Approximate analytical and numerical solutions to the damped pendulum oscillator: Newton-Raphson and moving boundary methods

In this work, some new approximate solutions to the damped pendulum equation are obtained. In addition, the Newton-Raphson method (NRM), moving boundary method, and fourth-order Runge Kutta forth-order (RK4) are introduced to analyze the problem under study numerically. With respect to the approximate analytic solutions, two schemes are devoted: in the first approach, we can solve our problem with specific values for the initial conditions (zero initial angle) and after that compare our analytic solution with numerical solutions and with some published solutions. Thereafter, some modifications and improvements for the analytic solution will be constructed in order to get high-accurate solutions. With respect to the second scheme, we can solve our problem with arbitrary initial conditions and then make a comparison between the obtained results and the mentioned numerical solutions. Moreover, the distance error for all obtained solutions is estimated with respect to the RK4 solution.