**Commenced**in January 2007

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**Edition:**International

**Paper Count:**30172

##### Solving One-dimensional Hyperbolic Telegraph Equation Using Cubic B-spline Quasi-interpolation

**Authors:**
Marzieh Dosti,
Alireza Nazemi

**Abstract:**

In this paper, the telegraph equation is solved numerically by cubic B-spline quasi-interpolation .We obtain the numerical scheme, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. The results of numerical experiments are presented, and are compared with analytical solutions by calculating errors L2 and L∞ norms to confirm the good accuracy of the presented scheme.

**Keywords:**
Cubic B-spline,
quasi-interpolation,
collocation method,
second-order hyperbolic telegraph equation.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1331887

**References:**

[1] D. M. Pozar, Microwave engineering, NewYork: Addison-Wesley, 1990.

[2] A. Mohebbi, M. Dehghan, High order compact solution of the one-spacedimensional linear hyperbolic equation, Numerical Methods for Partial Differential Equations, 24 (2008) 1222-1235.

[3] A. Jeffrey, Advanced engineering mathematics, Harcourt Academic Press, 2002.

[4] A. Jeffrey, Applied partial differential equations, NewYork: Academic Press, 2002.

[5] R. K. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, Computer Mathematics, 86 (2008) 2061-2071.

[6] H. Pascal, Pressure wave propagation in a fluid flowing through a porous medium and problems related to interpretation of Stoneley-s wave attenuation in acoustical well logging, Engineering Science, 24 (1986) 1553-1570.

[7] G. Bohme, Non-Newtonian fluid mechanics, NewYork: North-Holland, 1987.

[8] D. J. Evans, H. Bulut, Thenumerical solution of thetelegraph equation by the alternating group explicit method, Computer Mathematics 80 (2003) 1289- 1297.

[9] P. M. Jordan, M. R. Meyer, A. Puri, Causal implications of viscous damping in compressible fluid flows, Physics Review, 62 (2000) 7918- 7926.

[10] L. L. Schumaker, Spline Functions: Basic Theory, Krieger Publishing Company, Florida, 1981.

[11] J. M. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, NewYork, 1967.

[12] P. M. Prenter, Splines and variational methods, New York: John Wiley, 1975.

[13] C. De Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978.

[14] G. Micula, Handbook of Splines, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.

[15] B. Saka, I. Da ╦ç G, Quartic B-spline Galerkin approach to the numerical solution of the KdVB equation, Applied Mathematics and Computation, 215 (2009) 746-758.

[16] I. Da ╦ç G, A. Canvar, A. Sahin, TaylorGalerkin and Taylor-collocation methods for the numerical solutions of Burgers equation using B-splines, communication in nonlinear science and numerical simulation, 16 (2010) 2696-2708.

[17] S. Chandra Sekhara Rao, M. Kumar, Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems, Applied Numerical Mathematics, 58 (2008) 1572-1581.

[18] M. K. Kadalbajoo, P. Arora, B-spline collocation method for the singular-perturbation problem using artificial viscosity, Computers and Mathematics with Applications, 57 (2009) 650-663.

[19] M. K. Kadalbajoo, V. Gupta, A. Awasthi, A uniformly convergent Bspline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convectiondiffusion problem, Journal of Computational and Applied Mathematics, 220 (2008) 271 - 289.

[20] M. K. Kadalbajoo, V. Gupta, Numerical solution of singularly perturbed convectiondiffusion problem using parameter uniform B-spline collocation method, Journal of Mathematical Analysis and Applications, 355 (2009) 439-452.

[21] B. Saka, I. Da ╦ç G, Quartic B-spline collocation method to the numerical solutions of the Burgers- equation, Chaos, Solitons & Fractals, 32 (2007) 1125-1137.

[22] F. Gao, C. M Chi, Solving third-order obstacle problems with quartic B-splines, Applied Mathematics and Computation, 180 (2006) 270-274.

[23] K. R. Raslan, Collocation method using quartic B-spline for the equal width (EW) equation, Applied Mathematics and Computation, 168 (2005) 795-805.

[24] F. i. Haq, S. u. Islam, I. A. Tirmizi, A numerical technique for solution of the MRLW equation using quartic B-splines, Applied Mathematical Modelling, 34 (2010) 4151-4160.

[25] M. Eck, D. Lasser, B-spline-B'ezier representation of geometric spline curves: Quartics and quintics, Computers & Mathematics with Applications, 23 (1992) 23-39.

[26] C. G. Zhu, R. H. Wang, Numerical solution of Burger-s equation by cubic B-spline quasi-interpolation, Applied Mathematics and Computation, 208 (2009) 260-272.

[27] C. G. Zhu, W. S. Kang, Numerical solution of Burgers-Fisher equation by cubic B-spline quasi-interpolation, Applied Mathematics and Computation, 216 (2010) 2679-2686.

[28] G. Farin, Curves and Surfaces for CAGD, fifth ed., Morgan Kaufman, San Francisco, 2001.

[29] P. Sablonni`ere, Quasi-interpolants splines sobre particiones uniforms, in: First Meeting in Approximation Theory of the University of Ja`en, Ubeda, June 29- July 2, 2000, Pr`epublication IRMAR 00-38, Rennes, June 2000.

[30] P. Sablonni`ere, Univariate spline quasi-interpolants and applications to numerical analysis, Rend. Sem. Mat. Univ. Pol. Torino 63 (2005) 211- 222.

[31] M. Dehghan, A. Ghesmati, Solution ofthesecond-orderone-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Engineering Analysis with Boundary Elements, 34 (2010) 51-59.