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55 lines
1.4 KiB
Python
55 lines
1.4 KiB
Python
from collections.abc import Callable
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import numpy as np
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def euler_modified(
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ode_func: Callable, y0: float, x0: float, step_size: float, x_end: float
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) -> np.ndarray:
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"""
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Calculate solution at each step to an ODE using Euler's Modified Method
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The Euler Method is straightforward to implement, but can't give accurate solutions.
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So, some changes were proposed to improve accuracy.
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https://en.wikipedia.org/wiki/Euler_method
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Arguments:
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ode_func -- The ode as a function of x and y
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y0 -- the initial value for y
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x0 -- the initial value for x
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stepsize -- the increment value for x
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x_end -- the end value for x
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>>> # the exact solution is math.exp(x)
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>>> def f1(x, y):
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... return -2*x*(y**2)
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>>> y = euler_modified(f1, 1.0, 0.0, 0.2, 1.0)
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>>> float(y[-1])
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0.503338255442106
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>>> import math
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>>> def f2(x, y):
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... return -2*y + (x**3)*math.exp(-2*x)
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>>> y = euler_modified(f2, 1.0, 0.0, 0.1, 0.3)
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>>> float(y[-1])
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0.5525976431951775
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"""
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n = int(np.ceil((x_end - x0) / step_size))
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y = np.zeros((n + 1,))
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y[0] = y0
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x = x0
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for k in range(n):
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y_get = y[k] + step_size * ode_func(x, y[k])
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y[k + 1] = y[k] + (
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(step_size / 2) * (ode_func(x, y[k]) + ode_func(x + step_size, y_get))
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)
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x += 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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