Exercises on Initial Value Problems for Ordinary Differential Equations¶
Exercise 1¶
Show that for the integration case \(f(t, u) = f(t)\), Euler’s method is the same as the composite left-hand endpoint rule, as in the section Definite Integrals, Part 2.
Exercise 2¶
A) Verify that for the simple case where \(f(t, u) = f(t)\), the explicit trapezoid method gives the same result as the composite trapezoid rule for integration.
B) Do one step of this method for the canonical example \(du/dt = ku\), \(u(t_0) = u_0\). It will have the form \(U_1 = G U_0\) where the growth factor \(G\) approximates the factor \(g=e^{kh}\) for the exact solution \(u(t_1) = g u(t_0)\) of the ODE.
C) Compare to \(G=1+kh\) seen for Euler’s method.
D) Use the previous result to express \(U_i\) in terms of \(U_0=u_0\), as done for Euler’s method.
Exercise 3 (a lot like Exercise 2)¶
A) Verify that for the simple case where \(f(t, u) = f(t)\), explicit midpoint method gives the same result as the composite midpoint rule for integration (same comment as above).
B) Do one step of this method for the canonical example \(du/dt = ku\), \(u(t_0) = u_0\). It will have the form \(U_1 = G U_0\) where the growth factor \(G\) approximates the factor \(g=e^{kh}\) for the exact solution \(u(t_1) = g u(t_0)\) of the ODE.
C) Compare to the growth factors \(G\) seen for Euler and explicit trapezoid methods, and to the growth factor \(g\) for the exact solution.
Exercise 4¶
A) Apply Richardson extrapolation to one step of Euler’s method, using the values given by step sizes \(h\) and \(h/2\).
B) This should give a second order accurate method, so compare it to the above two methods.
Exercise 5¶
A) Verify that for the simple case where \(f(t, u) = f(t)\), this gives the same result as the Composite Simpson’s Rule for integration.
B) Do one step of this method for the canonical example \(du/dt = ku\), \(u(t_0) = u_0\). It will have the form \(U_1 = G U_0\) where the growth factor \(G\) approximates the factor \(g=e^{kh}\) for the exact solution \(u(t_1) = g u(t_0)\) of the ODE.
C) Compare to the growth factors \(G\) seen for previous methods, and to the growth factor \(g\) for the exact solution.
Exercise 6¶
Write a formula for \(U_h\) and \(e_h\) if one starts from the point \((t_i, U_i)\), so that \((t_i + h, U^h)\) is the proposed value for the next point \((t_{i+1}, U_{i+1})\) in the approximate solution — but only if \(e_h\) is small enough!
Exercise 7¶
Implement the error control version of the explicit trapezoid method from section on Error Control and Variable Step Sizes and test on the two familiar examples
(\(K=1\) is enough.)
This work is licensed under Creative Commons Attribution-ShareAlike 4.0 International