Euler method and Runge Kutta 4 Burden example

euler.py

r"""
Euler method 5.9
w(0) = alpha
w(i + 1) = w(i) + h * f(t_i, w(i))
w(0 + 1) = w(0) + h * f(t_0, w(0))
t_0 =  a + 0 * h = a
t_i = a + i * h, i = 0,..., N
"""

a = 0
b = 2
h = 1 / 2
N = int((b - a) / h)
y0 = 0.5


def f(t, y):
    return y - t**2 + 1


w = y0  # w0

for i in range(1, N + 1):
    w = w + h * f(a + (i - 1) * h, w)
    print(w)

rk4.py

r"""
Runge Kutta 4
k1 = h * f(t, w)
k2 = h * f(t + h / 2, w + 1 / 2 * k1)
k3 = h * f(t + h / 2 , w + 1 / 2 * k2)
k4 = h * f(t + h, w + k3)
w = w + 1 / 6 * (k1 + 2*k2 + 2*k3 + k4)
"""

r""""
Punto medio (Runge Kutta 2)
w = w + h * f(t + h / 2, h / 2 * f(t, w))
"""
a = 0
b = 2
h = 1 / 5
N = int((b - a) / h)
y0 = 0.5


def f(t, y):
    return y - t**2 + 1


w = y0  # w0

for i in range(1, N + 1):
    t = a + (i - 1) * h
    k1 = h * f(t, w)
    k2 = h * f(t + h / 2, w + 1 / 2 * k1)
    k3 = h * f(t + h / 2, w + 1 / 2 * k2)
    k4 = h * f(t + h, w + k3)
    w = w + 1 / 6 * (k1 + 2 * k2 + 2 * k3 + k4)
    print(w)