defmodule Fluxion.ODE do
  @moduledoc """
  An Ordinary Differential Equation which can be used to solve initial value
  problems.

  @y'(t) = f(t, y(t)), t \\ge 0@

  ## Examples

  Define an ODE module for the equation @y'(t) = te^t@.

      iex> defmodule Exp do
      ...>   @behaviour Fluxion.ODE
      ...>   @impl true
      ...>   def f(t, y_t), do: t * :math.exp(t)
      ...> end
      iex> IO.puts(Exp.f(1, 2))
      :ok
  """

  @type t() :: module()

  @doc """
  Compute the derivative at a given point in time.
  e.g. compute @f(t, y\\_t)@ for a given value of @t@.
  """
  @callback f(t :: number(), y_t :: number()) :: number()

  @doc """
  Evaluate an ODE module with the provided time and value.

  ## Examples

      iex> defmodule Const do
      ...>   @behaviour Fluxion.ODE
      ...>   @impl true
      ...>   def f(t, y_t), do: 2.0
      ...> end
      iex> Fluxion.ODE.f!(Const, 10.0, -87.0)
      2.0

  """
  @spec f!(ode :: t(), t :: number, y_t :: number) :: number
  def f!(ode, t, y_t) do
    ode.f(t, y_t)
  end
end