464 lines
13 KiB
Plaintext
464 lines
13 KiB
Plaintext
{
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"cells": [
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"cell_type": "markdown",
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"metadata": {
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}
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"source": [
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"# CDS: Numerical Methods Assignments\n",
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"\n",
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"- See lecture notes and documentation on Brightspace for Python and Jupyter basics. If you are stuck, try to google or get in touch via Discord.\n",
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"\n",
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"- Solutions must be submitted via the Jupyter Hub.\n",
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"\n",
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"- Make sure you fill in any place that says `YOUR CODE HERE` or \"YOUR ANSWER HERE\".\n",
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"\n",
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"## Submission\n",
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"\n",
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"1. Name all team members in the the cell below\n",
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"2. make sure everything runs as expected\n",
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"3. **restart the kernel** (in the menubar, select Kernel$\\rightarrow$Restart)\n",
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"4. **run all cells** (in the menubar, select Cell$\\rightarrow$Run All)\n",
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"5. Check all outputs (Out[\\*]) for errors and **resolve them if necessary**\n",
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"6. submit your solutions **in time (before the deadline)**"
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]
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},
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{
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"cell_type": "raw",
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"metadata": {},
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"source": [
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"team_members = \"\""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "62fbcf7aaa9e165abd27319c5bd35555",
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"grade": false,
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"grade_id": "cell-e9dc85dd9ad77a5e",
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"locked": true,
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"solution": false,
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}
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},
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"source": [
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"## Dynamic Behavior from Hyperbolic PDEs\n",
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"\n",
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"In the following you will derive and implement finite-difference methods to study generalized hyperbolic partial differential equations."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "910692c2f4b93f251a3a128caf2b0c37",
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"grade": true,
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"grade_id": "cell-acb87410eaef9fe1",
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"locked": false,
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"points": 0,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# Import packages here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "7238a56701aaf868838ffdb706f5eb8b",
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"grade": false,
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"grade_id": "cell-a27c54a734bf2232",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"### Task 1: finite-difference hyperbolic PDE solver\n",
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"\n",
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"Our aim is to implement a Python function to find the solution of the following PDE:\n",
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"\n",
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"\\begin{align*}\n",
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" \\frac{\\partial^2}{\\partial t^2} u(x,t)\n",
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" - \\alpha^2 \\frac{\\partial^2}{\\partial x^2} u(x,t) = 0,\n",
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" \\qquad\n",
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" 0 \\leq x \\leq l,\n",
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" \\qquad\n",
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" 0 \\leq t\n",
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"\\end{align*}\n",
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"\n",
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"with the boundary conditions\n",
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"\n",
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"\\begin{align*}\n",
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" u(0,t) = u(l,t) &= 0, \n",
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" &&\\text{for } t > 0 \\\\\n",
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" u(x,0) &= f(x) \\\\\n",
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" \\frac{\\partial}{\\partial t} u(x,0) &= g(x)\n",
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" &&\\text{for } 0 \\leq x \\leq l.\n",
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"\\end{align*}\n",
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" \n",
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"By approximating the partial derivatives with finite differences we can recast the problem into the following form\n",
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"\t\t\n",
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"\\begin{align*}\n",
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" \\vec{w}_{j+1} = \\mathbf{A} \\vec{w}_{j}\n",
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" - \\vec{w}_{j-1}.\n",
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"\\end{align*}\n",
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"\n",
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"Here $\\vec{w}_j$ is a vector of length $m$ in the discretized spatial coordinate $x_i$ at time step $t_j$. The spatial coordinates are defined as $x_i = i h$, where $i=0,1,\\dots,m-1$ and $h = l/(m-1)$. The time steps are defined as $t_j = j k$, where $j = 0, 1, \\dots, n-1$.\n",
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"\n",
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"The tri-diagonal matrix $\\mathbf{A}$ has size $(m-2)\\times(m-2)$ and is defined by\n",
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"\n",
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"\\begin{align*}\n",
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" \\mathbf{A} =\n",
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" \\left( \\begin{array}{cccc}\n",
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" 2(1-\\lambda^2) & \\lambda^2 & & 0\\\\\n",
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" \\lambda^2 & \\ddots & \\ddots & \\\\\n",
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" & \\ddots & \\ddots & \\lambda^2 \\\\\n",
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" 0 & & \\lambda^2 & 2(1-\\lambda^2) \n",
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" \\end{array} \\right),\n",
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"\\end{align*}\n",
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" \n",
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"where $\\lambda = \\alpha k / h$. This $(m-2)\\times(m-2)$ structure accounts for the first set of boundary conditions. Note that the product $\\mathbf{A} \\vec{w}_{j}$ is thus only performed over the $m-2$ subset, i.e. $i=1,2,\\dots,m-2$. The other boundary conditions are accounted for by initializing the first two time steps with\n",
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"\n",
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"\\begin{align*}\n",
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" w_{i,j=0} &= f(x_i) \\\\\n",
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" w_{i,j=1} &= (1-\\lambda^2) f(x_i)\n",
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" + \\frac{\\lambda^2}{2} f(x_{i+1})\n",
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" + \\frac{\\lambda^2}{2} f(x_{i-1})\n",
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" + kg(x_i).\n",
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"\\end{align*}\n",
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" \n",
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"Implement a Python function of the form $\\text{pdeHyperbolic(a, x, t, f, g)}$, where $\\text{a}$ represents the PDE parameter $\\alpha$, $\\text{x}$ and $\\text{t}$ are the discretized spatial and time grids, and $\\text{f}$ and $\\text{g}$ are the functions defining the boundary conditions. This function should return a two-dimensional array $\\text{w[:,:]}$, which stores the spatial vector $\\vec{w}_j$ at each time coordinate $t_j$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "95d463713a216d57cbf814a0f40a482d",
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"grade": true,
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"grade_id": "cell-1f9655333f2dc3cb",
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"locked": false,
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"points": 3,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"def pdeHyperbolic(a, x, t, f, g):\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "d5ee65aba16eef98296a6a66af3c0888",
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"grade": false,
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"grade_id": "cell-047f2dfab5489a88",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"### Task 2 \n",
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"\n",
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"Use your implementation to solve the following problems. Compare in the first problem your numerical solution to the analytic one and use it to debug your code.\n",
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"\n",
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"#### Problem 1:\n",
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"\\begin{align*}\n",
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" \\frac{\\partial^2}{\\partial t^2} u(x,t)\n",
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" - \\frac{\\partial^2}{\\partial x^2} u(x,t) &= 0,\n",
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" &&\\text{for }0 \\leq x \\leq 1 \\text{ and } 0 \\leq t \\leq 1\\\\\n",
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" u(0,t) = u(l,t) &= 0, \n",
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" &&\\text{for } t > 0 \\\\\n",
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" u(x,0) &= \\operatorname{sin}\\left(2 \\pi x \\right), \\\\\n",
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" \\frac{\\partial}{\\partial t} u(x,0) &= 2 \\pi \\operatorname{sin}\\left(2 \\pi x \\right)\n",
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" &&\\text{for } 0 \\leq x \\leq 1.\n",
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"\\end{align*}\n",
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"\n",
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"Compare your numerical solution to the analytic one and use it to debug your code. The corresponding analytic solution is\n",
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"\n",
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"\\begin{equation}\n",
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" u(x,t) = \\operatorname{sin}(2 \\pi x) (\\operatorname{cos}(2 \\pi t) + \\operatorname{sin}(2 \\pi t)).\n",
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"\\end{equation}\n",
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"\n",
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"Then write a unit test for your function based on this problem."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "17a4067a37c5edfaffd6ebffa47942d6",
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"grade": true,
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"grade_id": "cell-8b8cb282dfe95a03",
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"locked": false,
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"points": 0,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# Do your own testing here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "fbb85c41f7de70517abcdc482a875fd5",
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"grade": true,
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"grade_id": "cell-e83b24067743e433",
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"locked": false,
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"points": 3,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"def test_pdeHyperbolic():\n",
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" # YOUR CODE HERE\n",
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" raise NotImplementedError()\n",
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" \n",
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"test_pdeHyperbolic()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "42880f6025ac92dc08f1c44a5bbf4312",
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"grade": false,
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"grade_id": "cell-99399435c164c96d",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"#### Problem 2:\n",
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"\n",
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"\\begin{align*}\n",
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" \\frac{\\partial^2}{\\partial t^2} u(x,t)\n",
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" - \\frac{\\partial^2}{\\partial x^2} u(x,t) &= 0,\n",
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" &&\\text{for }0 \\leq x \\leq 1 \\text{ and } 0 \\leq t \\leq 2\\\\\n",
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" u(0,t) = u(l,t) &= 0, \n",
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" &&\\text{for } t > 0 \\\\\n",
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" u(x,0) &= \\left\\{ \\begin{array}{cc} +1 & \\text{for } x<0.5 \\\\ -1 & \\text{for } x \\geq 0.5 \\end{array} \\right., \\\\\n",
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" \\frac{\\partial}{\\partial t} u(x,0) &= 0\n",
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" &&\\text{for } 0 \\leq x \\leq 1.\n",
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"\\end{align*}\n",
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"\n",
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"Use $m=200$ and $n=400$ to discretize the spatial and time grids, respectively."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "1b6e620f32a96a23c736b827e282bc6b",
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"grade": true,
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"grade_id": "cell-ea865d2efbc574d6",
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"locked": false,
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"points": 2,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"deletable": false,
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"editable": false,
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"nbgrader": {
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"cell_type": "markdown",
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"checksum": "3bb27614fe36d18302f4ace6442c67d9",
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"grade": false,
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"grade_id": "cell-a783908a0db32a24",
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"locked": true,
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"schema_version": 3,
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"solution": false,
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"task": false
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}
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},
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"source": [
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"### Task 3\n",
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"\n",
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"Animate your solutions! To this end you can use the following code:\n",
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"\n",
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"```python\n",
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"\n",
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"# use matplotlib's animation package\n",
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"import matplotlib.pylab as plt\n",
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"import matplotlib\n",
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"import matplotlib.animation as animation\n",
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"# set the animation style to \"jshtml\" (for the use in Jupyter)\n",
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"matplotlib.rcParams['animation.html'] = 'jshtml'\n",
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"\n",
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"# create a figure for the animation\n",
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"fig = plt.figure()\n",
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"plt.grid(True)\n",
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"plt.xlim( ... ) # fix x limits\n",
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"plt.ylim( ... ) # fix y limits\n",
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"\n",
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"# Create an empty plot object and prevent its showing (we will fill it each frame)\n",
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"myPlot, = plt.plot([0], [0])\n",
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"plt.close()\n",
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"\n",
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"# This function is called each frame to generate the animation (f is the frame number)\n",
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"def animate(f): \n",
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" myPlot.set_data( ... ) # update plot\n",
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"\n",
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"# Show the animation\n",
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"frames = np.arange(1, np.size(t)) # t is the time grid here\n",
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"myAnimation = animation.FuncAnimation(fig, animate, frames, interval = 20)\n",
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"myAnimation\n",
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"\n",
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"```"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
|
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"deletable": false,
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"nbgrader": {
|
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"cell_type": "code",
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"checksum": "5fa181b7e8df93bd0dd35613bc553701",
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"grade": true,
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"grade_id": "cell-0bd05d93759bd652",
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"locked": false,
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"points": 3,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# Animate problem 1 here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"deletable": false,
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"nbgrader": {
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"cell_type": "code",
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"checksum": "d4f93f8615bd3576e0248a27d5c1d664",
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"grade": true,
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"grade_id": "cell-43b3222743c428f5",
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"locked": false,
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"points": 0,
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"schema_version": 3,
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"solution": true,
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"task": false
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}
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},
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"outputs": [],
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"source": [
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"# Animate problem 2 here ...\n",
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"\n",
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"# YOUR CODE HERE\n",
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"raise NotImplementedError()"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.10"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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