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id	title	challengeType	forumTopicId	dashedName
59880443fb36441083c6c20e	Euler method	5	302258	euler-method

--description--

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 [the wikipedia page](https://en.wikipedia.org/wiki/Euler method "wp: Euler method").

The ODE has to be provided in the following form:

$\frac{dy(t)}{dt} = f(t,y(t))$

with an initial value

$y(t_0) = y_0$

To get a numeric solution, we replace the derivative on the LHS with a finite difference approximation:

$\frac{dy(t)}{dt} \approx \frac{y(t+h)-y(t)}{h}$

then solve for y(t+h):

$y(t+h) \approx y(t) + h \, \frac{dy(t)}{dt}$

which is the same as

$y(t+h) \approx y(t) + h \, f(t,y(t))$

The iterative solution rule is then:

$y_{n+1} = y_n + h \, f(t_n, y_n)$

where h 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.

Example: Newton's Cooling Law

Newton's cooling law describes how an object of initial temperature T(t_0) = T_0 cools down in an environment of temperature T_R:

$\frac{dT(t)}{dt} = -k \, \Delta T$

$\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$

It says that the cooling rate \\frac{dT(t)}{dt} of the object is proportional to the current temperature difference \\Delta T = (T(t) - T_R) to the surrounding environment.

The analytical solution, which we will compare to the numerical approximation, is

$T(t) = T_R + (T_0 - T_R) \; e^{-k t}$

--instructions--

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:

2 s
5 s and
10 s

and compare with the analytical solution.

Initial values:

initial temperature $T_0$ shall be 100 °C
room temperature $T_R$ shall be 20 °C
cooling constant $k$ shall be 0.07
time interval to calculate shall be from 0 s to 100 s

First parameter to the function is initial time, second parameter is initial temperature, third parameter is elapsed time and fourth parameter is step size.

--hints--

eulersMethod should be a function.

assert(typeof eulersMethod === 'function');

eulersMethod(0, 100, 100, 2) should return a number.

assert(typeof eulersMethod(0, 100, 100, 2) === 'number');

eulersMethod(0, 100, 100, 2) should return 20.0424631833732.

assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732);

eulersMethod(0, 100, 100, 5) should return 20.01449963666907.

assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907);

eulersMethod(0, 100, 100, 10) should return 20.000472392.

assert.equal(eulersMethod(0, 100, 100, 10), 20.000472392);

--seed--

--seed-contents--

function eulersMethod(x1, y1, x2, h) {

}

--solutions--

function eulersMethod(x1, y1, x2, h) {
  let x = x1;
  let y = y1;

  while ((x < x2 && x1 < x2) || (x > x2 && x1 > x2)) {
    y += h * (-0.07 * (y - 20));
    x += h;
  }

  return y;
}

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