added rkf45 method (#10438)

* added rkf45 method

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Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

maths/rkf45.py

@@ -0,0 +1,112 @@
+"""
+Use the Runge-Kutta-Fehlberg method to solve Ordinary Differential Equations.
+"""
+
+from collections.abc import Callable
+
+import numpy as np
+
+
+def runge_futta_fehlberg_45(
+    func: Callable,
+    x_initial: float,
+    y_initial: float,
+    step_size: float,
+    x_final: float,
+) -> np.ndarray:
+    """
+    Solve an Ordinary Differential Equations using Runge-Kutta-Fehlberg Method (rkf45)
+    of order 5.
+
+    https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method
+
+    args:
+    func: An ordinary differential equation (ODE) as function of x and y.
+    x_initial: The initial value of x.
+    y_initial: The initial value of y.
+    step_size: The increment value of x.
+    x_final: The final value of x.
+
+    Returns:
+        Solution of y at each nodal point
+
+    # exact value of y[1] is tan(0.2) = 0.2027100937470787
+    >>> def f(x, y):
+    ...     return 1 + y**2
+    >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, 1)
+    >>> y[1]
+    0.2027100937470787
+    >>> def f(x,y):
+    ...     return x
+    >>> y = runge_futta_fehlberg_45(f, -1, 0, 0.2, 0)
+    >>> y[1]
+    -0.18000000000000002
+    >>> y = runge_futta_fehlberg_45(5, 0, 0, 0.1, 1)
+    Traceback (most recent call last):
+        ...
+    TypeError: 'int' object is not callable
+    >>> def f(x, y):
+    ...     return x + y
+    >>> y = runge_futta_fehlberg_45(f, 0, 0, 0.2, -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: The final value x must be greater than initial value of x.
+    >>> def f(x, y):
+    ...     return x
+    >>> y = runge_futta_fehlberg_45(f, -1, 0, -0.2, 0)
+    Traceback (most recent call last):
+        ...
+    ValueError: Step size must be positive.
+    """
+    if x_initial >= x_final:
+        raise ValueError("The final value x must be greater than initial value of x.")
+
+    if step_size <= 0:
+        raise ValueError("Step size must be positive.")
+
+    n = int((x_final - x_initial) / step_size)
+    y = np.zeros(
+        (n + 1),
+    )
+    x = np.zeros(n + 1)
+    y[0] = y_initial
+    x[0] = x_initial
+    for i in range(n):
+        k1 = step_size * func(x[i], y[i])
+        k2 = step_size * func(x[i] + step_size / 4, y[i] + k1 / 4)
+        k3 = step_size * func(
+            x[i] + (3 / 8) * step_size, y[i] + (3 / 32) * k1 + (9 / 32) * k2
+        )
+        k4 = step_size * func(
+            x[i] + (12 / 13) * step_size,
+            y[i] + (1932 / 2197) * k1 - (7200 / 2197) * k2 + (7296 / 2197) * k3,
+        )
+        k5 = step_size * func(
+            x[i] + step_size,
+            y[i] + (439 / 216) * k1 - 8 * k2 + (3680 / 513) * k3 - (845 / 4104) * k4,
+        )
+        k6 = step_size * func(
+            x[i] + step_size / 2,
+            y[i]
+            - (8 / 27) * k1
+            + 2 * k2
+            - (3544 / 2565) * k3
+            + (1859 / 4104) * k4
+            - (11 / 40) * k5,
+        )
+        y[i + 1] = (
+            y[i]
+            + (16 / 135) * k1
+            + (6656 / 12825) * k3
+            + (28561 / 56430) * k4
+            - (9 / 50) * k5
+            + (2 / 55) * k6
+        )
+        x[i + 1] = step_size + x[i]
+    return y
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()