Integration and ODEs

Methods

====================================================== ============================== Method Description ====================================================== ============================== solveIvp_ (fun, t0, t_bound, y0, method, atol, rtol) Solve initial value problem ====================================================== ==============================

Description

.. _solveIvp:

.. function:: solveIvp(fun, t0, t_bound, y0, method, atol, rtol)

:param fun: (Float64, Array<Float64>, Int64) -> Float64, specifies the ODE
:param t0: initial time
:param t_bound: maximum time for solving ODE
:param y0: initial state
:param method: method for solving ODE, ``rk45`` or ``rk23``
:param atol: absolute error tolerance (currently unused)
:param rtol: relative error tolerance (currently unused)
:return: (Vector<Float64>, Array<Vector<Float64>>)

Solve initial value problem. Currently implements Runge-Kutta method (RK45 and RK23).

Example:

1. Solve the ODE :math:`y'(t) = y^2(t)\cos(t+y(t))`.

::

    func fun3(x: Float64, y: Array<Float64>, j: Int64): Float64 {
        return y[0] * y[0] * cos(x + y[0])
    }

    var (vec_x, vec_y) = solveIvp(
        fun3, 0.0, 100.0, Array<Float64>([0.2]), "rk45", 1e-3, 1e-6)
    plot(vec_x, vec_y[0])
    xlabel("t")
    ylabel("y(t)")
    title("Solution to ODE y'=y^2cos(t+y)")

Result:

.. image:: ../../tests/imgs/ode/rk_45.png
    :align: center
    :width: 480