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On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method

Year 2017, Volume: 5 Issue: 3, 273 - 283, 01.07.2017

Amr M. S. Mahdy Emad M.h. Mohamed

Abstract

In this paper, shifted Chebyshev polynomials of the third kind method is presented to solve numerically the Fredholm, Volterra-Hammerstein integral equations. The proposed method converts the equation system of linear or non-linear algebraic equations, which can be solved. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique. All computations are done using Mathematica 7.

Keywords

Fredholm-Hammerstein integral equations Volterra integral equation shifted Chebyshev polynomials of the third kind method

References

  • R. P. Agarwal, Boundary value problems for higher order integro- differential equations, Nonlinear Anal. Theory Methods Appl., 9 (1983), 259-270.
  • L. C. Andrews, Special Functions For Engineers and Applied Mathematicians, Macmillan publishing company, New York, (1985).
  • E. Babolian, F. Fattahzadeh, E. Golpar Raboky, A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type, Applied Mathematics and Computation, 189 (2007), 641-46.
  • A. K. Borzabadi, A. V. Kamyad and H. H. Mehne, A different approach for solving the nonlinear Fredholm integral equations of the second kind, Applied Mathematics and Computation, 173 (2006), 724-735.
  • L. M. Delves and J. L. Mohamad, Computational Methods for Integral Equations, Cambridge University press, (1985).
  • M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method, International Journal of Pure and Applied Mathematics, 74(3) (2012), 287-297.
  • M. M. Khader, N. H. Sweilam and A.M.S. Mahdy, An efficient numerical method for solving the fractional diffusion equation, Journal of Applied Mathematics and Bioinformatics, 1 (2011), 1-12.
  • M. M. Khader, Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials, Arab Journal of Mathematical Sciences, 18 (2012), 61-71.
  • M. M. Khader, Numerical solution of nonlinear multi-order fractional dif- ferential equations by implementation of the operational matrix of frac- tional derivative, Studies in Nonlinear Sciences, 2(1) (2011), 5-12.
  • S. T. Mohamed and M. M. Khader, Numerical solutions to the second order Fredholm integro-differential equations using the spline functions expansion, Global Journal of Pure and Applied Mathematics, 34 (2011), 21-29.
  • N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coeffcients, Journal of the Franklin Institute, 345 (2008), 839-850.
  • M. Shahrezaee, Solving an integro-differential equation by Legendre polynomials and Block-pulse functions, Dynamical Systems and Applications, (2004), 642-647.
  • N. H. Sweilam, Fourth order integro-differential equations using variational iteration method, Comput. Maths. Appl., 54 (2007), 1086-1091.
  • N. H. Sweilam, M. M. Khader and R.F. Al-Bar, Homotopy perturbation method for linear and nonlinear system of fractional integro-differential equations, International Journal of Computational Mathematics and Numerical Simulation, 1 (2008), 73-87.
  • N. H. Sweilam and M.M. Khader, A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations, ANZIAM, 51 (2010), 464-475.
  • N. H. Sweilam and M. M. Khader, Semi exact solutions for the bi-harmonic equation using homotopy analysis method, World Applied Sciences Journal, 13 (2011), 1-7.
  • N. H. Sweilam, M. M. Khader, and W. Y. Kota, On the Numerical Solution of Hammerstein Integral Equations using Legendre Approximation International Journal of Applied Mathematical, Research.1(1) (2012), 65-76
  • S. Youse and M. Razzaghi, Legendre wavelet method for the nonlinear Volterra-Fredholm integral equations, Math. Comp. Simul., 70 (2005), 1-8.
  • Z. Esmailzadeh and A. Jafarian, On the Numerical Solution of Urysohn Integral Equations using Legendre Approximation, Journal of Mathematical Modeling, 1(2013),76-84.
  • N. H. Sweilam, A. M. Nagy and A. A. El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind, Journal of King Saud University Science, 28, (2016), 41-47.

Year 2017, Volume: 5 Issue: 3, 273 - 283, 01.07.2017

Amr M. S. Mahdy Emad M.h. Mohamed

Abstract

References

  • R. P. Agarwal, Boundary value problems for higher order integro- differential equations, Nonlinear Anal. Theory Methods Appl., 9 (1983), 259-270.
  • L. C. Andrews, Special Functions For Engineers and Applied Mathematicians, Macmillan publishing company, New York, (1985).
  • E. Babolian, F. Fattahzadeh, E. Golpar Raboky, A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type, Applied Mathematics and Computation, 189 (2007), 641-46.
  • A. K. Borzabadi, A. V. Kamyad and H. H. Mehne, A different approach for solving the nonlinear Fredholm integral equations of the second kind, Applied Mathematics and Computation, 173 (2006), 724-735.
  • L. M. Delves and J. L. Mohamad, Computational Methods for Integral Equations, Cambridge University press, (1985).
  • M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method, International Journal of Pure and Applied Mathematics, 74(3) (2012), 287-297.
  • M. M. Khader, N. H. Sweilam and A.M.S. Mahdy, An efficient numerical method for solving the fractional diffusion equation, Journal of Applied Mathematics and Bioinformatics, 1 (2011), 1-12.
  • M. M. Khader, Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials, Arab Journal of Mathematical Sciences, 18 (2012), 61-71.
  • M. M. Khader, Numerical solution of nonlinear multi-order fractional dif- ferential equations by implementation of the operational matrix of frac- tional derivative, Studies in Nonlinear Sciences, 2(1) (2011), 5-12.
  • S. T. Mohamed and M. M. Khader, Numerical solutions to the second order Fredholm integro-differential equations using the spline functions expansion, Global Journal of Pure and Applied Mathematics, 34 (2011), 21-29.
  • N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coeffcients, Journal of the Franklin Institute, 345 (2008), 839-850.
  • M. Shahrezaee, Solving an integro-differential equation by Legendre polynomials and Block-pulse functions, Dynamical Systems and Applications, (2004), 642-647.
  • N. H. Sweilam, Fourth order integro-differential equations using variational iteration method, Comput. Maths. Appl., 54 (2007), 1086-1091.
  • N. H. Sweilam, M. M. Khader and R.F. Al-Bar, Homotopy perturbation method for linear and nonlinear system of fractional integro-differential equations, International Journal of Computational Mathematics and Numerical Simulation, 1 (2008), 73-87.
  • N. H. Sweilam and M.M. Khader, A Chebyshev pseudo-spectral method for solving fractional order integro-differential equations, ANZIAM, 51 (2010), 464-475.
  • N. H. Sweilam and M. M. Khader, Semi exact solutions for the bi-harmonic equation using homotopy analysis method, World Applied Sciences Journal, 13 (2011), 1-7.
  • N. H. Sweilam, M. M. Khader, and W. Y. Kota, On the Numerical Solution of Hammerstein Integral Equations using Legendre Approximation International Journal of Applied Mathematical, Research.1(1) (2012), 65-76
  • S. Youse and M. Razzaghi, Legendre wavelet method for the nonlinear Volterra-Fredholm integral equations, Math. Comp. Simul., 70 (2005), 1-8.
  • Z. Esmailzadeh and A. Jafarian, On the Numerical Solution of Urysohn Integral Equations using Legendre Approximation, Journal of Mathematical Modeling, 1(2013),76-84.
  • N. H. Sweilam, A. M. Nagy and A. A. El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind, Journal of King Saud University Science, 28, (2016), 41-47.
There are 20 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Amr M. S. Mahdy This is me

Emad M.h. Mohamed This is me

Publication Date July 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 3

Cite

APA Mahdy, A. M. S., & Mohamed, E. M. (2017). On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method. New Trends in Mathematical Sciences, 5(3), 273-283.
AMA Mahdy AMS, Mohamed EM. On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method. New Trends in Mathematical Sciences. July 2017;5(3):273-283.
Chicago Mahdy, Amr M. S., and Emad M.h. Mohamed. “On the Numerical Solution of Hammerstein Integral Equations Using Shifted Chebyshev Polynomials of the Third Kind Method”. New Trends in Mathematical Sciences 5, no. 3 (July 2017): 273-83.
EndNote Mahdy AMS, Mohamed EM (July 1, 2017) On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method. New Trends in Mathematical Sciences 5 3 273–283.
IEEE A. M. S. Mahdy and E. M. Mohamed, “On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method”, New Trends in Mathematical Sciences, vol. 5, no. 3, pp. 273–283, 2017.
ISNAD Mahdy, Amr M. S. - Mohamed, Emad M.h. “On the Numerical Solution of Hammerstein Integral Equations Using Shifted Chebyshev Polynomials of the Third Kind Method”. New Trends in Mathematical Sciences 5/3 (July 2017), 273-283.
JAMA Mahdy AMS, Mohamed EM. On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method. New Trends in Mathematical Sciences. 2017;5:273–283.
MLA Mahdy, Amr M. S. and Emad M.h. Mohamed. “On the Numerical Solution of Hammerstein Integral Equations Using Shifted Chebyshev Polynomials of the Third Kind Method”. New Trends in Mathematical Sciences, vol. 5, no. 3, 2017, pp. 273-8.
Vancouver Mahdy AMS, Mohamed EM. On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method. New Trends in Mathematical Sciences. 2017;5(3):273-8.
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