On approximate analytical solutions of transcendental equations

Abstract

In this paper, we provide a simple method to analytically solve general problems in science and engineering, which involve transcendetal functions. To validate the technique, we first compared our results to the exact solutions of two well-known problems, which are written in terms of transcendental equations: (1) circuit analysis of a resistor-diode association; (2) obtaining an analytical expression for the charge control in junctionless-nanowire FET devices. Next, to demonstrate the versatility of the method, we address the pendulum differential equation, to obtain an analytical expression for the period of the simple pendulum, considering any possible initial oscillation amplitude. None of these problems has a closed-form analytical solution. Alternatively, in all cases, simplified approximate analytical expressions were obtained, presenting low relative error when contrasted with the respective benchmark numerical results, thereby indicating that the approach can be considered quite accurate for most practical problems of interest.

Keywords
Transcendental equations; differential equations; analytical modeling; simple pendulum; nanoelectronic devices