Heat Equation (1D)

import os
import numpy as np
import matplotlib.pyplot as plt
os.chdir(r'D:\projects\wordpress\ex41')
os.getcwd()
N = 5
u0 = 1
u1 = 2
dx = 1.0 / (N + 1)
A = (np.eye(N, k=-1) - 2 * np.eye(N) + np.eye(N, k=1)) / dx**2
print(A)
d = -5 * np.ones(N)
d[0] -= u0 / dx**2
d[N-1] -= u1 / dx**2
u = np.linalg.solve(A, d)
x = np.linspace(0, 1, N+2)
U = np.hstack([[u0], u, [u1]])
fig, ax = plt.subplots(figsize=(8, 4))
ax.plot(x, U)
ax.plot(x[1:-1], u, 'ks')
ax.set_xlim(0, 1)
ax.set_xlabel(r"$x$", fontsize=18)
ax.set_ylabel(r"$u(x)$", fontsize=18)
fig.savefig("ch11-fdm-1d.pdf")
fig.tight_layout()
plt.savefig("example41.png", dpi=100)
plt.show()
plt.close()

<img class="  wp-image-201 aligncenter" src="https://gantovnik.com/bio-tips/wp-content/uploads/2019/01/example41.png" alt="example41" width="644" height="322" />

import os

import numpy as np

import matplotlib.pyplot as plt

os.chdir(r'D:\projects\wordpress\ex41')

os.getcwd()

N = 5

u0 = 1

u1 = 2

dx = 1.0 / (N + 1)

A = (np.eye(N, k=-1) - 2 * np.eye(N) + np.eye(N, k=1)) / dx**2

print(A)

d = -5 * np.ones(N)

d[0] -= u0 / dx**2

d[N-1] -= u1 / dx**2

u = np.linalg.solve(A, d)

x = np.linspace(0, 1, N+2)

U = np.hstack([[u0], u, [u1]])

fig, ax = plt.subplots(figsize=(8, 4))

ax.plot(x, U)

ax.plot(x[1:-1], u, 'ks')

ax.set_xlim(0, 1)

ax.set_xlabel(r"$x$", fontsize=18)

ax.set_ylabel(r"$u(x)$", fontsize=18)

fig.savefig("ch11-fdm-1d.pdf")

fig.tight_layout()

plt.savefig("example41.png", dpi=100)

plt.show()

plt.close()

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Heat equation (1D)

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