Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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USING CROOKED LINES FOR THE HIGHER ACCURACY IN SYSTEM OF INTEGRAL EQUATIONS

  • Hashemiparast, S.M. (Department of Mathematics, Faculty of science, K.N. Toosi University of Technology) ;
  • Sabzevari, M. ;
  • Fallahgoul, H.
  • Received : 2010.03.12
  • Accepted : 2010.06.14
  • Published : 2011.01.30
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Abstract

The numerical solution to the linear and nonlinear and linear system of Fredholm and Volterra integral equations of the second kind are investigated. We have used crooked lines which includ the nodes specified by modified rationalized Haar functions. This method differs from using nominal Haar or Walsh wavelets. The accuracy of the solution is improved and the simplicity of the method of using nominal Haar functions is preserved. In this paper, the crooked lines with unknown coefficients under the specified conditions change the system of integral equations to a system of equations. By solving this system the unknowns are obtained and the crooked lines are determined. Finally, error analysis of the procedure are considered and this procedure is applied to the numerical examples, which illustrate the accuracy and simplicity of this method in comparison with the methods proposed by these authors.

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References

  1. S.M. Hashemiparast, H. Fallahgoul, Fourier series approximation for periodic solution of system of integral equations using Szego-Bernstein weights, IJCM. (2009) is appeared.
  2. B. Asadi, M. Tavassoli Kajani, A. Hadi Vencheh, Solving second kind integral equations with hybrid fourier and block-pulse functions, Appl. Math. comput. 160 (2005) 517-522 . https://doi.org/10.1016/j.amc.2003.11.038
  3. M. Tavassoli Kajani, A. Hadi Vencheh, Solving second kind integral equations with hybrid Chebyshev and block-pulse functions, Appl. Math. comput. 163 (2005) 71-77. https://doi.org/10.1016/j.amc.2003.11.044
  4. M. Ohkita, Y. Kobayashi, An application of rationalized Haar functions to solution of linear differential equations, IEEE Trans. Circuits Syst. 9 (1986) 853-862.
  5. M. Ohkita, Y. Kobayashi, An application of rationalized Haar functions to solution of linear partial differential equations, Math. Comput. Simulation 30 (1988) 419-428. https://doi.org/10.1016/0378-4754(88)90055-9
  6. E. Babolian, A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, comput. Appl. Math. in press, (2008)
  7. K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solving linear integral equations, Appl. Math. comput. 162 (2) (2005) 1119-1129.
  8. K. Maleknejad, S. Sohrabi, Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Appl. Math. comput. 188 (2007) 123-128. https://doi.org/10.1016/j.amc.2006.09.099
  9. Y. Ordokhani, M. Razzaghi, Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rationalized Haar functions, Appl. Math. Letters. 21 (2008) 4-9. https://doi.org/10.1016/j.aml.2007.02.007
  10. M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, comput. Appl. Math. 200 (1) (2007) 12-20.
  11. M. Razzaghi, J. Nazarzadeh, Walsh functions, Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 23, 19999, pp. 429-440.
  12. R.T. Lynch, J.J. Reis, Haar transform image conding, in: Proceedings of the National Telecommunications Conference, Dallas, TX, 1976, pp. 44.3-1-44.3.
  13. J.J. Reis, R.T. Lynch, J. Butman, Adaptive Haar transform video bandwidth reduction system for RPV's, in: Proceeding of Annual Meeting of Society of Photo-Optic Institute of Engineering (SPIE), San Diago, CA, 1976, pp. 24-35.
  14. J. Xiao, L. Wen, D. Zhang, Solving second kind Fredholm integral equations by periodic wavelet Galerkin method, Appl. Math. comput. (2005).
  15. K. Atkinson, W. Han, Theoretical Numerical Analysis, Springer. (2005).
  16. K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, (1997).
  17. P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press. (1997).