How can I integrate this ODE numerically?

In summary, The problem is to integrate the given ODE with boundary conditions of f(0)=0, f'(inf)=1, and f'(-inf)= (a-1)/a. The individual asking for help has considered using a shooting method but is unsure of how to apply it to these specific boundary conditions. They are seeking assistance or a reference for further guidance.
  • #1
michujo
4
0
Hello to everybody I'm trying to integrate this ODE but I don't know how:

f'''+a*f*f''=0 where a is a parameter.
With the boundary conditions f(0)=0; f(-inf)=1; f(inf)=1.

I had thought about using a shooting method but I don't know how to apply it with these boundary conditions. It would be nice if somebody could help me or give me a reference where I can find it( I have already looked for it in some books but I find nothing).

Thank you.

Michujo.
 
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  • #2
I'm sorry,there's a mistake in the boundary conditions. They are as follows:

f(0)=0; f'(inf)=1; f'(-inf)=(a-1)/a; where a is a parameter.
 

1. How do I choose the appropriate numerical integration method for my ODE?

The most important factor in selecting a numerical integration method for your ODE is the order of the ODE. For first-order ODEs, simple methods like Euler's method or Heun's method are often sufficient. However, for higher-order ODEs, more sophisticated methods like Runge-Kutta or Adams-Bashforth methods may be necessary. Other factors to consider include the desired accuracy and stability of the solution, as well as any specific features of the ODE such as stiffness.

2. What is the difference between explicit and implicit numerical integration methods?

In explicit methods, the solution at the next time step is calculated using only information from the current time step. This makes them simpler to implement and faster to compute, but they may be less stable for certain ODEs. On the other hand, implicit methods use information from both the current and next time step, making them more stable but also more computationally expensive.

3. How do I determine the step size for numerical integration?

The appropriate step size for numerical integration depends on the desired accuracy of the solution and the stability of the method being used. In general, smaller step sizes will result in a more accurate solution, but they also require more computational time. It is important to strike a balance between accuracy and efficiency when selecting the step size.

4. Can I use the same numerical integration method for all types of ODEs?

No, different numerical integration methods are better suited for different types of ODEs. For example, stiff ODEs require methods specifically designed for stiff systems, while non-stiff ODEs can often be solved using simpler methods. It is important to understand the properties of your ODE in order to choose the most appropriate numerical integration method.

5. How do I know if the numerical solution is accurate?

The accuracy of a numerical solution can be evaluated by comparing it to an analytical solution, if one is available. If not, the solution can be compared to solutions obtained using different numerical methods and/or different step sizes. Additionally, error metrics such as the global error or local error can be calculated to assess the accuracy of the solution.

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