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A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations

Ali Saleh Alshomrani (Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia)
Sapna Pandit (Department of Mathematics, Indian Institute of Technology Roorkee, India)
Abdullah K. Alzahrani (Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia)
Metib Said Alghamdi (Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia)
Ram Jiwari (Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India)

Engineering Computations

ISSN: 0264-4401

Article publication date: 12 June 2017

Abstract

Purpose

The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc.

Design/methodology/approach

Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed.

Findings

A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations.

Originality/value

To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).

Keywords

Acknowledgements

The authors are very thankful to the editor and reviewers for their valuable suggestions to improve the quality of the paper.

Citation

Alshomrani, A.S., Pandit, S., Alzahrani, A.K., Alghamdi, M.S. and Jiwari, R. (2017), "A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations", Engineering Computations, Vol. 34 No. 4, pp. 1257-1276. https://doi.org/10.1108/EC-05-2016-0179

Publisher

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Emerald Publishing Limited

Copyright © 2017, Emerald Publishing Limited