The Telegraph Equation and Its Solution by Reduced Differential Transform Method

One-dimensional second-order hyperbolic telegraph equation was formulated using Ohm’s law and solved by a recent and reliable semianalytic method, namely, the reduced differential transform method (RDTM). Using this method, it is possible to find the exact solution or a closed approximate solution of a differential equation. Three numerical examples have been carried out in order to check the effectiveness, the accuracy, and convergence of the method. The RDTM is a powerful mathematical technique for solving wide range of problems arising in science and engineering fields. 1. Introduction In this modern era communication system plays a key role in the worldwide society. High frequency communication systems continue to benefit from significant industrial attention, triggered by a host of radio frequency (RF) and microwave communication (MW) systems. These systems use the transmission media for transferring the information carrying signal from one point to another point. This transmission media can be categorized into two groups, namely, guided and unguided. In guided medium the signal is transferred through the coaxial cable or transmission line. These guided media are capable of transporting the high frequency voltage and current waves. While in unguided media electromagnetic waves carry the signal over part of or the entire communication path through RF and MW channels. These electromagnetic waves are transmitted and received through antenna. In guided transmission media, specifically cable transmission medium is investigated to address the problem of efficient telegraphic transmission. A cable transmission medium can be classified as a guided transmission medium and represents a physical system that directly propagates the information between two or more locations. In order to optimize the guided communication system it is necessary to determine or project power and signal losses in the system, because all the systems have such losses. To determine these losses and eventually ensure a maximum output, it is necessary to formulate some kind of equation with which to calculate these losses. In this paper a mathematical derivation for the telegraph equation in terms of voltage and current for a section of a transmission line has been formulated and the obtained mathematical equation is solved by a very recent approximate analytical method, namely, the reduced differential transform method (RDTM). Let us consider an infinitesimal piece of telegraph cable wire as an electrical circuit shown in Figure 1. Further assume that the cable is imperfectly insulated