Numerical Analysis

Lotka-Volterra Equations : The " Lotka-Volterra equations " refer to two coupled differential equations There is one critical point which occurs when and it is . The Runge-Kutta method is used to numerically solve O.D.E.'s over an interval .

Nonlinear Pendulum: A simple pendulum consists of a point mass m attached to a rod of negligible weight. The torque is (1) , where denotes the the angle of the rod measured downward from a vertical axis. The moment of inertia for the point mass is where l is the length of the rod. The torque can also be expressed as , where is the angular acceleration, using Newton's second law , and the second derivative, this can be written as (2) . Equating (1) and (2) results in the nonlinear D. E. (3) . Linear Pendulum : Introductory courses discuss the pendulum with small oscillations as an example of a simple harmonic oscillator . If the angle of oscillation is small, use the approximation in equation (3) and obtain the familiar linear D. E. for simple harmonic motion : (4) , Using the substitution , the solution to (4) is known to be (5) , which has period . When the solution (5) is written with a phase shift, it

In calculus, a model for projectile motion with no friction is considered, and a "parabolic trajectory" is obtained. If the initial velocity is and is the initial angle to the horizontal, then the parametric equations for the horizontal and vertical components of the position vector are (1) , and (2) . Solve equation (1) for t and get , then replace this value of t in equation (2) and the result is , which is an equation of a parabola. The time required to reach the maximum height is found by solving : , yields , and the maximum height is . The time till impact is found by solving , which yields , and for this model, . The range is found by calculating : . For a fixed initial velocity , the range is a function of , and is maximum when . Numerical solution of second order D. E.'s This module illustrates numerical solutions of a second order differential equatio