Numerical methods: Midpoint rule for integration

Jan 2019
2
0
Quebec
Hi. I want to solve an ode using some numerical integration methods. I have an equation of the form \(\displaystyle y'(t)=f(t,y(t))\).

Let's say my equation is: \(\displaystyle y'(t)=\mu y(t)+g(t)\), with \(\displaystyle \mu\) a constant, and g an arbitrary function.

If I use Euler method, I have that \(\displaystyle y_{n+1}=y_n+hf(t_n,y_n)\).

So I would have: \(\displaystyle y_{n+1}=y_n+h[\mu y_n+g_n]\)

Now, if I want to use the midpoint rule, I would have:

\(\displaystyle y_{n+1}=y_n+hf(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))\).

The problem I have is with how to intepret the term \(\displaystyle f(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))\), for my example, what would it explicitly be? Would be it ok to take:

\(\displaystyle f(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))=\mu y_n+\mu\frac{h}{2}[\mu y_n +q_n]+q(t_n+\frac{h}{2})\)?

Thanks.
 
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Jan 2019
2
0
Quebec
Yes, you are right, it should read \(\displaystyle f(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))=\mu y_n+\mu\frac{h}{2}[\mu y_n +g_n]+g(t_n+\frac{h}{2})\)

That would be ok?

Great, just made a numerical testing and seems to work! thanks.
 
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