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.
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 $T_R$:
:: $\frac{dT(t)}{dt} = -k \, \Delta T$
or
:: $\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$
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
:: $T(t) = T_R + (T_0 - T_R) \; e^{-k t}$
Task:Implement a routine of Euler's method and then to use it to solve the given example of Newton's cooling law with it for three different step sizes of:
::* 2 s
::* 5 s and
::* 10 s
and to 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 ──► 100 s
eulersMethod is a function.
testString: assert(typeof eulersMethod === 'function', 'eulersMethod is a function.');
- text: eulersMethod(0, 100, 100, 10) should return a number.
testString: assert(typeof eulersMethod(0, 100, 100, 10) === 'number', 'eulersMethod(0, 100, 100, 10) should return a number.');
- text: eulersMethod(0, 100, 100, 10) should return 20.0424631833732.
testString: assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732, 'eulersMethod(0, 100, 100, 10) should return 20.0424631833732.');
- text: eulersMethod(0, 100, 100, 10) should return 20.01449963666907.
testString: assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907, 'eulersMethod(0, 100, 100, 10) should return 20.01449963666907.');
- text: eulersMethod(0, 100, 100, 10) should return 20.000472392.
testString: assert.equal(eulersMethod(0, 100, 100, 10), 20.000472392, 'eulersMethod(0, 100, 100, 10) should return 20.000472392.');
```