61 lines
3.8 KiB
Markdown
61 lines
3.8 KiB
Markdown
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---
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title: Euler method
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id: 59880443fb36441083c6c20e
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challengeType: 5
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videoUrl: ''
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localeTitle: 欧拉方法
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---
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## Description
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<section id="description"><p>欧拉方法在数值上近似具有给定初始值的一阶常微分方程(ODE)的解。它是解决初始值问题(IVP)的一种显式方法,如<a href="https://en.wikipedia.org/wiki/Euler method" title="wp:欧拉方法">维基百科页面中所述</a> 。 </p><p> ODE必须以下列形式提供: </p><p> :: <big>$ \ frac {dy(t)} {dt} = f(t,y(t))$</big> </p><p>具有初始值</p><p> :: <big>$ y(t_0)= y_0 $</big> </p><p>为了得到数值解,我们用有限差分近似替换LHS上的导数: </p><p> :: <big>$ \ frac {dy(t)} {dt} \ approx \ frac {y(t + h)-y(t)} {h} $</big> </p><p>然后解决$ y(t + h)$: </p><p> :: <big>$ y(t + h)\ about y(t)+ h \,\ frac {dy(t)} {dt} $</big> </p><p>这是一样的</p><p> :: <big>$ y(t + h)\ about y(t)+ h \,f(t,y(t))$</big> </p><p>然后迭代解决方案规则是: </p><p> :: <big>$ y_ {n + 1} = y_n + h \,f(t_n,y_n)$</big> </p><p>其中<big>$ h $</big>是步长,是解决方案准确性最相关的参数。较小的步长会提高精度,但也会增加计算成本,因此必须根据手头的问题手工挑选。 </p><p>示例:牛顿冷却法</p><p> Newton的冷却定律描述了在温度<big>$ T_R $</big>的环境中初始温度<big>$ T(t_0)= T_0 $</big>的对象如何冷却: </p><p> :: <big>$ \ frac {dT(t)} {dt} = -k \,\ Delta T $</big> </p><p>要么</p><p> :: <big>$ \ frac {dT(t)} {dt} = -k \,(T(t) - T_R)$</big> </p><p>它表示物体的冷却速率<big>$ \ frac {dT(t)} {dt} $</big>与周围环境的当前温差<big>$ \ Delta T =(T(t) - T_R)$成正比</big> 。 </p><p>我们将与数值近似进行比较的解析解是</p><p> :: <big>$ T(t)= T_R +(T_0 - T_R)\; Ë^ { -克拉} $</big> </p>任务: <p>实现欧拉方法的一个例程,然后用它来解决牛顿冷却定律的给定例子,它有三种不同的步长: </p><p> :: * 2秒</p><p> :: * 5秒和</p><p> :: * 10秒</p><p>并与分析解决方案进行比较。 </p>初始值: <p> :: *初始温度<big>$ T_0 $</big>应为100°C </p><p> :: *室温<big>$ T_R $</big>应为20°C </p><p> :: *冷却常数<big>$ k $</big>应为0.07 </p><p> :: *计算的时间间隔应为0s──►100s </p></section>
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## Instructions
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<section id="instructions">
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</section>
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>eulersMethod</code>是一个函数。
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testString: 'assert(typeof eulersMethod === "function", "<code>eulersMethod</code> is a function.");'
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- text: '<code>eulersMethod(0, 100, 100, 10)</code>应该返回一个数字。'
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testString: 'assert(typeof eulersMethod(0, 100, 100, 10) === "number", "<code>eulersMethod(0, 100, 100, 10)</code> should return a number.");'
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- text: '<code>eulersMethod(0, 100, 100, 10)</code>应返回20.0424631833732。'
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testString: 'assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732, "<code>eulersMethod(0, 100, 100, 10)</code> should return 20.0424631833732.");'
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- text: '<code>eulersMethod(0, 100, 100, 10)</code>应返回20.01449963666907。'
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testString: 'assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907, "<code>eulersMethod(0, 100, 100, 10)</code> should return 20.01449963666907.");'
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- text: '<code>eulersMethod(0, 100, 100, 10)</code>应返回20.000472392。'
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testString: 'assert.equal(eulersMethod(0, 100, 100, 10), 20.000472392, "<code>eulersMethod(0, 100, 100, 10)</code> should return 20.000472392.");'
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function eulersMethod (x1, y1, x2, h) {
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// Good luck!
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}
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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// solution required
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```
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</section>
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