Reducing third order ODE to a system of first order equs + 4th order runge-kutta

In summary, the conversation discusses reducing a given equation to a system of three first order equations and applying boundary conditions. The equations are f' = g, g' = h, and h' = -f*h, with boundary conditions g(0) = 0, g(infinity) = 0, and f(infinity) = 0. The process of applying the boundary conditions is also explained.
  • #1
TTM
18
0
Hi,
I am stuck on the initial steps of this problem - its been a while since my diff. eq course.
I need to reduce this into a system of three equations then apply a 4th order runge kutta method to solve.

Homework Statement


f'''+f*f''=0
Boundary conditions:
f'(0)=f(0)=0
f'(infinity)=0

A. Reduce this equation to a system of three first order equations, with associated boundary conditions.

The Attempt at a Solution


Am I on the right track?
(equation 1) f'=g
(2) g'=h
(3) h'=-f*h

how to apply boundary conditions?
 
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  • #2


Hello,

Yes, you are on the right track. To apply the boundary conditions, you can simply substitute the values into your system of equations.

For the first boundary condition, f'(0) = 0, you can substitute this into your first equation: 0 = g(0). This means that g(0) must also be equal to 0.

For the second boundary condition, f'(infinity) = 0, you can use the fact that as x approaches infinity, the function f(x) must also approach 0. This means that g(infinity) must be equal to 0.

So your final system of equations, with the associated boundary conditions, would be:
(equation 1) f'=g, with boundary condition g(0) = 0
(2) g'=h, with boundary condition g(infinity) = 0
(3) h'=-f*h, with boundary condition f(infinity) = 0

I hope this helps. Good luck with your problem!
 

1. Why is it necessary to reduce a third order ODE to a system of first order equations?

Reducing a third order ODE to a system of first order equations allows for easier and more efficient numerical solutions. It also allows for easier implementation of numerical methods, such as the 4th order Runge-Kutta method.

2. How does reducing a third order ODE to a system of first order equations make it easier to solve?

By converting the third order ODE into a system of first order equations, the problem is broken down into simpler equations that can be solved using standard numerical methods. This also allows for the use of more advanced numerical methods, such as the 4th order Runge-Kutta method, which is not applicable to higher order ODEs.

3. Can any third order ODE be reduced to a system of first order equations?

Yes, any third order ODE can be reduced to a system of first order equations. This is possible because all higher order ODEs can be rewritten as a system of first order equations by introducing new variables.

4. What is the 4th order Runge-Kutta method and why is it commonly used in solving ODEs?

The 4th order Runge-Kutta method is a numerical method for solving ODEs. It is commonly used because it provides a good balance between accuracy and computational efficiency. It also has a higher order of convergence compared to other numerical methods, meaning it can produce more accurate results with fewer computations.

5. Are there any limitations to reducing a third order ODE to a system of first order equations and using the 4th order Runge-Kutta method?

One limitation is that the 4th order Runge-Kutta method is not suitable for stiff systems of equations, where the solution changes rapidly over a small interval. In these cases, other numerical methods may be more appropriate. Additionally, reducing a higher order ODE to a system of first order equations can result in a larger system of equations, which may be more computationally demanding.

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