EULER FORWARD METHOD-NUMERICAL SOLUTIONS OF ODE OF 1st ORDER
As we have already observed there are many analytic methods present for finding the solution of any given equation. But we do find some (rather a large number) of ODEs whose results we fail to obtain by these analytic methods(for exact solutions). For such differential equation we seek support from numerical methods for approximate solutions using initial conditions(IVP).
Such problems gave birth to numerical methods namely Euler, Rung-Kutta, Taylor Series, Milne, Picard etc.
Euler’s method is the starting discussion whenever numerical analysis for IVPs come in the picture. Though, it is not the most accurate but ,by far, it is the simplest.
Note: We need to assume that the given equation (in the form of
with an initial condition
should satisfy Lipschitz conditions and the solution obtained must be unique(well-posedness).
Euler Forward Method
Also called Euler Method or Euler Explicit method.
It is a simple and single step numerical method where we obtain the solutions in the tabular form
**Discretization: Mesh or grid of variable(s)
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 yields our final formula for the Approximation,
Euler Forward Method depends on the step length “h”. Smaller the step length better is our approximation. But this method becomes very tedious at times to get the desired accuracy. To avoid such labour, Euler Modified Method or Euler Backward method comes in the picture.
***{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.
Euler Method is one of the fundamental numerical methods used for differential equations. This a reasonable and easy to follow derivation but it does not admit of much error analysis, it also gives room to lot of questions. I hope to cover some applications pretty soon so bear with me :) In the coming blog posts, I also plan to cover some terms I used in this blog in detail so don’t forget to follow & subscribe for email notifications!!
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 ❤
