Numerical Analysis - Backward Euler Method

Simple derivation of the Backward Euler method for numerically approximating the solution of a first-order ordinary differential equation (ODE). Builds upon knowledge presented in lesson on the Forward Euler method. Comparison between the Forward Euler and Backward Euler methods is presented, as it pertains to solving first order ordinary differential equations.

[@deborahfranza2925]

I'm utilizing this for my summer research and was looking for a clear and concise explanation, very well done! Thank you so much!

[@engineeringmadeeasy4480]

You're very welcome!

[@gulshankarim7344]

Please solve any question using this method, through which we clearly understand.

[@djoqo]

Great Video, Thank You!!

[@yizhang7027]

Very short yet very clear explanation. Thank you very much.

[@engineeringmadeeasy4480]

Glad it was helpful!

[@jacielespinosa-severo7134]

Thank you, this made things clearer

[@ceroorec2153]

Thank you for the video, it was very helpful.

[@icee562]

Thank you sooooo much.
Best explanation on the yt!!

[@engineeringmadeeasy4480]

You are very welcome! Glad you found it so helpful

[@user-zg3ii4oz4s]

Thank you very much for the clear explanation. Now I can do my tutorial sheet!
May I ask what '0(deltat^2)' represents when describing the local error?
Thanks!

[@jesusalfredohernandezorozc3836]

Well explained. Thank you

[@engineeringmadeeasy4480]

Glad it was helpful!

[@MH-ix2nh]

How do you plug y_n+1 into the equation if it's the thing you're looking for????

[@engineeringmadeeasy4480]

That's a great question! Exactly why it's called an "implicit" numerical scheme
I would recommend checking out my video where I do an example MATLAB code with the backward Euler scheme and walk through the process, or my video on the trapezoidal Euler method.
Basically, if your differential equation is simple and linear, you can usually rearrange terms and still solve for y_n+1. But if you have a more complex or nonlinear differential equation, you can move all of your terms to the right side and use a function like "fsolve" or "fzero" in MATLAB to find y_n+1.
Hope that helps!

[@icee562]

You don't.
If you're using the Backwards Euler Formula, we obviously don't have the next point y_{n+1} right off the bat,
however, when possible, you can solve the D.E (Recurrence Relation) for y_{n+1} in terms of current terms y_n and t_n.
Also use the fact that delta t, aka, , t_{n+1} - t_n = h --> t_{n+1} = t_n + h.
The advantage of the Regular(forward) Euler formula is that it is straight-forward, no pun intended, but it can become unstable quick as hell.
i.e it diverges from the real solution pretty fast through error.
The advantage of the Reverse Euler formula is that it is slightly more stable but it involves way more calculations.

[@deborahfranza2925]

@@engineeringmadeeasy4480 Subscribed

[@solo_dezaldivar]

Can't thank you enough!!

[@TheRainHarvester]

If I'm updating position,y, based on a force of gravity, G m1 m2 G/ r^2, how do solve for y n+1? (Trying to put this in code for a gravity simulation)

[@summerxia7378]

Thank you!

[@engineeringmadeeasy4480]

You're welcome!

[@geniusmathematics9123]

Sir, please upload more video 🙏😪

[@moritzpfurtscheller4248]

Great video as always keep it up!

[@engineeringmadeeasy4480]

Appreciate it!

[@anishadevi94]

how to implement this for nth order system of ODE?

[@rhot2963]

How do you choose stopping criterion if you use euler method in while loop?

[@royallad2597]

You can do that by setting a maxiter.(Someone in our class asked the same thing and this was the professors answer)