An approximation to the solution of one-dimensional hyperbolic telegraph equation based on the collocation of quadratic B-spline functions

Zarebnia, Mohammad; Parvaz, Reza

doi:10.22034/cmde.2020.40112.1749

An approximation to the solution of one-dimensional hyperbolic telegraph equation based on the collocation of quadratic B-spline functions

Document Type : Research Paper

Authors

Department of Mathematics, University of Mohaghegh Ardabili, 56199-11367 Ardabil, Iran.

10.22034/cmde.2020.40112.1749

Abstract

In this work, the collocation method based on B-spline functions is used to obtain a numerical solution for a one-dimensional hyperbolic telegraph equation. The proposed method is consists of two main steps. As the first step, by using a finite difference scheme for the time variable, a partial differential equation is converted to an ordinary differential equation by the space variable. In the next step, for solving this equation collocation method is used. In the analysis section of the proposed method, the convergence of the method is studied. Also, some numerical results are given to demonstrate the validity and applicability of the presented technique. The L∞, L2, and Root-Mean Square(RMS) in the solutions show the efficiency of the method computationally.

Keywords

20.1001.1.23453982.2021.9.4.18.0

References

[1] J. Biazar, H. Ebrahimi, and Z. Ayati, An approximation to the solution of telegraph equation by variational iteration method, Numerical Methods for Partial Differential Equations, 25 (2009), 797-801.
[2] I. Dag, A. Dogan, and B. Saka, B-spline collocation methods for numerical solutions of the RLW equation, International journal of computer mathematics, 6(80) (2003), 743-757.
[3] M. Dehghan and A. Ghesmati, Solution of the second-order one-dimensional hyperbolic tele- graph equation by using the dual reciprocity boundary integral equation (DRBIE) method, En- gineering Analysis with Boundary Elements, 34 (2010), 51-59.
[4] H. F. Dinga, Y. X. Zhangb, J. X. Caoa, and J. H. Tianb, A class of difference scheme for solving telegraph equation by new non-polynomial spline methods, Applied Mathematics and Computation, 278 (2012), 4671-4683.
[5] D. J. Evans and H. Bulut, The numerical solution of the telegraph equation by the alternating group explicit method, Computer Mathematics, 80 (2003), 1289-1297.
[6] F. Gao and C. Chi, Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation, Applied Mathematics and Computation, 187 (2007), 1272-1276.
[7] M. Garshasbi and M. Khakzad, The RBF collocation method of lines for the numerical solution of the CH-γ equation, Journal of Advanced Research in Dynamical and Control Systems, 4 (2015), 65-83.
[8] M. Garshasbi and F. Momeni, Numerical solution of Hirota-Satsuma coupled mKdV equation with quantic B-spline collocation method, Journal of Computer Science & Computational Math- ematics, 3(1) (2011), 13-18.
[9] M. M. Hosseini, S. T Mohyud-Din, S. M. Hosseini, and M. Heydari, Study on Hyperbolic Telegraph Equations by Using Homotopy Analysis Method, Studies in Nonlinear Sciences, 2(1) (2010), 50-56.
[10] A. Jeffrey, Advanced engineering mathematics, Harcourt Academic Press, 2002.
[11] D. Kincad and W. Cheny, Numerical analysis, Brooks/COLE, 1991.
[12] R.C. Mittal and R. Bhatia, Numerical solution of second order one dimensional hyperbolic tele- graph equation by cubic B-spline collocation method, Applied Mathematics and Computation, 220 (2013), 496-506.
[13] S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Applied Mathematics and Computation, 170 (2005), 1126-1134.
[14] R. Parvaz and M. Zarebnia, Cubic B-spline collocation method for numerical solution of the one-dimensional hyperbolic telegraph equation, Journal of Advanced Research in Scientific Com- puting, 4(4) (2012), 46-60.
[15] P. M. Prenter, Spline and Variational Methods, Wiley, New York, 1975.
[16] L. Schumaker, Spline functions: basic theory, Cambridge University Press, 2007.
[17] S. Sharifi and J. Rashidinia, Numerical solution of hyperbolic telegraph equation by cubic B- spline collocation method, Applied Mathematics and Computation, 281 (2016), 28-38.
[18] J. N. Sharma, K. Singh, and J. N. Sharma, Partial differential equations for engineers and scientists, second edition, Alpha Science International, 2009.
[19] J. Stoer and R. Bulirsch, Introduction to numerical analysis, third edition, Springer-Verlg, 2002.
[20] S. Toubaei, M. Garshasbi, and M. Jalalvand, A numerical treatment of a reaction-diffusion model of spatial pattern in the embryo, Computational Methods for Differential Equations, 2(4) (2016), 116-127.
[21] M. Zarebnia and R. Parvaz, On the numerical treatment and analysis of Benjamin Bona Ma- hony Burgers equation, Applied Mathematics and Computation, 284 (2016), 79-88.