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Systems of nonlinear Volterra integro-differential equations

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Abstract

An efficient method based on operational Tau matrix is developed, to solve a type of system of nonlinear Volterra integro-differential equations (IDEs). The presented method is also modified for the problems with separable kernel. Error estimation of the new schemes are analyzed and discussed. The advantages of this approach and its modification is that, the solution can be expressed as a truncated Taylor series, and the error function at any stage can be estimated. Methods are applied on the four problems with separable kernel to show the applicability and efficiency of our schemes, specially for those problems at broad intervals.

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References

  1. Dehghan, M., Shakeri, F.: Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method. Prog. Electromagn. Res. 78, 361–376 (2008)

    Article  Google Scholar 

  2. Siddiqui, A.M., Mahmood, R., Ghori, G.K.: Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 352, 404–410 (2006)

    Article  MATH  Google Scholar 

  3. Mariani, M.C., Salas, M., Vivas, A.: Solutions to integro-differential systems arising in hydrodynamic models for charged transport in semiconductors. Nonlinear Anal. RWA 11, 3912–3922 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Tamizhmani, K.M., Satsuma, J., Grammaticos, B., Ramani, V.: Nonlinear integro differential equations as a discreet systems. Inverse Probl. 15, 787–791 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Saberi-Nadjafi, J., Tamamgar, M.: Variational iteration method: a highly promising method for solving the system of integro-differential equations. Comput. Math. Appl. 56, 346–351 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Arikoglu, A., Oskol, I.: Solution of integral and integro-differential equation systems by using differential transform method. Comput. Math. Appl. 56, 2411–2417 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Biazar, J., Ghazevini, H., Eslami, M.: He’s homotopy perturbation method for systems of integro-differential equations. Chaos Solitons Fractals 39, 1253–1258 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Aminikhah, H., Salahi, M.: A new homotopy perturbation method for system of nonlinear integro-differential equations. Int. J. Comput. Math. 87(5), 1186–1194 (2010)

    Article  MATH  Google Scholar 

  9. Golbabai, A., Mammadov, M., Seifollahi, S.: Solving a system of nonlinear integral equations by an RBF network. Comput. Math. Appl. 57, 1651–1658 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Abbasbandy, S., Taati, A.: 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 

  11. Zarebnia, M., Abadi, M.G.: Numerical solutions of system of nonlinear second-order integro-differential equations. Comput. Math. Appl. 60, 591–601 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Berenguer, M.I., Garralda-Guillem, A.I., Ruiz Galán, M.: An approximation method for solving systems of Volterra integro-differential equations. Appl. Numer. Math. (2011, in press). doi:10.1016/j.apnum.2011.03.007

  13. Kanwal, R.P., Liu, K.C.: A Taylor expansion approach for solving integral equations. J. Math. Educ. Sci. Technol. 20, 411–414 (1989)

    Article  MATH  Google Scholar 

  14. Gülsu, M., Sezer, M.: The approximate solution of high-order linear difference equations with variable coefficients in terms of Taylor polynomials. Appl. Math. Comput. 168, 76–88 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gülsu, M., Sezer, M.: A Taylor polynomial approach for solving differential-difference equations. J. Comput. Appl. Math. 186, 349–364 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gülsu, M., Sezer, M.: Approximations to the solution of linear Fredholm integro differential-difference equation of high-order. J. Franklin Inst. 343, 720–737 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Akyüz-Daşcıoǧlu, A., Yaslan, H.C.: An approximation method for the solution of nonlinear integral equations. Appl. Math. Comput. 174, 619–629 (2006)

    Article  MathSciNet  Google Scholar 

  18. Gülsu, M., Sezer, M.: A Taylor collocation method for solving high-order linear Pantograph equations with linear function argument. Numer. Methods Partial Differ. Equ. (2011). doi:10.1002/num.20600

    Google Scholar 

  19. Sezer, M., Yalçinbaş, S., Şahin, N.: Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations. Appl. Math. Comput. 214, 406–416 (2008)

    Article  MATH  Google Scholar 

  20. Maleknejad, K., Mahmoudi, Y.: Taylor polynomial solution of high-order nonlinear Volterra–Fredholm integro-differential equation. Appl. Math. Comput. 145, 641–653 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kurt, N., Sezer, M.: Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients. J. Franklin Inst. 345, 839–850 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ghasemi, M., Kajani, M.T., Babolian, E.: Numerical solutions of the nonlinear Volterra-Fredholm integral equations by using homotopy perturbation method. Appl. Math. Comput. 188, 446–449 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Sezer, M.: Taylor polynomial solution of Volterra integral equations. Int. J. Math. Educ. Sci. Technol. 25, 625–633 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Yalçinbas, S.: Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations. Appl. Math. Comput. 127, 196–206 (2002)

    Article  Google Scholar 

  25. Ortiz, E.L., Samara, H.: 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 

  26. Rasidinia, J., Tahmasebi, A.: A Taylor expansion approach for numerical solution of nonlinear Volterra integro-differential equations. Comput. Math. Appl. (2011, under consideration)

  27. Ebadi, G., Rahimi, M.Y., Shahmorad, S.: Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method. Appl. Math. Comput. 188, 1580–1586 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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  1. School of Mathematics, Iran University of Science and Technology, Narmak, 16846-13114, Tehran, Iran

    Jalil Rashidinia & Ali Tahmasebi

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  1. Jalil Rashidinia
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  2. Ali Tahmasebi
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Correspondence to Jalil Rashidinia.

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Rashidinia, J., Tahmasebi, A. Systems of nonlinear Volterra integro-differential equations. Numer Algor 59, 197–212 (2012). https://doi.org/10.1007/s11075-011-9484-3

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  • DOI: https://doi.org/10.1007/s11075-011-9484-3

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