130 lines
4.3 KiB
Markdown
130 lines
4.3 KiB
Markdown
---
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title: Euler method
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id: 59880443fb36441083c6c20e
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challengeType: 5
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forumTopicId: 302258
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---
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## Description
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<section id='description'>
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Euler's method numerically approximates solutions of first-order ordinary differential equations (ODEs) with a given initial value. It is an explicit method for solving initial value problems (IVPs), as described in <a href="https://en.wikipedia.org/wiki/Euler method" title="wp: Euler method" target="_blank">the wikipedia page</a>.
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The ODE has to be provided in the following form:
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<ul style="list-style: none;">
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<li><big>$\frac{dy(t)}{dt} = f(t,y(t))$</big></li>
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</ul>
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with an initial value
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<ul style="list-style: none;">
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<li><big>$y(t_0) = y_0$</big></li>
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</ul>
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To get a numeric solution, we replace the derivative on the LHS with a finite difference approximation:
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<ul style="list-style: none;">
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<li><big>$\frac{dy(t)}{dt} \approx \frac{y(t+h)-y(t)}{h}$</big></li>
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</ul>
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then solve for $y(t+h)$:
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<ul style="list-style: none;">
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<li><big>$y(t+h) \approx y(t) + h \, \frac{dy(t)}{dt}$</big></li>
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</ul>
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which is the same as
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<ul style="list-style: none;">
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<li><big>$y(t+h) \approx y(t) + h \, f(t,y(t))$</big></li>
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</ul>
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The iterative solution rule is then:
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<ul style="list-style: none;">
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<li><big>$y_{n+1} = y_n + h \, f(t_n, y_n)$</big></li>
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</ul>
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where <big>$h$</big> is the step size, the most relevant parameter for accuracy of the solution. A smaller step size increases accuracy but also the computation cost, so it has always has to be hand-picked according to the problem at hand.
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<strong>Example: Newton's Cooling Law</strong>
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Newton's cooling law describes how an object of initial temperature <big>$T(t_0) = T_0$</big> cools down in an environment of temperature <big>$T_R$</big>:
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<ul style="list-style: none;">
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<li><big>$\frac{dT(t)}{dt} = -k \, \Delta T$</big></li>
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</ul>
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or
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<ul style="list-style: none;">
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<li><big>$\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$</big></li>
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</ul>
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It says that the cooling rate <big>$\frac{dT(t)}{dt}$</big> of the object is proportional to the current temperature difference <big>$\Delta T = (T(t) - T_R)$</big> to the surrounding environment.
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The analytical solution, which we will compare to the numerical approximation, is
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<ul style="list-style: none;">
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<li><big>$T(t) = T_R + (T_0 - T_R) \; e^{-k t}$</big></li>
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</ul>
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</section>
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## Instructions
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<section id='instructions'>
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Implement a routine of Euler's method and then use it to solve the given example of Newton's cooling law for three different step sizes of:
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<ul>
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<li><code>2 s</code></li>
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<li><code>5 s</code> and</li>
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<li><code>10 s</code></li>
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</ul>
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and compare with the analytical solution.
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<strong>Initial values:</strong>
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<ul>
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<li>initial temperature <big>$T_0$</big> shall be <code>100 °C</code></li>
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<li>room temperature <big>$T_R$</big> shall be <code>20 °C</code></li>
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<li>cooling constant <big>$k$</big> shall be <code>0.07</code></li>
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<li>time interval to calculate shall be from <code>0 s</code> to <code>100 s</code></li>
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</ul>
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First parameter to the function is initial time, second parameter is initial temperature, third parameter is elapsed time and fourth parameter is step size.
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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> should be a function.
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testString: assert(typeof eulersMethod === 'function');
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- text: <code>eulersMethod(0, 100, 100, 2)</code> should return a number.
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testString: assert(typeof eulersMethod(0, 100, 100, 2) === 'number');
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- text: <code>eulersMethod(0, 100, 100, 2)</code> should return 20.0424631833732.
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testString: assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732);
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- text: <code>eulersMethod(0, 100, 100, 5)</code> should return 20.01449963666907.
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testString: assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907);
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- text: <code>eulersMethod(0, 100, 100, 10)</code> should return 20.000472392.
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testString: assert.equal(eulersMethod(0, 100, 100, 10), 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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}
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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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function eulersMethod(x1, y1, x2, h) {
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let x = x1;
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let y = y1;
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while ((x < x2 && x1 < x2) || (x > x2 && x1 > x2)) {
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y += h * (-0.07 * (y - 20));
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x += h;
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}
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return y;
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}
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```
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</section>
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