euler method for simple motion
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Solving an ODE for simple motion using Euler method
intro
Euler method, which is also called forward Euler method, is a first-order numerical procedure for solving ordinary differential equation (ODE) as initial value problem (IVP) 1. This method can be described in only five lines of pseudocode using variables with index 2 or without index 3. And it is an an explicit method for solving IVP 4. The use of Euler method for some simple motions is discussed here.
ode
An ODE in the form of
$$\tag{1} \frac{dx(t)}{dt} = f(t, x(t)) $$
must be supplied with initial value
$$\tag{2} x(t_0) = x_0 $$
to be an IVP.
solution
With time step $\Delta t$, Eqn (1) will have
$$\tag{3} x(t + \Delta t) = x(t) + \Delta t f(t, x(t)). $$
as an iterative solution with Eqn (2) as the initial value. It is convenient to use indexed variables for Eqn (2) as follow
$$\tag{4} x_{n+1} = x_n + h f(t_n, x_n), $$
where $h = \Delta t$, $t_n = t_0 + nh$, $x_n = x(t_n)$.
notes
Sharad Bhardwaj, “Euler Method for solving differential equation”, GeeksforGeeks, 23 Nov 2022, url https://www.geeksforgeeks.org/euler-method-solving-differential-equation/ [20240423]. ↩︎
fCC, “Euler’s Method Explained with Examples”, freeCodeCamp, 26 Jan 2020, url https://www.freecodecamp.org/news/eulers-method-explained-with-examples/ [20240423]. ↩︎
Paul Dawkins, “Euler’s Method”, 16 Nov 2022, url https://tutorial.math.lamar.edu/classes/de/eulersmethod.aspx [20240423]. ↩︎
Rosetta Code contributors, “Euler method”, 14 Apr 2024, url https://rosettacode.org/wiki/?oldid=363664 [20240423]. ↩︎