Kybernetika 48 no. 5, 939-957, 2012

Numerical solution of second order one-dimensional linear hyperbolic equation using trigonometric wavelets

Mahmood Jokar and Mehrdad Lakestani

Abstract:

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.

Keywords:

telegraph equation, trigonometric wavelets, hermite interpolation, operational matrix of derivative

Classification:

65T60, 65T40, 65L60, 35L20

paper.pdf

References:

  1. B. Alpert, G. Beylkin, R. Coifman and V. Rokhlin: Wavelet-like bases for the fast solution of second-kind integral equation. SIAM J. Sci. Comput. 14 (1993), 159-184.   CrossRef
  2. C. K. Chui and H. N. Mhaskar: On trigonometric wavelets. Constr. Approx. 9 (1993), 167-190.   CrossRef
  3. C. K. Chui: An Introduction to Wavelets. Academic Press, Boston 1992.   CrossRef
  4. W. Dahmen, S. Prössdorf and R. Schneider: Wavelet approximation methods for pseudodifferential equations. In: Stability and Convergence, Math. Z. 215 (1994), 583-620.   CrossRef
  5. M. Dehghan: 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.   CrossRef
  6. M. Dehghan: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simulation 71 (2006), 16-30.   CrossRef
  7. M. Dehghan: Implicit collocation technique for heat equation with non-classic initial condition. Internat. J. Non-Linear Sci. Numer. Simul. 7 (2006), 447-450.   CrossRef
  8. M. Dehghan and A. Shokri: A numerical method for solving the hyperbolic telegraph equation. Numer. Methods Partial Differential Equations 24 (2008), 1080-1093.   CrossRef
  9. M. Dehghan and M. Lakestani: 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.   CrossRef
  10. J. Gao and Y. L. Jiang: Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel. J. Comput. Appl. Math. 215 (2008), 242-259.   CrossRef
  11. R. Kress: Linear Integral Equations. Springer, New York 1989.   CrossRef
  12. M. Lakestani and M. Dehghan: 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.   CrossRef
  13. M. Lakestani, M. Razzaghi and M. Dehghan: Semiorthogonal wavelets approximation for Fredholm integro-differential equations. Math. Prob. Engrg. (2006), 1-12.   CrossRef
  14. M. Lakestani and B. N. Saray: Numerical solution of telegraph equation using interpolating scaling functions. Comput. Math. Appl. 60 (2010), 7, 1964-1972.   CrossRef
  15. M. Lakestani, M. Jokar and M. Dehghan: Numerical solution of nth-Order Integro-Differential equations using trigonometric wavelets. Numer. Math. Methods Appl. Sci. 34 (2011), 11, 1317-1329.   CrossRef
  16. L. Lapidus, G. F. Pinder and : Numerical Solution of Partial Differential Equations in Science and Engineering. Wiley, New York 1982.   CrossRef
  17. G. G. Lorentz: Convergence theorems for polynomials with many zeros. Math. Z. 186 (1984), 117-123.   CrossRef
  18. G. G. Lorentz and R. A. Lorentz: Mathematics from Leningrad to Austin. In: Selected Works In Real, Functional And Numerical Analysis, (1997).   CrossRef
  19. 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 non-linear hyperbolic equations with variable coefficients. J. Comput. Appl. Math. 72 (1996), 421-431.   CrossRef
  20. R. K. Mohanty: An unconditionally stable difference scheme for the one-space dimensional linear hyperbolic equation. Appl. Math. Lett. 17 (2004), 101-105.   CrossRef
  21. 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.   CrossRef
  22. A. Mohebbi and M. Dehghan: High order compact solution of the one-space-dimensional linear hyperbolic equation. Numer. Methods Partial Differential Equations 24 (2008), 1222-1235.   CrossRef
  23. T. V. Petersdorff and C. Schwab: Wavelet approximation for first kind integral equations on polygons. Numer. Math. 74 (1996), 479-516.   CrossRef
  24. E. Quak: Trigonometric wavelets for hermite interpolation. J. Math. Comput. 65 (1996), 683-722.   CrossRef
  25. M. Shamsi and M. Razzaghi: Solution of Hallen's integral equation using multiwavelets. Comput. Phys. Comm. 168 (2005), 187-197.   CrossRef
  26. Z. Shan and Q. Du: Trigonometric wavelet method for some elliptic boundary value problems. J. Math. Anal. Appl. 344 (2008), 1105-1119.   CrossRef
  27. E. H. Twizell: An explicit difference method for the wave equation with extended stability range. BIT 19 (1979), 378-383.   CrossRef