On Exact Solutions for the Vibrations of a Nonlinear Continuous Model String

In this paper, we consider a nonlinear boundary value problem for the vibrations of a nonlinear continuous model string, which describes the standing vibrations of a finite, continuous, and nonlinear string. Under the assumption that the nonlinear string is proportional to the derivative of the stress with respect to the strain, exact solutions of this problem are provided. To obtain the solution, the given nonlinear partial differential equation is transformed into a first-order nonlinear system by introducing the dependent variables transformation u = yx, v = yt. The modified method of separation of variables is applied to the system and the solutions for t and x are written as general implicit formulas. It is shown that the technique used here offers advantages in finding solutions of such problems.