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Numerical solution of second order one-dimensional linear hyperbolic equation using trigonometric wavelets
Kybernetika, Tome 48 (2012) no. 5, pp. 939-957 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses the trigonometric wavelets. The method consists of expanding the required approximate solution as the elements of trigonometric wavelets. Using the operational matrix of derivative, we reduce the problem to a set of algebraic linear equations. Some numerical example is included to demonstrate the validity and applicability of the technique. The method produces very accurate results. An estimation of error bound for this method is presented and it is shown that in this method the matrix of coefficients is a sparse matrix.
A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses the trigonometric wavelets. The method consists of expanding the required approximate solution as the elements of trigonometric wavelets. Using the operational matrix of derivative, we reduce the problem to a set of algebraic linear equations. Some numerical example is included to demonstrate the validity and applicability of the technique. The method produces very accurate results. An estimation of error bound for this method is presented and it is shown that in this method the matrix of coefficients is a sparse matrix.
Classification : 35L20, 65L60, 65T40, 65T60
Keywords: telegraph equation; trigonometric wavelets; hermite interpolation; operational matrix of derivative 
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Jokar, Mahmood; Lakestani, Mehrdad. Numerical solution of second order one-dimensional linear hyperbolic equation using trigonometric wavelets. Kybernetika, Tome 48 (2012) no. 5, pp. 939-957. http://geodesic.mathdoc.fr/item/KYB_2012_48_5_a7/

[1] Alpert, B., Beylkin, G., Coifman, R., Rokhlin, V.: Wavelet-like bases for the fast solution of second-kind integral equation. SIAM J. Sci. Comput. 14 (1993), 159-184. | DOI | MR

[2] Chui, C. K., Mhaskar, H. N.: On trigonometric wavelets. Constr. Approx. 9 (1993), 167-190. | DOI | MR | Zbl

[3] Chui, C. K.: An Introduction to Wavelets. Academic Press, Boston 1992. | MR | Zbl

[4] Dahmen, W., Prössdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations. In: Stability and Convergence, Math. Z. 215 (1994), 583-620. | MR | Zbl

[5] 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 (2005), 24-40. | DOI | MR | Zbl

[6] Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simulation 71 (2006), 16-30. | DOI | MR | Zbl

[7] Dehghan, M.: Implicit collocation technique for heat equation with non-classic initial condition. Internat. J. Non-Linear Sci. Numer. Simul. 7 (2006), 447-450.

[8] Dehghan, M., Shokri, A.: A numerical method for solving the hyperbolic telegraph equation. Numer. Methods Partial Differential Equations 24 (2008), 1080-1093. | DOI | MR | Zbl

[9] 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 (2009), 931-938. | DOI | MR | Zbl

[10] Gao, J., Jiang, Y. L.: Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel. J. Comput. Appl. Math. 215 (2008), 242-259. | DOI | MR

[11] Kress, R.: Linear Integral Equations. Springer, New York 1989. | MR | Zbl

[12] Lakestani, M., Dehghan, M.: The solution of a second-order nonlinear differential equation with Neumann boundary conditions using semi-orthogonal B-spline wavelets. Internat. J. Comput. Math. 83 (2006), 8-9, 685-694. | DOI | MR | Zbl

[13] Lakestani, M., Razzaghi, M., Dehghan, M.: Semiorthogonal wavelets approximation for Fredholm integro-differential equations. Math. Prob. Engrg. (2006), 1-12. | DOI

[14] Lakestani, M., Saray, B. N.: Numerical solution of telegraph equation using interpolating scaling functions. Comput. Math. Appl. 60 (2010), 7, 1964-1972. | DOI | MR | Zbl

[15] Lakestani, M., Jokar, M., Dehghan, M.: Numerical solution of nth-Order Integro-Differential equations using trigonometric wavelets. Numer. Math. Methods Appl. Sci. 34 (2011), 11, 1317-1329. | DOI | MR

[16] Lapidus, L., Pinder, G. F.: Numerical Solution of Partial Differential Equations in Science and Engineering. Wiley, New York 1982. | MR | Zbl

[17] Lorentz, G. G.: Convergence theorems for polynomials with many zeros. Math. Z. 186 (1984), 117-123. | DOI | MR | Zbl

[18] Lorentz, G. G., Lorentz, R. A.: Mathematics from Leningrad to Austin. In: Selected Works In Real, Functional And Numerical Analysis, (1997). | Zbl

[19] 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 (1996), 421-431. | DOI | MR

[20] Mohanty, R. K.: An unconditionally stable difference scheme for the one-space dimensional linear hyperbolic equation. Appl. Math. Lett. 17 (2004), 101-105. | DOI | MR | Zbl

[21] 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 (2005), 229-236. | DOI | MR | Zbl

[22] Mohebbi, A., Dehghan, M.: High order compact solution of the one-space-dimensional linear hyperbolic equation. Numer. Methods Partial Differential Equations 24 (2008), 1222-1235. | DOI | MR | Zbl

[23] Petersdorff, T. V., Schwab, C.: Wavelet approximation for first kind integral equations on polygons. Numer. Math. 74 (1996), 479-516. | DOI | MR

[24] Quak, E.: Trigonometric wavelets for hermite interpolation. J. Math. Comput. 65 (1996), 683-722. | DOI | MR | Zbl

[25] Shamsi, M., Razzaghi, M.: Solution of Hallen's integral equation using multiwavelets. Comput. Phys. Comm. 168 (2005), 187-197. | DOI | Zbl

[26] Shan, Z., Du, Q.: Trigonometric wavelet method for some elliptic boundary value problems. J. Math. Anal. Appl. 344 (2008), 1105-1119. | DOI | MR | Zbl

[27] Twizell, E. H.: An explicit difference method for the wave equation with extended stability range. BIT 19 (1979), 378-383. | DOI | MR | Zbl