Help setting up a tricky system of ODEs

In summary, the conversation discusses the problem of setting up a system of ODEs for Burgers equation with a source term. The system includes equations for x, v, and H, where H does not have an explicit formula but can be calculated using the chain rule and knowledge of H_t and (int H)'. The problem is solved by introducing H into the system and using a numerical PDE solver to solve for H_t and then calculate f(H,H_x,s,s_x). This approach allows for the use of ODE45 in solving the system.
  • #1
Weezix1
1
0
Hi there
I have been trying to set up a system of ODEs that are ultimately a solution to Burgers equation with a source term, and it boils down to:
x' = 11v
v' = f(H,H_x,s,s_x)
where x = x(t), H = H(x,t), s = s(x) and H_x,s_x are the partial derivatives wrtx.

The problem comes that I do not have an explicit formula for H, all I have is an equation for H_t, and the knowledge that (int H)' = int s

By the chain rule,
H' = x'H_x + H_t
I know everything on the RHS except the H_x, so I thought since H also depends on time this needs to go into the system, giving
x' = 11v
v' = f(H,H_x,s,s_x)
H' = 11vH_x + H_t
which solves the problem of having H in the v' equation, but I am stumped as to how to deal with the H_x equation.

I am looking to be able to solve this with ODE45 on matlab, which I have never used before.

Any ideas?
 
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  • #2
One approach you could try is to use a numerical PDE solver, such as finite difference or finite volume methods. Such a solver would solve the equation for H_t as a function of H_x and x, and then you can use the solution to calculate f(H,H_x,s,s_x). This should allow you to solve your system of ODEs using ODE45.
 

Related to Help setting up a tricky system of ODEs

1. What are ODEs and why are they important in scientific research?

ODEs, or ordinary differential equations, are mathematical equations that involve derivatives of one or more dependent variables with respect to one or more independent variables. They are important in scientific research because they allow us to model and understand complex systems and phenomena in fields such as physics, engineering, biology, and economics.

2. How do I approach setting up a system of ODEs?

The first step in setting up a system of ODEs is to clearly define the dependent and independent variables in your system. Then, you can use your knowledge of the system's behavior and any relevant equations or models to write out the differential equations for each variable. It may also be helpful to break the system into smaller, more manageable subsystems.

3. What are some common challenges in setting up a system of ODEs?

One common challenge is determining the appropriate initial conditions for the system. These initial conditions specify the values of the dependent variables at the starting point of the system. Additionally, identifying all relevant equations and accurately representing the relationships between variables can be difficult.

4. How can I ensure that my system of ODEs is accurate and reliable?

One way to ensure accuracy and reliability is to compare the results of your system to real-world data or experimental results. This can help you identify any discrepancies and make adjustments to your model. It may also be helpful to consult with other scientists or experts in the field.

5. Are there any software or tools that can assist with setting up a system of ODEs?

Yes, there are various software and tools available that can assist with setting up and solving systems of ODEs, such as MATLAB, Wolfram Mathematica, and Python libraries like SciPy and SymPy. These tools can help you visualize and analyze your system, as well as solve the equations numerically or analytically.

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