The discussion focuses on the application of the backward Euler method for solving second-order differential equations (d.e) and its potential extension to systems of four ordinary differential equations (odes). Participants explore the formulation of the method and its implications for specific equations.
Participants generally agree on the approach of rewriting second-order equations as first-order systems and extending the method to multiple equations. However, the specific implementation details and the application to the example provided remain open for further discussion.
There are unresolved details regarding the specific formulation of the backward Euler scheme for the proposed system and the implications of the initial conditions on the solution process.
pgioun said:Hi,
How can one use back Euler method for 2nd order d.e?
Yes. Just write the 4 variables as a vector, so your equiations become one equation like $$\frac{d}{dx}\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} = f\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} + g(x)$$Is it possible this method to be expanded for a system of 4 odes?
AlephZero said: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 $$\frac{d}{dx}\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} = f\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4\end{pmatrix} + g(x)$$
pgioun said:Ok.. and then how the back Euler scheme will be like..?