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91 lines
2.4 KiB
Python
91 lines
2.4 KiB
Python
"""
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Use the Runge-Kutta-Gill's method of order 4 to solve Ordinary Differential Equations.
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https://www.geeksforgeeks.org/gills-4th-order-method-to-solve-differential-equations/
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Author : Ravi Kumar
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"""
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from collections.abc import Callable
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from math import sqrt
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import numpy as np
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def runge_kutta_gills(
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func: Callable[[float, float], float],
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x_initial: float,
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y_initial: float,
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step_size: float,
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x_final: float,
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) -> np.ndarray:
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"""
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Solve an Ordinary Differential Equations using Runge-Kutta-Gills Method of order 4.
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args:
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func: An ordinary differential equation (ODE) as function of x and y.
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x_initial: The initial value of x.
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y_initial: The initial value of y.
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step_size: The increment value of x.
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x_final: The final value of x.
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Returns:
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Solution of y at each nodal point
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>>> def f(x, y):
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... return (x-y)/2
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>>> y = runge_kutta_gills(f, 0, 3, 0.2, 5)
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>>> float(y[-1])
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3.4104259225717537
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>>> def f(x,y):
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... return x
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>>> y = runge_kutta_gills(f, -1, 0, 0.2, 0)
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>>> y
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array([ 0. , -0.18, -0.32, -0.42, -0.48, -0.5 ])
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>>> def f(x, y):
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... return x + y
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>>> y = runge_kutta_gills(f, 0, 0, 0.2, -1)
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Traceback (most recent call last):
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...
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ValueError: The final value of x must be greater than initial value of x.
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>>> def f(x, y):
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... return x
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>>> y = runge_kutta_gills(f, -1, 0, -0.2, 0)
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Traceback (most recent call last):
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...
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ValueError: Step size must be positive.
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"""
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if x_initial >= x_final:
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raise ValueError(
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"The final value of x must be greater than initial value of x."
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)
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if step_size <= 0:
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raise ValueError("Step size must be positive.")
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n = int((x_final - x_initial) / step_size)
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y = np.zeros(n + 1)
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y[0] = y_initial
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for i in range(n):
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k1 = step_size * func(x_initial, y[i])
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k2 = step_size * func(x_initial + step_size / 2, y[i] + k1 / 2)
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k3 = step_size * func(
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x_initial + step_size / 2,
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y[i] + (-0.5 + 1 / sqrt(2)) * k1 + (1 - 1 / sqrt(2)) * k2,
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)
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k4 = step_size * func(
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x_initial + step_size, y[i] - (1 / sqrt(2)) * k2 + (1 + 1 / sqrt(2)) * k3
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)
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y[i + 1] = y[i] + (k1 + (2 - sqrt(2)) * k2 + (2 + sqrt(2)) * k3 + k4) / 6
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x_initial += step_size
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return y
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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