×

Numerical solution of telegraph equation using interpolating scaling functions. (English) Zbl 1205.65288

Summary: A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses interpolating scaling functions. The method consists of expanding the required approximate solution as the elements of interpolating scaling functions. Using the operational matrix of derivatives, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] El-Azab, M. S.; El-Gamel, M., A numerical algorithm for the solution of telegraph equations, Appl. Math. Comput., 190, 757-764 (2007) · Zbl 1132.65087
[2] Metaxas, A. C.; Meredith, R. J., Industrial Microwave, Heating (1993), Peter Peregrinus: Peter Peregrinus London
[3] Roussy, G.; Pearcy, J. A., Foundations and Industrial Applications of Microwaves and Radio Frequency Fields (1995), John Wiley: John Wiley New York
[4] Dehghan, M., On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Methods Partial Differential Equations, 21, 24-40 (2005) · Zbl 1059.65072
[5] Mohanty, R. K.; Jain, M. K.; George, K., On the use of high order difference methods for the system of one space second order non-linear hyperbolic equations with variable coefficients, J. Comput. Appl. Math., 72, 421-431 (1996) · Zbl 0877.65066
[6] Twizell, E. H., An explicit difference method for the wave equation with extended stability range, BIT, 19, 378-383 (1979) · Zbl 0441.65066
[7] Gao, F.; Chi, C., Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation, Appl. Math. Comput., 187, 1272-1276 (2007) · Zbl 1114.65347
[8] Saadatmandi, A.; Dehghan, M., Numerical solution of the one-dimensional wave equation with an integral condition, Numer. Methods Partial Differential Equations, 23, 282-292 (2007) · Zbl 1112.65097
[9] Borhanifar, A.; Abazari, Reza, An unconditionally stable parallel difference scheme for telegraph equation scheme for telegraph equation, Math. Probl. Eng. (2009), Article ID 969610 · Zbl 1181.78018
[10] Saadatmandi, A.; Dehghan, M., Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differential Equations, 26, 1, 239-252 (2010) · Zbl 1186.65136
[11] Abdou, M. A., Adomian decomposition method for solving the telegraph equation in charged particle transport, J. Quant. Spectrosc. Radiat. Transfer, 95, 407-414 (2005)
[12] Mohanty, R. K., An unconditionally stable difference scheme for the one-space dimensional linear hyperbolic equation, Appl. Math. Lett., 17, 101-105 (2004) · Zbl 1046.65076
[13] Mohanty, R. K., An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput., 165, 229-236 (2005) · Zbl 1070.65076
[14] Lapidus, L.; Pinder, G. F., Numerical Solution of Partial Differential Equations in Science and Engineering (1982), Wiley: Wiley New York · Zbl 0584.65056
[15] Mohebbi, A.; Dehghan, M., High order compact solution of the one-space-dimensional linear hyperbolic equation, Numer. Methods Partial Differential Equations, 24, 1222-1235 (2008) · Zbl 1151.65071
[16] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simulation, 71, 16-30 (2006) · Zbl 1089.65085
[17] Dehghan, M., Implicit collocation technique for heat equation with non-classic initial condition, Int. J. Nonlinear Sci. Numer. Simul., 7, 447-450 (2006)
[18] Dehghan, M.; Shokri, A., A numerical method for solving the hyperbolic telegraph equation, Numer. Methods Partial Differential Equations, 24, 1080-1093 (2008) · Zbl 1145.65078
[19] Dehghan, M.; Lakestani, M., The use of Chebyshev cardinal functions for solution of the second-order one-dimensional telegraph equation, Numer. Methods Partial Differential Equations, 25, 931-938 (2009) · Zbl 1169.65102
[20] Shamsi, M.; Razzaghi, M., Numerical solution of the controlled Duffing oscillator by the interpolating scaling functions, J. Electromagn. Waves Appl., 18, 5, 691-705 (2004)
[21] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamic (1987), Springer-Verlag
[22] Shamsi, M.; Razzaghi, M., Solution of Hallen’s integral equation using multiwavelets, Comput. Phys. Comm., 168, 187-197 (2005) · Zbl 1196.65203
[23] Alpert, B.; Beylkin, G.; Gines, D.; Vozovoi, L., Adaptive solution of partial differential equations in multiwavelet bases, J. Comput. Phys., 182, 149-190 (2002) · Zbl 1015.65046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.