7. Partial Differential Equations: Exercise

7. Partial Differential Equations: Exercise#

Name:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.integrate import odeint

1. Diffusion Equation#

For the diffusion equation with Dirichlet boundary condition, take initial states with different spatial frequencyes, such as $\( y(x, 0) = \sin(\frac{nx}{L}\pi ) \)\( with different \)n$, and see how quickly they decay in time.

L = 1
x = np.linspace(0, L)
n = 3
y = np.sin(n*x*np.pi/L)
plt.plot(x, y)

[<matplotlib.lines.Line2D at 0x116019ac0>]
_images/c1bafbdd020668b79e3ec035285ddc0f2aa315f0c512b99ad520ae9e422cae02.png

2. Wave Equation#

While the wave equation with Dirichlet boundary condition simulates oscillation of a string, that with Neumann condition $\( \left.\frac{\partial y(x,t)}{\partial x}\right|_{x_0}=\left.\frac{\partial y(x,t)}{\partial x}\right|_{x_N}=0 \)$ can simulate water wave.

  1. Implement a wave equation with a decay term $\( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} - d \frac{\partial u}{\partial t} \)$ with the Neumann boundary conditions and see how the wave ripples.

  1. See how the waves vary with the initial condition or stimulum.

Optional: Wave equation in 2D#

Try simulating waves in a 2D space with a cyclic boundary condition.

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