Research

Numerical Methods for ODEs

This example is used to demonstrate the event location facilities available with the MATLAB Ordinary Differential Equation (ODE) solvers in the book 'Solving ODEs with MATLAB' written by Professors L.F. Shampine and I. Gladwell of SMU and Professor S. Thompson of Radford University. The figure shows the path of a ball bouncing down an inclined plane. The motion of the ball under the action of gravity is described by a simple system of two second order ODEs. It is easy enough to solve the ODEs analytically, but following the ball down the ramp is complicated because it is necessary to determine the timing of the ``events'' (when the ball makes contact with the plane), determine new initial conditions for the integration after the ball bounces, and deal with the fact that eventually the ball rolls, at which point the model breaks down. A more realistic model would incorporate air resistance in the equations of motion and stiction in the determination of the timing and effects of the impacts. These ODEs are nonlinear and it is necessary to solve them numerically. The accuracy of the plotted solution would then depend on the error tolerances used in the solution of the differential equations and the solution of the algebraic equations for the time of impact.