Analytical technique for solving strongly nonlinear oscillator differential equations

Abstract

The main motivation of this paper is to apply general formulas of the gamma function to find the analytical solutions of some types of nonlinear differential equations with recognizable simplicity and less calculation compared to other existing analytical methods in the literature. The simplicity and accuracy of the Gamma function method (GFM) in solving strongly nonlinear differential equations can be displayed through three practical examples arising in many different fields of engineering and science. The results obtained analytically are compared with the well-known analytical methods already published in the literature, such as Akbari-Ganji's method (AGM), homotopy perturbation method (HPM), and exact numerical solutions to confirm the efficiency of GFM. Excellent agreement is found between the analytical solutions and the exact numerical ones. Accordingly, the GFM is more reliable for exploring approximate solutions for strongly nonlinear oscillators. Moreover, the GFM can be easily extended to other nonlinear problems and is, therefore, very applicable in engineering and applied sciences.

Keywords

• Gamma function method
• Circular sector oscillator
• Tapered beams
• Nonlinear equations
• Analytical solutions
• Numerical solutions