Euler Backward Method-Numerical Solution for ODE First Order
Euler’s Backward method is an extension of Euler Forward method(If you haven't, click on the link to check the post). It is also called Euler Implicit Method.
As we are all aware the General form of First Order ODE can be written as
Numerically approximating the term we get the following approximation
Then we put it in our original ODE and we get
Approximating for the next index we get ‘
This brings us to our final formula for the Approximation,
This is a implicit solution scheme
{Keynotes}
{Explicit Solution} This attempts at finding a solution to the non-linear system of equations iteratively by considering the current state of the system as well as its subsequent(or previous) time state.
{Implicit Solution} : The implicit integration techniques include Backward Euler, trapezoidal etc. because the value being calculated is a function of other unknown variables at the same time.
Just like forward Euler method the error is proportional to step size
We want a technique with a great absolute stability that allows us to take a big “h” even when the situation is stiff.
The local error for this method is $O({h²})$
The method is first order accurate(i.e it quantifies the rate of convergence of a differential equation to the exact solution)
Conclusion
The Euler method is a first-order numerical strategy for solving ordinary differential equations (ODEs) with a given initial value in computational mathematics. To draw a conclusion we can say that it is the simplest Runge-Kutta technique and the most fundamental explicit approach of numerical integration of ordinary differential equations.
Reference
https://www.cs.usask.ca/~spiteri/M314/notes/AP/chap3.pdf
https://services.math.duke.edu/~holee/math361-2020/lectures/ode-2-euler.pdf
https://www.youtube.com/watch?v=cdUpAuGTIfE
Numerical Methods by S.A Molla
Numerical Methods for scientific and engineering computation by M.K Jain, R.K Jain, Iyenger
Till then, Bbye!! Take Care ❤
