A new approach for solution of telegraph equation

Document Type : Research Paper


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

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


‎In this paper‎, ‎B-spline collocation method is developed for‎ the solution of one-dimensional hyperbolic telegraph equation‎. ‎The‎ convergence of the method is proved‎. ‎Also the method is applied on‎ some test examples and the numerical results have been compared‎ with the analytical solutions‎. ‎The $L_\infty$,$L_2$ and Root-Mean-Square‎ errors (RMS) in the solutions show the efficiency of the method‎ ‎computationally‎.


[1] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys. 293 (2015) 104–114.
[2] R.C. Cascaval, E. Eckstein, C.L. Frota, J.A. Goldstein, Fractional telegraph equations, J. Math. Ana. Appl. 276(1) (2002) 145–159.
[3] V. Devi, R.K. Maurya, S. Singh, V.K. Singh, Lagranges operational approach for the approximate solution of two-dimensional hyperbolic telegraph equation subject to Dirichlet boundary conditions, Appl. Math. Comput. 367 (2020) 124717.
[4] H.F. Ding, 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, Appl. Math. Comput. 278 (2012) 4671–4683.
[5] F. Gao and C. Chi, Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation, Appl. Math. Comput. 187 (2007) 1272–1276.
[6] R. Jiwari, S. Pandit and R.C. Mittal, A differential quadrature algorithm for solution of the second order one-dimensional hyperbolic telegraph equation, IJNS 13 (2012) 259–266.
[7] S.A. Khuri and A. Sayfy, A spline collocation approach for the numerical solution of a generalized nonlinear Klein Gordon equation, Appl. Math. Comput. 216 (2010) 1047–1056.
[8] A.C. Metaxas and R.J. Meredith, Industrial Microwave, Heating, Peter Peregrinus, London, 1993.
[9] R.C. Mittal and R. Bhatia, A numerical study of two-dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method, Appl. Math. Comput. 244 (2014) 976–997.
[10] R.K. Mohanty, M.K. Jain and K. George, On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients, J. Comput. Appl. Math. 72 (1996)421–431.
[11] R.K. Mohanty, An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput. 165 (2005) 229–236.
[12] G. Pittaluga and L. Sacripante, Quantic spline interpolation on uniform meshes, Acta. Math. Hung. 72 (1996) 4167–175.
[13] P.M. Prenter, Spline and Variational Methods, Wiley, New York, 1975.
[14] G. Roussy and J. A. Pearcy, Foundations and Industrial Applications of Microwaves and Radio Frequency Fields, John Wiley, New York, 1995.
[15] P. Sancho, Relativistic Extension of the Quantum Telegraph Equation, Open. Syst. Inf. Dyn. 7(2) (2000) 157–164.
[16] J. N. Sharma, K. Singh and J. N. Sharma, Partial Differential Equations for Engineers and Scientists, second edition, Alpha Science International, 2009.
[17] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, third edition, Springer-Verlg, 2002.
[18] M. Zarebnia and R. Parvaz, Cubic B-spline collocation method for numerical solution of the one-dimensional hyperbolic telegraph equation, JARSC 4(4) (2012) 46–60.
Volume 12, Issue 1
May 2021
Pages 385-396
  • Receive Date: 29 August 2017
  • Revise Date: 04 November 2019
  • Accept Date: 21 July 2020