<p>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 <ahref="https://en.wikipedia.org/wiki/Euler method"title="wp: Euler method">the wikipedia page</a>.</p><p>The ODE has to be provided in the following form:</p><p>:: <big>$\frac{dy(t)}{dt} = f(t,y(t))$</big></p><p>with an initial value</p><p>:: <big>$y(t_0) = y_0$</big></p><p>To get a numeric solution, we replace the derivative on the LHS with a finite difference approximation:</p><p>:: <big>$\frac{dy(t)}{dt} \approx \frac{y(t+h)-y(t)}{h}$</big></p><p>then solve for $y(t+h)$:</p><p>:: <big>$y(t+h) \approx y(t) + h \, \frac{dy(t)}{dt}$</big></p><p>which is the same as</p><p>:: <big>$y(t+h) \approx y(t) + h \, f(t,y(t))$</big></p><p>The iterative solution rule is then:</p><p>:: <big>$y_{n+1} = y_n + h \, f(t_n, y_n)$</big></p><p>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.</p>
<p>Example: Newton's Cooling Law</p><p>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>:</p><p>:: <big>$\frac{dT(t)}{dt} = -k \, \Delta T$</big></p>
<p>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.</p><p>The analytical solution, which we will compare to the numerical approximation, is</p>
<p>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:</p>
<p>::* 2 s</p>
<p>::* 5 s and </p>
<p>::* 10 s </p>
<p>and to compare with the analytical solution.</p>
Initial values:
<p>::* initial temperature <big>$T_0$</big> shall be 100 °C</p>
<p>::* room temperature <big>$T_R$</big> shall be 20 °C</p>
<p>::* cooling constant <big>$k$</big> shall be 0.07 </p>
<p>::* time interval to calculate shall be from 0 s ──► 100 s</p>