Laplace Transform Collocation Method for Solving Hyperbolic Telegraph Equation

Saheed O Akindeinde,Babatunde S Ogundare,Adebayo A Aderogba,Adebayo O Adewumi

doi:10.1155/2017/3504962

Saheed O Akindeinde, Babatunde S Ogundare + Show 2 more

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

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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.

Highlights

In this paper, we consider the second-order one-dimensional telegraph equation ∂2u ∂t2 + α ∂u ∂t βu = ∂2u ∂x2 f (x, t), (1)where α, β are known constants and f(x, t) 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 and it is still useful for describing long distance power lines and cable TV systems [1]
Motivated by the works of Odejide and Binuyo [6] where the weighted residual method was applied to the onedimensional telegraph equation, in this work, a new and efficient collocation method based on the Laplace transform is proposed to approximate the solution of (1)
We adopted a combination of Laplace transform scheme and collocation method to develop a new numerical method for solving one-dimensional linear hyperbolic telegraph equation

Summary

Introduction

Where α, β are known constants and f(x, t) 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]. Motivated by the works of Odejide and Binuyo [6] where the weighted residual method was applied to the onedimensional 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). By solving (17) subject to the homogeneous conditions above, we obtain the error function en(x, t) This allows us to compute u(x, t) = un(x, t)+en(x, t) even for problems without known exact solutions

Numerical Examples

Conclusion