International Journal of Engineering Mathematics

Volume 2017 (2017), Article ID 3504962, 9 pages

https://doi.org/10.1155/2017/3504962

## Laplace Transform Collocation Method for Solving Hyperbolic Telegraph Equation

Research Group in Computational Mathematics (RGCM), Department of Mathematics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria

Correspondence should be addressed to Babatunde S. Ogundare

Received 13 July 2016; Revised 11 February 2017; Accepted 28 February 2017; Published 10 April 2017

Academic Editor: Bhabani S. Dandapat

Copyright © 2017 Adebayo O. Adewumi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This article presents a new numerical scheme to approximate the solution of one-dimensional telegraph equations. With the use of Laplace transform technique, a new form of trial function from the original equation is obtained. The unknown coefficients in the trial functions are determined using collocation method. The efficiency of the new scheme is demonstrated with examples and the approximations are in excellent agreement with the analytical solutions. This method produced better approximations than the ones produced with the standard weighted residual methods.

#### 1. Introduction

In this paper, we consider the second-order one-dimensional telegraph equationwhere are known constants and is continuous in the displayed arguments.

Equation (1) describes an electrical signal traveling along a transmission cable; this was first derived in the horse and buggy days of the telegraph (from where it derived its name) and it is still useful for describing long distance power lines and cable TV systems [1].

The study of electric signal in a transmission line, dispersive wave propagation, pulsating blood flow in arteries, and random motion of bugs along a hedge is amongst a host of physical and biological phenomena which can be described by (1). For expository details on the abovementioned phenomena, readers are advised to see [1–4].

Recently, telegraph equation is found to be more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of science [5]. Without any doubt, (1) and its solution are of great importance in many areas of application. 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. Borhanifar and Abazari [16] developed an unconditionally stable parallel difference scheme for telegraph equation. Lakestani and Saray [8] developed a numerical technique for the solution of second-order one-dimensional linear hyperbolic equation. The method consists of expanding the required approximate solution as the elements of interpolating scaling function. In their technique, by using operational matrix of derivatives, they reduced the problem to a set of algebraic equations [7]. Hesameddini and Asadolahifard [17] applied the Sinc-Collocation Method to approximate the solution of (1). Mittal and Bhatia [18] and Rashidinia et al. [19] employed the Cubic B-spline Collocation Method (CuBSCM) to approximate the solution of (1). In [20], the authors employed the Fibonacci Polynomials approach to approximate solution of telegraph equations.

Motivated by the works of Odejide and Binuyo [6] where the weighted residual method was applied to the one-dimensional telegraph equation, in this work, a new and efficient collocation method based on the Laplace transform is proposed to approximate the solution of (1). This new method shall be called Laplace Transform Collocation Method (LTCM).

The rest of this paper is organized as follows: In Section 2, brief description of the method is presented, and Section 3 is devoted to the error analysis of the method. Implementation of the method using numerical examples is presented in Section 4 while the last section presents our conclusion.

#### 2. Laplace Transform Collocation Method (LTCM)

To put emphasis on the essential mathematical details of the new method, we consider the following one-dimensional hyperbolic telegraph equation: with the initial conditions

and Dirichlet boundary condition where and are known constant coefficients, , , , and are known continuous functions in their respective domains, and the function is unknown.

Taking the Laplace transform of (2), we have After simple algebraic simplification, we get

The function and its derivatives in (6) are thereafter replaced with a trial function of the form where are constants to be determined which satisfy the given conditions (3) and (4). Thus, we have the following:

Taking the inverse Laplace transform of (8), we haveSubstituting (9) into (2), we get

Now, collocating (10) at points , we have where

Thus, (11) constitutes -equations in -unknowns which can be determined by using Gaussian elimination method. Substituting these coefficients into (9) gives the approximate solutions.

#### 3. Error Analysis of Laplace Transform Collocation Method (LTCM)

Let us define the error function , where and denote, respectively, the exact and approximate solution obtained via our proposed method. In line with [20], we define the residual function where

It then follows that subject to initial conditions

Now since is a linear operator, we obtain for the error function with the homogeneous conditions

By solving (17) subject to the homogeneous conditions above, we obtain the error function This allows us to compute even for problems without known exact solutions.

#### 4. Numerical Examples

In this section, we implement the new method on some examples to test its efficiency and applicability.

*Example 1. *We consider the case in which , , and , and (2) becomes (see [6]) subject to The exact solution is given by .

We assume the trial function of the form:Taking the Laplace transform of (19), we get

Rearranging (22), we have Taking the inverse Laplace transform of (23), we have the following new trial solution:

Substituting (24) into (19), we have the following residual function: Collocating (25) at equally spaced points and for and equating to zero, we then use the Gaussian elimination method to solve the two systems of equations and obtained and .

Substituting these values into (24), we obtain the following approximate solution:

Table 1 gives the comparison between the new method (LTCM) and the Method of Weighted Residual (MWR) for Example 1. Figure 1 shows the comparison of approximate and exact solution for Example 1. The error plot for Example 1 is shown in Figure 2.