学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.4(簡単な微分方程式)、2階微分方程式、例(数直線上を運動する点、速度に比例する抵抗、減衰振動)を取り組んでみる。
よって、関数は微分方程式を満たす。
コード(Emacs)
Python 3
#!/usr/bin/env python3 from sympy import pprint, symbols, sqrt, exp, cos, Derivative t, n, ε, A, α, σ = symbols('t, n, ε, A, α, σ') σ0 = sqrt(n ** 2 - ε ** 2) x = A * exp(-ε * t) * cos(σ * t + α) D1 = Derivative(x, t, 1) D2 = Derivative(x, t, 2) x1 = D1.doit() x2 = D2.doit() for s in [D1, x1, D2, x2]: pprint(s.factor()) print() eq = x2 + 2 * ε * x1 + n ** 2 * x print(eq.subs({σ: σ0}).expand() == 0)
入出力結果(Terminal, Jupyter(IPython))
$ ./sample02.py ∂ ⎛ -t⋅ε ⎞ ──⎝A⋅ℯ ⋅cos(t⋅σ + α)⎠ ∂t -t⋅ε -A⋅(ε⋅cos(t⋅σ + α) + σ⋅sin(t⋅σ + α))⋅ℯ 2 ∂ ⎛ -t⋅ε ⎞ ───⎝A⋅ℯ ⋅cos(t⋅σ + α)⎠ 2 ∂t ⎛ 2 2 ⎞ -t⋅ε A⋅⎝ε ⋅cos(t⋅σ + α) + 2⋅ε⋅σ⋅sin(t⋅σ + α) - σ ⋅cos(t⋅σ + α)⎠⋅ℯ True $
HTML5
<div id="graph0"></div> <pre id="output0"></pre> <label for="r0">r = </label> <input id="r0" type="number" min="0" value="0.5"> <label for="dx">dx = </label> <input id="dx" type="number" min="0" step="0.0001" value="0.005"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-10"> <label for="x2">x2 = </label> <input id="x2" type="number" value="10"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-10"> <label for="y2">y2 = </label> <input id="y2" type="number" value="10"> <br> <label for="n0">n = </label> <input id="n0" type="number" min="0" value="1"> <label for="ε0">ε = </label> <input id="ε0" type="number" min="0" value="0.1"> <label for="a0">A = </label> <input id="a0" type="number" value="1"> <label for="α0">α = </label> <input id="α0" type="number" value="2"> <button id="draw0">draw</button> <button id="clear0">clear</button> <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script> <script src="sample02.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'), pre0 = document.querySelector('#output0'), width = 600, height = 600, padding = 50, btn0 = document.querySelector('#draw0'), btn1 = document.querySelector('#clear0'), input_r = document.querySelector('#r0'), input_dx = document.querySelector('#dx'), input_x1 = document.querySelector('#x1'), input_x2 = document.querySelector('#x2'), input_y1 = document.querySelector('#y1'), input_y2 = document.querySelector('#y2'), input_n0 = document.querySelector('#n0'), input_ε0 = document.querySelector('#ε0'), input_a0 = document.querySelector('#a0'), input_α0 = document.querySelector('#α0'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_n0, input_a0, input_α0], p = (x) => pre0.textContent += x + '\n', range = (start, end, step=1) => { let res = []; for (let i = start; i < end; i += step) { res.push(i); } return res; }; let draw = () => { pre0.textContent = ''; let r = parseFloat(input_r.value), dx = parseFloat(input_dx.value), x1 = parseFloat(input_x1.value), x2 = parseFloat(input_x2.value), y1 = parseFloat(input_y1.value), y2 = parseFloat(input_y2.value), n0 = parseFloat(input_n0.value), ε0 = parseFloat(input_ε0.value), a0 = parseFloat(input_a0.value), α0 = parseFloat(input_α0.value); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return; } let points = [], lines = [], σ = Math.sqrt(n0 ** 2 - ε0 ** 2), f = (x) => a0 * Math.exp(-ε0 * x) * Math.cos(σ * x + α0), f1 = (x) => -a0 * Math.exp(-ε0 * x) * (Math.E * Math.cos(σ * x + α0) + σ * Math.sin(σ * x + α0)), f2 = (x) => -a0 * Math.exp(-ε0 * x) * (σ ** 2 + ε0 ** 2) * Math.cos(σ * x + α0) + n0 ** 2 + f(x), fns = [[f, 'red'], [f1, 'green'], [f2, 'blue']], fns1 = [], fns2 = []; fns.forEach((o) => { let [fn, color] = o; for (let x = x1; x <= x2; x += dx) { let y = fn(x); if (Math.abs(y) < Infinity) { points.push([x, y, color]); } } }); fns1.forEach((o) => { let [fn, color] = o; lines.push([x1, fn(x1), x2, fn(x2), color]); }); fns2.forEach((o) => { let [fn, color] = o; for (let x = x1; x <= x2; x += dx0) { let g = fn(x); lines.push([x1, g(x1), x2, g(x2), color]); } }); let xscale = d3.scaleLinear() .domain([x1, x2]) .range([padding, width - padding]); let yscale = d3.scaleLinear() .domain([y1, y2]) .range([height - padding, padding]); let xaxis = d3.axisBottom().scale(xscale); let yaxis = d3.axisLeft().scale(yscale); div0.innerHTML = ''; let svg = d3.select('#graph0') .append('svg') .attr('width', width) .attr('height', height); svg.selectAll('line') .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines)) .enter() .append('line') .attr('x1', (d) => xscale(d[0])) .attr('y1', (d) => yscale(d[1])) .attr('x2', (d) => xscale(d[2])) .attr('y2', (d) => yscale(d[3])) .attr('stroke', (d) => d[4] || 'black'); svg.selectAll('circle') .data(points) .enter() .append('circle') .attr('cx', (d) => xscale(d[0])) .attr('cy', (d) => yscale(d[1])) .attr('r', r) .attr('fill', (d) => d[2] || 'green'); svg.append('g') .attr('transform', `translate(0, ${height - padding})`) .call(xaxis); svg.append('g') .attr('transform', `translate(${padding}, 0)`) .call(yaxis); [fns, fns1, fns2].forEach((fs) => p(fs.join('\n'))); }; inputs.forEach((input) => input.onchange = draw); btn0.onclick = draw; btn1.onclick = () => pre0.textContent = ''; draw();
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