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Sparse representation of system of Fredholm integro-differential equations by using alpert multiwavelets

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Abstract

A numerical technique is presented for the solution of system of Fredholm integro-differential equations. The method consists of expanding the required approximate solution as the elements of Alpert multiwavelet functions (see Alpert B. et al. J. Comput. Phys. 2002, vol. 182, pp. 149–190). Using the operational matrix of integration and wavelet transform matrix, we reduce the problem to a set of algebraic equations. This system is large. We use thresholding to obtain a new sparse system; consequently, GMRES method is used to solve this new system. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results.

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References

  1. J. Pour-Mahmoud and M. Y. Rahimi-Ardabili, S. Shahmorad, “Numerical solution of the system of Fredholm integro-differential equations by the Tau method,” Appl. Math. Comput. 168, 465–478 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Rashidinia and M. Zarebnia, “Convergence of approximate solution of system of Fredholm integral equations,” J. Math. Anal. Appl. 333, 1216–1227 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Davari and M. Khanian, “Solution of system of Fredholm integro-differential equations by Adomian decomposition method,” Austral. J. Basic Appl. Sci. 5 (12), 2356–2361 (2011).

    Google Scholar 

  4. P. Oja and D. Saveljeva, “Cubic spline collocation for Volterra integral equations,” Computing, 69, 319–337 (2001).

    Article  MathSciNet  Google Scholar 

  5. B. Zhang, T. Lin, Y. Lin and M. Rao, “Defect correction and a posteriori error estimation of Petrov–Galerkin methods for nonlinear Volterra integro-differential equation,” Appl. Math. 45, 241–263 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  6. E. L. Ortiz and L. Samara, “An operational approach to the tau method for the numerical solution of nonlinear differential equations,” Computing 27, 15–25 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Lakestani, M. Razzaghi, and M. Dehghan, “Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations, Math. Probl. Eng., Article ID 96184, 1–12 (2006).

    Article  MathSciNet  Google Scholar 

  8. M. Lakestani, B. N. Saray, and M. Dehghan, “Numerical solution for the weakly singular Fredholm integrodifferential equations using Legendre multiwavelets,” J. Comput. Appl. Math. 235, 3291–3303 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Razzaghi and S. Yousefi, “Legendre wavelets method for the nonlinear Volterra Fredholm integral equations,” Math. Comput. Simul. 70, 1–8 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Lakestani and M. Dehghan, “Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions,” Int. J. Comput. Math. 87 (6), 1389–1394 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  11. Y. Ren, Â. Zhang, and H. Qiao, “A simple Taylor-series expansion method for a class of second kind integral equations,” J. Comput. Appl. Math. 110, 15–24 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Linz, Analytical and Numerical Methods for Volterra Equations (SIAM, Philadelphia, PA, 1985).

    Book  MATH  Google Scholar 

  13. J. Abdul Jerri, Introduction to Integral Equations with Applications (Wiley, New York, 1999).

    MATH  Google Scholar 

  14. S. Abbasbandy and A. Taati, “Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation,” J. Comput. Appl. Math. 231, 106–113 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Khani, M. Mohseni Moghadam, and S. Shahmorad, “Numerical solution of special class of system of nonlinear Volterra integro-differential equations by a simple high accuracy method,” Bull. Iran. Math. Soc. 34 (2), 141–152 (2008).

    MATH  Google Scholar 

  16. G. Ebadi, M. Y. Rahimi, and S. Shahmorad, “Numerical solution of the system of nonlinear Fredholm integrodifferential equations by the operational Tau method with an error estimation,” Sci. Iran. 14, 546–554 (2007).

    MATH  MathSciNet  Google Scholar 

  17. M. Zarebnia and M. G. Ali Abadi, “Numerical solution of system of nonlinear second-order integro-differential equations,” Comput. Math. Appl. 60, 591–601 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Dai and J. E. Cochran Jr., “Wavelet collocation method for optimal control problems,” J. Optim. Theory Appl. 143, 265–287 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  19. I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  20. Â. Alpert, G. Beylkin, D. Gines, and L. Vozovoi, “Adaptive solution of partial differential equations in multiwavelet bases,” J. Comput. Phys. 182, 149–190 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  21. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992).

    Book  MATH  Google Scholar 

  22. M. Shamsi and M. Razzaghi, “Solution of Hallen’s integral equation using multiwavelets,” Comput. Phys. Commun. 168, 187–197 (2005).

    Article  MATH  Google Scholar 

  23. E. G. Quak and N. Weyrich, “Wavelet on the interval,” Ed. by S. P. Singh (Toim) Approximation Theory, Wavelets, and Applications (Kluwer, 1995), pp. 247–283.

    Chapter  Google Scholar 

  24. M. Shamsi and M. Razzaghi, “Numerical solution of the controlled Duffing oscillator by the interpolating scaling functions,” Electromagn. Waves Appl. 18 (5), 691–705 (2004).

    Article  MathSciNet  Google Scholar 

  25. M. Lakestani and B. N. Saray, “Numerical solution of telegraph equation using interpolating scaling functions,” Comput. Math. Appl. 60, 1964–1972 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Hanwei, L. Kecheng, H. Jianguo, Y. Jiaxian, and L. Peiguo, “A novel wavelet transform matrix for efficient solutions of electromagnetic integral equations,” Proceedings of 1999 International Conference on Computational Electromagnetics and Its Applications, ICCEA’99 (1999).

    Google Scholar 

  27. M. Dehghan, Â. N. Saray, and Ì. Lakestani, “Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers–Huxley equation,” Math. Comput. Model. 55, 1129–1142 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  28. Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual method for solving nonsymmetric linear systems” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  29. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, Philadelphia, 2003).

    Book  MATH  Google Scholar 

  30. J. C. Goswami, A. K. Chan, and Ñ. K. Chui, “On solving first-kind integral equations using wavelets on bounded interval,” IEEE Trans. Antennas Propag. 43 (6), 614–622 (1995).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

    Behzad Nemati Saray & Mehrdad Lakestani

  2. Department of Mathematics and Statistics Mississippi State University, Mississippi State, MS 39762, Washington D.C., USA

    Mohsen Razzaghi

Authors
  1. Behzad Nemati Saray
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  2. Mehrdad Lakestani
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  3. Mohsen Razzaghi
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Correspondence to Behzad Nemati Saray.

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Saray, B.N., Lakestani, M. & Razzaghi, M. Sparse representation of system of Fredholm integro-differential equations by using alpert multiwavelets. Comput. Math. and Math. Phys. 55, 1468–1483 (2015). https://doi.org/10.1134/S0965542515090031

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