11 September 2007 KKKQ 3013 PENGIRAAN BERANGKA Week 10 – Ordinary Differential Equations 11 September am – 9.00 am

11 September 2007 Week 10 Page 2 Topics

11 September 2007 Week 10 Page 3 Tutorial Example 1 (adapted courtesy of ref. [1]) [1] KQ3013 lecture notes Evaluate the values of y numerically for the following differential equation: Given that the initial condition is y(0) = 1, perform the calculation from x = 0 to x = 2 with a step size of h = 0.5. [Note that the exact solution could be obtained by integrating the above differential equation or derivative, to give y = -0.5x 4 + 4x 3 – 10x x + 1 ]. Use: (i) Ralston method (a 2 nd order Runge-Kutta method) (ii) Classical 4 th order Runge Kutta method

11 September 2007 Week 10 Page 4 Tutorial Example 1 (adapted courtesy of ref. [1])

11 September 2007 Week 10 Page 5 Tutorial Example 1 (adapted courtesy of ref. [1])

11 September 2007 Week 10 Page 6 Tutorial Example 1 (adapted courtesy of ref. [1])

11 September 2007 Week 10 Page 7 Tutorial Example 1 (adapted courtesy of ref. [1])

11 September 2007 Week 10 Page 8 Tutorial Example 1 (adapted courtesy of ref. [1])

11 September 2007 Week 10 Page 9 Tutorial Example 1 (adapted courtesy of ref. [1]) [1] KQ3013 lecture notes Summarising: Notice that a 4 th order Runge-Kutta method has 0% error i.e. since the exact ODE solution for y is a 4 th order polynomial, therefore, a 4 th order RK should be exact.