The modeling of many phenomena in various fields such as mathematics, physics, chemistry,
engineering, biology, and astronomy is done by the nonlinear partial differential
equations (PDE). The hyperbolic telegraph equation is one of them, where it describes
the vibrations of structures (e.g., buildings, beams, and machines) and are the basis
for fundamental equations of atomic physics. There are several analytical and numerical
methods are used to solve the telegraph equation. An analytical solution considers
framing the problem in a well-understood form and calculating the exact resolution.
It also helps to understand the answers to the problem in terms of accuracy and convergence.
These analytic methods have limitations with accuracy and convergence. Therefore,
a novel analytic approximate method is proposed to deal with constraints in this paper.
This method uses the Taylors' series in its derivation. The proposed method has used
for solving the second-order, hyperbolic equation (Telegraph equation) with the initial
condition. Three examples have presented to check the effectiveness, accuracy, and
convergence of the method. The solutions of the proposed method also compared with
those obtained by the Adomian decomposition method (ADM), and the Homotopy analysis
method (HAM). The technique is easy to implement and produces accurate results. In
particular, these results display that the proposed method is efficient and better
than the other methods in terms of accuracy and convergence.