It is recommended to consult with a numerical analyst or refer to literature to determine the stability criterion for a particular method.

Can I use any programming language for implementing numerical methods for coupled differential equations?

Yes, most programming languages have built-in functions or libraries for implementing numerical methods for coupled differential equations. Some commonly used languages include MATLAB, Python, and C++. It is important to choose a language that you are comfortable with and that has the necessary tools for implementing the desired method.

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