The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems Muhammad Abbas a,b,⇑ , Ahmad Abd. Majid b , Ahmad Izani Md. Ismail b , Abdur Rashid c a Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan b School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia c Department of Mathematics, Gomal University, 29050 Dera Ismail Khan, Pakistan article info Keywords: One-dimensional wave equation Non-local conservation constraints Cubic trigonometric B-spline basis functions Cubic trigonometric B-spline collocation method Stability abstract In this paper, a collocation finite difference scheme based on new cubic trigonometric B-spline is developed and analyzed for the numerical solution of a one-dimensional hyperbolic equation (wave equation) with non-local conservation condition. The usual finite difference scheme is used to discretize the time derivative while a cubic trigonomet- ric B-spline is utilized as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann (Fourier) method. The accuracy of the proposed scheme is tested by using it for several test problems. The numerical results are found to be in good agreement with known exact solutions and with existing schemes in literature. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction There are quite a number of phenomena in science and engineering which can be modeled by the use of hyperbolic partial differential equations subject to non-local conservation condition instead of traditional boundary conditions [1] and these arise in the study of chemical heterogeneity [2,3], medical science, viscoelasticity, plasma physics [4] and thermoelasticity [5,6]. This type of problems also arises in non-local reactive transport in underground water flows in porous media, semi-conductor modeling, non-Newtonian fluid flows and radioactive nuclear decay in fluid flows [7]. The temperature distribution of air near the ground over time during calm clear nights is a good example of such models [8]. The analysis, development and implementation of numerical methods for the solution of such problems have received wide attention in the literature. In this study, we discuss the numerical solution of the wave equation subject to non-local conservation condition, using cubic trigonometric B-spline collocation method (CuTBSM). Consider a vibrating elastic string of length L which is located on the x-axis of the interval [0, L]. Then, the vertical displacement u(x, t) of the elastic string at point x units from the origin after a time t elapsed is given by the one-dimensional wave equation @ 2 u @t 2 ðx; tÞ a 2 @ 2 u @x 2 ðx; tÞ¼ qðx; tÞ; 0  x  L; 0  t  T : ð1Þ http://dx.doi.org/10.1016/j.amc.2014.04.031 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author at: Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan. E-mail addresses: m.abbas@uos.edu.pk (M. Abbas), majid@cs.usm.my (A.A. Majid). Applied Mathematics and Computation 239 (2014) 74–88 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc