The numerical solution of initial-value problems that involve nonlinear ordinary differential equations is considered in this chapter. In Section 1.1 some numerical methods, especially the Runge-Kutta methods, are introduced for solving the firstand second-order equations. They are applied in Section 1.2 for finding the motion of a free-falling sphere through air and in Section 1.3 to simulate the motions of a simple pendulum and an aeroelastic system.
To extend the applications from one-dimensional to two-dimensional motions, Runge-Kutta formulas for solving simultaneous second-order equations are deduced in Section 1.4. Simultaneously, we have implemented MATLAB initial value solver ODE45 in the programs developed in this chapter and elsewhere in the book. After the motion of a spherical projectile in the presence of a fluid has been computed, the numerical integration procedure of Section 1.5 is combined with the half-interval method to find the maximum range of such a body. Section 1.6 deals with the computation of the trajectory of a glider, and Section 1.7 is an example from aerodynamics concerning the vortex sheet trailing behind a finite wing.
1.1 NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS: INITIAL-VALUE PROBLEMS
Consider the simplest case of a first-order ordinary differential equation having the general form
where f is an analytic function. If, at a starting point t = t 0 , the function x has a given value x 0 , it is desired to find x (t) for t > t 0 that satisfies both (1.1.1) 1