Back Euler method for 2nd order d.e

[pgioun]

Hi,
How can one use back Euler method for 2nd order d.e?

Is it possible this method to be expanded for a system of 4 odes?

Thanks

 

[AlephZero]

Hi,
How can one use back Euler method for 2nd order d.e?

Write $dy/dx = p$ and $d^2y/dx^2 = dp/dx$, then solve two first order equations for $y$ and $p$. (The first equation is your original DE rewritten using $y$, $p$, and $dp/dx$. The second equation is just $dy/dx = p$).

Is it possible this method to be expanded for a system of 4 odes?

Yes. Just write the 4 variables as a vector, so your equiations become one equation like $$\frac{d}{dx}\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} = f\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} + g(x)$$

 

[pgioun]

Write $dy/dx = p$ and $d^2y/dx^2 = dp/dx$, then solve two first order equations for $y$ and $p$. (The first equation is your original DE rewritten using $y$, $p$, and $dp/dx$. The second equation is just $dy/dx = p$).


Yes. Just write the 4 variables as a vector, so your equiations become one equation like $$\frac{d}{dx}\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} = f\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} + g(x)$$

Ok.. and then how the back Euler scheme will be like..?
If it was one ode it would be: $y_{n+1}$=$y_{n}$+f( $y_{n+1}$, $t_{n+1}$).
To be more specific I want to solve the system:$y^{4}$=1/$y^{2}$, with y(0)=0,y''(0)=0, applying this method..
Thanks.

 

[AlephZero]

Ok.. and then how the back Euler scheme will be like..?

It looks exactly the same. Just replace the scalar y with the vector.

 

[blue_raver22]

for your question. The back Euler method is a numerical method used to approximate solutions to ordinary differential equations (ODEs). It is a type of implicit method, meaning that the solution at a given time step is dependent on the solution at the previous time step. This method is particularly useful for stiff ODEs, where the solution changes rapidly over a small interval of time.

To use the back Euler method for a second order ODE, we first need to rewrite the equation as a system of first order ODEs. This can be done by introducing a new variable to represent the derivative of the original variable. For example, if we have the second order ODE y'' = f(x,y,y'), we can rewrite it as a system of first order ODEs: y' = z and z' = f(x,y,z). We can then use the back Euler method to approximate the solution for both y and z at each time step.

As for your second question, it is possible to expand the back Euler method to solve a system of four ODEs. The process would be the same as for a second order ODE - rewrite the system as a system of first order ODEs and use the back Euler method to approximate the solutions for each variable at each time step. However, it is important to note that the accuracy of the method may decrease as the number of equations increases, so it is important to consider other numerical methods as well.