2017年11月3日金曜日

学習環境

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。

dx dt =A e et ( e )cos( σt+α )+A e et ( sin( σt+α ) )σ =A e et ( ecos( σt+α )+σsin( σt+α ) ) d 2 x d t 2 =A( e et ( e )( ecos( σt+α )+σsin( σt+α ) )+ e et ( esin( σt+α )σ+σcos( σt+α )σ ) ) =A e et ( e 2 cos( σt+α )eσsin( σt+α )eσsin( σt+α )+ σ 2 cos( σt+α ) ) =A e et ( ( σ 2 e 2 )cos( σt+α )2eσsin( σt+α ) ) d 2 x d t 2 +2ϵ dx dt + n 2 x =A e et ( ( σ 2 e 2 )cos( σt+α )2eσsin( σt+α ) )2ϵA e et ( ecos( σt+α )+σsin( σt+α ) )+ n 2 x =A e et ( ( σ 2 e 2 )cos( σt+α )2eσsin( σt+α )+2 ϵ 2 cos( σt+α )+2ϵσsin( σt+α ) )+ n 2 x =A e et ( σ 2 + e 2 )cos( σt+α )+ n 2 x n 2 = σ 2 + ϵ 2 A e et ( σ 2 + e 2 )cos( σt+α )+ n 2 x =A e et ( σ 2 + e 2 )cos( σt+α )+( σ 2 + ϵ 2 )A e et cos( σt+α ) =0 d 2 x d t 2 +2ϵ dx dt + n 2 x=0

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();








0 コメント:

コメントを投稿