fluxion v0.2.0 Fluxion.ODE.Euler View Source

The Euler method for solving the initial value problem for ordinary differential equations of the form @y’(t) = f(t, y(t))@.

Euler’s method defines a series of @y@ values according to the following:

@y_{k+1} = y_k + hf(t_k, y_k)@ with @k = 0, 1, …, n@

In this module, the value for @y_0@ will be referred to as “initial” and the value for @h@ will be referred to as the “stride”.

The stride is a crucial parameter for Euler’s method. Euler’s method is convergent because as @h \to 0@ the series of values @y_k@ approaches @y(t_k)@.

Convergence rate is @O(h)@ and numeric error accumulates as @O(1/h)@, so properly selected stride is important to balance convergence and numeric error.

Link to this section Summary

Functions

step(ode, initial, stride, n \\ 1)

Step Euler’s solution @n@ times

Link to this section Functions

step(ode, initial, stride, n \\ 1)

step(
  ode :: Fluxion.ODE.t(),
  initial :: number(),
  stride :: number(),
  n :: number()
) :: number()

Step Euler’s solution @n@ times.

Example

For the purpose of example, start from a known function for @y@ like @y(t) = sin(t)@. This gives @y’(t) = cos(t)@. Given this, it’s possible to approximate the real solution with Euler’s method.

iex> defmodule TSq do
...>   @behaviour Fluxion.ODE
...>   @impl true
...>   def f(x, _y_k), do: :math.cos(x)
...> end
iex> steps = 10
iex> stride = 0.01
iex> # Compute the real value of y(t) after 10 steps
iex> real_y_1 = :math.sin(stride*steps)
iex> # Approximate the value of y(t) after 10 steps
iex> approx_y_1 = Fluxion.ODE.Euler.step(TSq, 0 , stride, steps)
iex> # Compare the results
iex> Fluxion.Math.is_equal(real_y_1, approx_y_1, 1.0e-3)
:true