A Novel Analytic Method for Solving Linear and Non-Linear Telegraph Equation

Abstract

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 secondorder, 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.

New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method

“…In this section, we introduce the basic definitions and properties of fractional calculus [21][22][23][24]. Definition 1.…”

“…Recently, many authors have studied different inequalities of Riemann-Liouville fractional integrals; for more details, see [21][22][23][24]. Some properties of the operator j α , which are needed here, are as follows:…”

On the solutions of some nonlinear fractional partial differential equations using an innovative and direct procedure

“…In this article, the author has extended the implementation of a new analytical technique, which is named the fractional novel analytical method (FNAM). This method FNAM was developed by Araya Wiwatwanich [43][44][45] and was used for the solutions of higher nonlinear fractional partial differential equations. In the present work, it is observed that the results obtained by FNAM are quite excellent as compared to other analytical techniques.…”