Accurate analytic approximation to the nonlinear pendulum problem

Turkyilmazoglu M.

PHYSICA SCRIPTA, vol.84, no.1, 2011 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 84 Issue: 1
  • Publication Date: 2011
  • Doi Number: 10.1088/0031-8949/84/01/015005
  • Journal Name: PHYSICA SCRIPTA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Hacettepe University Affiliated: No


This paper is concerned with the accurate analytic solution of the nonlinear pendulum differential equation. Instead of the traditional Taylor series or asymptotic methods, the homotopy analysis technique is used, which does not require a small perturbation parameter or a large asymptotic parameter. It is shown here that such a method is extremely powerful in gaining the pendulum solution in terms of purely trigonometric cosine functions. The obtained explicit analytical expressions for the frequency, period and displacement generate results that compare excellently with the numerically computed ones for moderate as well as sufficiently high values of the initial amplitudes. For larger initial amplitudes close to 180 degrees Pade approximants of the found series solutions lead to fairly accurate results whose relative errors as compared with the exact values are less than 1%.