Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach

This paper presents a new approach and methodology to solve the second-order one-dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions using the cubic trigonometric B-spline collocation method. The usual finite difference scheme is used to discretize the time derivative. The cubic trigonometric B-spline basis functions are utilized as an interpolating function in the space dimension, with a weighted scheme. The scheme is shown to be unconditionally stable for a range of values using the von Neumann (Fourier) method. Several test problems are presented to confirm the accuracy of the new scheme and to show the performance of trigonometric basis functions. The proposed scheme is also computationally economical and can be used to solve complex problems. The numerical results are found to be in good agreement with known exact solutions and also with earlier studies. Subjects: Computer Mathematics; Mathematical Modeling; Mathematical Physics *Corresponding author: Muhammad Abbas, Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan E-mail: [email protected] Reviewing editor: Shaoyong Lai, Southwestern University of Finance and Economics, China Additional information is available at the end of the article ABOUT THE AUTHORS Tahir Nazir is a PhD student in Department of Mathematics, University of Sargodha, Sargodha. He has obtained his MPhil degree in Mathematics from University of Sargodha since July 2011 and master’s degree in Mathematics from Department of Mathematics, University of the Punjab, Lahore, Pakistan. His research interests are Numerical methods and spline approximations. Muhammad Abbas is an assistant professor of Mathematics at University of Sargodha, Sargodha, Pakistan. He completed his bachelor and masters from the University of the Punjab, Lahore-Pakistan in the years 2001 and 2003, respectively. In 2012, he obtained his Doctorate in Computer Graphics at School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia. His research focus is in the area of Computer Aided Graphic Design, Numerical methods and spline approximations. Muhammad Yaseen is an assistant professor of Mathematics at University of Sargodha, Pakistan. He received his MSc and MPhil degrees from Quaide-Azam University Islamabad, Pakistan. His area of interest is Numerical Analysis. He is currently doing his PhD from University of Sargodha. PUBLIC INTEREST STATEMENT The trigonometric B-spline functions were used extensively in Computer Aided Geometric Design (CAGD) as tools to generate curves and surfaces. An advantage of these piecewise functions is its local support properties where the functions are said to have support in specific interval. Due to these properties, trigonometric B-splines have been used to generate the numerical solutions of linear and non-linear partial differential equations. In this paper, the cubic trigonometric B-spline basis function is considered. Collocation method based on the proposed basis functions and finite difference approximation are developed to solve the one-dimensional telegraph equation. Trigonometric B-splines are used to interpolate the solution in x-dimension and finite difference approximations are used to discretize the time derivatives. The proposed method has been proved to be unconditionally stable. Received: 22 May 2017 Accepted: 04 August 2017 First Published: 23 September 2017 © 2017 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Page 1 of 17 Tahir Nazir