68 lines
1.5 KiB
Python
68 lines
1.5 KiB
Python
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import numpy as np
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import matplotlib.pyplot as plt
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from matplotlib.scale import FuncScale
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# f: (t, y) -> R, dy/dt = f
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def euler(f, a, b, y0, h):
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xs = [a]
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ys = [y0]
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x = a
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y = y0
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while x < b:
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y += h * f(x, y)
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x += h
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if abs(y) > threshold:
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break
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xs.append(x)
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ys.append(y)
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return xs, ys
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def improvedEuler(f, a, b, y0, h):
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xs = [a]
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ys = [y0]
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x = a
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y = y0
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while x <= b:
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slope1 = f(x, y)
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y_predictor = y + h * slope1
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slope2 = f(x + h, y_predictor)
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y = y + (h/2) * (slope1 + slope2)
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x = x + h
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if abs(y) > threshold:
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break
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xs.append(x)
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ys.append(y)
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return xs, ys
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f = lambda t, y: t**3 + y**3
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threshold = 100
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a = 0
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b = 2
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lims = [
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(0, 1.4, 1.8, 1, threshold),
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(1, 0.4, 0.6, 1, threshold),
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(-1, 0.4, 0.6, -threshold, -1),
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]
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y0, xm, xM, ym, yM = lims[0]
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fig, ax= plt.subplots(1, 2)
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for h in [0.1, 0.01, 0.001, 0.0001]:
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ax[0].plot(*euler(f, a, b, y0, h), label=f'$\\Delta t$={h}', linewidth=1)
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T, Y = improvedEuler(f, a, b, y0, h)
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ax[1].plot(T, Y, label=f'$\\Delta t$={h}', linewidth=1)
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print(f"Max x: {T[-1]}")
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for axe in ax:
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axe.legend()
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axe.set_xlim(xm, xM)
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axe.set_ylim(ym, yM)
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#axe.set_yscale(FuncScale(axe, (lambda y: np.log(-y), lambda y: -np.exp(y))))
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ax[0].set_title("Euler Method")
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ax[1].set_title("Improved Euler Method")
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#fig.suptitle(r"Numerically solving $y'=y^3 + t^3$ starting at (0, 0)")
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plt.show()
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