The Telegraph Equation and Its Solution by Reduced Differential Transform Method

Abstract

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.

A computational modelling of micro strip patch antenna and its solution by RDTM

“…In this section, the fundamental of the reduced differential transformation is described [3][4][5][6][7][8][9][10][11]. Consider a function of two variables wðx; tÞ, and assume that it can be represented as a product of two single-variable functions, i.e.…”

“…The RDTM is proposed by Keskin and Oturanc [5] for the fractional differential equations and they showed that it is a powerful mathematical tool for solving a wide range of problems including communication system. Thereafter, RDTM is used by various authors of [6,[8][9][10][11]22]. Aghajania et al [14] used this method for solving Black-Scholes equation for European option valuation.…”

Laplace Transform Collocation Method for Solving Hyperbolic Telegraph Equation

“…Various analytical and numerical methods have been developed and employed to solve this equation. These include the Method of Weighted Residuals [6], Laplace transform inversion technique with homotopy perturbation method [7], radial basis function method [8], Chebyshev tau method [9], Legendre multiwavelet Galerkin method [10], reciprocity boundary integral equation method [3], Adomian decomposition method [11], unconditionally stable difference scheme [12], and the Reduced Differential Transform Method (RDTM) [13] to mention just a few. Other researchers have also proposed different numerical schemes for solving telegraph equation; for example, Dehghan and Lakestani [14] proposed a method based on Chebyshev cardinal functions to solve one-dimensional hyperbolic telegraph equation, and Javidi [15] used Chebyshev spectral collocation method for computing numerical solution of telegraph equation.…”

High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

“…The numerical solution of second order hyperbolic PDEs has been studied extensively by a variety of techniques such as the finite element methods , finite‐difference schemes , combined finite difference scheme and Haar wavelets , discrete eigenfunctions method , Legendre multiwavelet approximations , the singular dynamic method , interpolating scaling functions , cubic and quartic B‐spline collocation methods , nonpolynomial spline methods , the reduced differential transform method , and so forth. In the present work, we present a shifted Gegenbauer pseudospectral method (SGPM) for the solution of Problem .…”

“…The telegraph equation is in particular important as it is commonly used in the study and modeling of signal analysis for transmission and propagation of electrical signals in a cable transmission line [7,8], and in reaction diffusion occurring in many branches of sciences [9,10]. The numerical solution of second order hyperbolic PDEs has been studied extensively by a variety of techniques such as the finite element methods [11,12], finite-difference schemes [3,[13][14][15], combined finite difference scheme and Haar wavelets [6], discrete eigenfunctions method [7], Legendre multiwavelet approximations [16], the singular dynamic method [17], interpolating scaling functions [18], cubic and quartic B-spline collocation methods [19,20], nonpolynomial spline methods [21], the reduced differential transform method [22], and so forth. In the present work, we present a shifted Gegenbauer pseudospectral method (SGPM) for the solution of Problem P. The numerical scheme exploits the stability and the well conditioning of the numerical integral operators, and collocates the integral formulation of Problem P in the physical (nodal) space using some novel operational matrices of integration (also called integration matrices) based on shifted Gegenbauer polynomials.…”