Solving the equation of the physical pendulum by some numerical methods of differential equations

The pendulum problem is one of the most studied topics in the field of Physics. In fact, in most cases, the simplest case is analyzed, so that in the less complex techniques given to describe the harmonic motion, small oscillation amplitudes are considered. However, the analysis of large oscillations presents a greater complexity and numerical methods are required to solve approximately the nonlinear differential equations that describe this type of problem. In this work, the nonlinear differential equations describing the damped oscillations of a rigid pendulum rotating on a pivot subject to a uniform gravitational force in the vertical direction will be solved numerically. Some initial conditions will be proposed and some classical numerical algorithms will be implemented, such as Euler and Euler Back methods both of first order, second order, Runge Kutta, Adams Bashforth and Trapezium methods, third order methods such as Adams Bashforth, Adams Moulton and fourth order methods such as Runge Kutta and Adams Moulton. Approximate solutions will be recommended for some values of the parameters of the differential equation studied and, finally, a comparison of the numerical algorithms with the solution of ordinary differential equations ODE45 of Matlab software will be made.