07_PDE_Ex.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 7. Partial Differential Equations: Exercise\n",
    "\n",
    "Name: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from scipy.integrate import odeint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Diffusion Equation\n",
    "For the diffusion equation with Dirichlet boundary condition, take initial states with different spatial frequencyes, such as \n",
    "\n",
    "$$ y(x, 0) = \\sin(\\frac{nx}{L}\\pi ) $$\n",
    "\n",
    "with different $n$, and see how quickly they decay in time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fdbe8eb4070>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "L = 1\n",
    "x = np.linspace(0, L)\n",
    "n = 3\n",
    "y = np.sin(n*x*np.pi/L)\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Wave Equation\n",
    "\n",
    "While the wave equation with Dirichlet boundary condition simulates oscillation of a string, that with Neumann condition \n",
    "\n",
    "$$ \\left.\\frac{\\partial y(x,t)}{\\partial x}\\right|_{x_0}=\\left.\\frac{\\partial y(x,t)}{\\partial x}\\right|_{x_N}=0 $$\n",
    "\n",
    "can simulate water wave.\n",
    "\n",
    "1) Implement a wave equation with a decay term\n",
    "\n",
    "$$ \\frac{\\partial^2 u}{\\partial t^2} = c^2 \\frac{\\partial^2 u}{\\partial x^2} - d \\frac{\\partial u}{\\partial t} $$\n",
    "\n",
    "with the Neumann boundary conditions and see how the wave ripples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "2) See how the waves vary with the initial condition or stimulum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optional: Wave equation in 2D\n",
    "Try simulating waves in a 2D space with a cyclic boundary condition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.2"
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 },
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}