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Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind

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Abstract

This paper presents a new numerical approximation method to solve a system of nonlinear Fredholm integral equations of second kind. Spectral collocation method and their properties are applied to determine the general solution procedure for nonlinear Fredholm integral equations (FIEs). The convergence and error analysis of spectral collocation method are incorporated for the given nonlinear model. Legendre–Gauss–Lobatto (LGL) points are used as collocation points with various Legendre–Gauss quadrature with weight functions. The use of Legendre polynomials, together with the Gauss quadrature collocation points is well known for the accurate approximations that converge exponentially. Finally, we validate our theoretical results with a number of numerical examples, which further enhance the efficiency of our proposed scheme.

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References

  • Adomian G (1991) A review of the decomposition method and some recent result for nonlinear equations. Comput Math Appl 21(5):101–127

    Article  MathSciNet  Google Scholar 

  • Adomian G (1994) Solution of physical problems by decomposition. Comput Math Appl 27(9/10):145–154

    Article  MathSciNet  Google Scholar 

  • Akyuz-Dascioglu A (2004) Chebyshev polynomial solutions of systems of linear integral equations. Appl Math Comput 151:221–232

    MathSciNet  MATH  Google Scholar 

  • Ali Liaqat, Islam Saeed, Gul Taza (2017) A simple algorithm for exact solutions of systems of linear and nonlinear integro-differential equations. Appl Math Comput 307:311–320

    MathSciNet  MATH  Google Scholar 

  • Babolian E, Biazar J (2000) Solution of a systems of nonlinear Volterra equations by Adomian decomposition method. Far East J Math Sci 2(6):945–945

    Google Scholar 

  • Babolian E, Biazar J (2001) Solution of a systems of linear Volterra equations by Adomian decomposition method. Far East J Math Sci 7(1):17–25

    MathSciNet  MATH  Google Scholar 

  • Babolian E, Biazar J, Vahidi AR (2004) The decomposition method applied to systems of Fredholm integral equations of the second kind. Appl Math Comput 148:443–452

    MathSciNet  MATH  Google Scholar 

  • Bakodah HO A comparison study between a Chebyshev collocation method and the Adomian decomposition method for solving linear system of Fredholm integral equations of the second kind, Journal of King AbdulAziz University, in press

  • Bakodah HO (2012) Some modifications of adomian decomposition method applied to nonlinear system of fredholm integral equations of the second kind. Int J Contemp Math Sci 7(19):929–942

    MathSciNet  MATH  Google Scholar 

  • Berenguer MI, Gámez D (2015) Study on convergence and error of a numerical method for solving systems of nonlinear Fredholm-Volterra integral equations of Hammerstein type. Appl Anal. https://doi.org/10.1080/00036811.2015.1096346

    Article  MATH  Google Scholar 

  • Biazar J, Babolian E, Kember G, Nouri A, Islam R (2003) An alternate algorithm for computing Adomian polynomials in special cases. Appl Math Comput 138:523–529

    MathSciNet  MATH  Google Scholar 

  • De Bonie MC, Laurita C (2008) Numerical treatment of second kind Fredholm integral equations systems on bounded intervals. J Comput Appl Math 217:64–87

    Article  MathSciNet  Google Scholar 

  • Delves LM, Mohamed JL (1985) Computational methods for integral equations. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Gabet L (1994) The theoretical foundation of Adomian method. Comput Math Appl 27(12):41–52

    Article  MathSciNet  Google Scholar 

  • Ivaz K, Mostahkam BS (2006) Newton-Taw numerical solution of a system of nonlinear Fredholm integral equations of second kind. Appl Comput Math 5(2):201–208

    MathSciNet  MATH  Google Scholar 

  • Maleknejad K, Shahrezaee M, Khatami H (2005) Numerical solution of integral equations system of the second kind by block-pulse functions. Appl Math Comput 166:15–24

    MathSciNet  MATH  Google Scholar 

  • Maleknejad K, Aghazadeh N, Rabbani M (2006) Numerical solution of the second kind Fredholm integral equations system by using a Taylor-series expansion method. Appl Math Comput 175:1229–1234

    MathSciNet  MATH  Google Scholar 

  • Saeed RK (2008) Homotopy perturbation method for solving systems of nonlinear Fredholm integral equations of the second kind. J Appl Sci Res 4(10):1166–1173

    Google Scholar 

  • Saeed Rostam K (2008) Homotopy perturbation method for solving system of nonlinear fredholm integral equations of the second kind. J Appl Sci Res 4(10):1166–1173

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, COMSATS University, Park Road Chak Shahzad, Islamabad, Pakistan

    Sami Ullah Khan & Ishtiaq Ali

  2. Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa, Kingdom of Saudi Arabia

    Ishtiaq Ali

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  1. Sami Ullah Khan
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  2. Ishtiaq Ali
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Correspondence to Ishtiaq Ali.

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Communicated by Hui Liang.

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Khan, S.U., Ali, I. Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind. Comp. Appl. Math. 38, 125 (2019). https://doi.org/10.1007/s40314-019-0897-2

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  • DOI: https://doi.org/10.1007/s40314-019-0897-2

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