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An Iterative Approach for Solving Nonlinear Volterra–Fredholm Integral Equation Using Tension Spline

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Abstract

According to mathematical literature, the parametric spline functions have not been used directly for solving the integral equations so far. In this work, we develop the tension spline approximation to obtain the numerical solution of Volterra–Fredholm integral equation. First, the tension spline method was obtained and then the combination of this method with the quasi-linearization method has been used for solving the nonlinear Volterra–Fredholm integral equation. The error analysis of the method is given. Illustrative examples are included to verify the effectiveness and applicability of the presented approach.

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References

  • Abdou M (2002) Fredholm–Volterra integral equation of the first kind and contact problem. Appl Math Comput 125(2–3):177–193

    MathSciNet  MATH  Google Scholar 

  • Bellman RE, Kalaba RE (1965) Quasilinearization and nonlinear boundary-value problems

  • Bloom F (1979) Asymptotic bounds for solutions to a system of damped integrodifferential equations of electromagnetic theory. South Carolina Univ Columbia Dept of Mathematics Computer Science and Statistics, Technical report

  • Brunner H (1990) On the numerical solution of nonlinear Volterra–Fredholm integral equations by collocation methods. SIAM J Numer Anal 27(4):987–1000

    MathSciNet  MATH  Google Scholar 

  • Brunner H (1992) Implicitly linear collocation methods for nonlinear Volterra equations. Appl Numer Math 9(3–5):235–247

    MathSciNet  MATH  Google Scholar 

  • Ezquerro J, Hernández-Verón M (2018) The majorant principle applied to Hammerstein integral equations. Appl Math Lett 75:50–58

    MathSciNet  MATH  Google Scholar 

  • Frankel JI (1995) A Galerkin solution to a regularized Cauchy singular integro-differential equation. Q Appl Math 53(2):245–258

    MathSciNet  MATH  Google Scholar 

  • Golbabai A, Nikan O, Tousi JR (2016) Note on using radial basis functions method for solving nonlinear integral equations. Commun Numer Anal 2:81–91

    MathSciNet  Google Scholar 

  • Lakshmikantham V, Shahzad N (1994) Further generalization of generalized quasilinearization method. Int J Stoch Anal 7(4):545–552

    MathSciNet  MATH  Google Scholar 

  • Lakshmikantham V, Leela S, Sivasundaram S (1995) Extensions of the method of quasilinearization. J Optim Theory Appl 87(2):379–401

    MathSciNet  MATH  Google Scholar 

  • Maleknejad K, Hadizadeh M (1999) A new computational method for Volterra–Fredholm integral equations. Comput Math Appl 37(9):1–8

    MathSciNet  MATH  Google Scholar 

  • Maleknejad K, Najafi E (2011) Numerical solution of nonlinear Volterra integral equations using the idea of quasilinearization. Commun Nonlinear Sci Numer Simul 16(1):93–100

    MathSciNet  MATH  Google Scholar 

  • Marzban H, Tabrizidooz H, Razzaghi M (2011) A composite collocation method for the nonlinear mixed Volterra-Fredholm–Hammerstein integral equations. Commun Nonlinear Sci Numer Simul 16(3):1186–1194

    MathSciNet  MATH  Google Scholar 

  • Mirzaee F (2017) Numerical solution of nonlinear Fredholm–Volterra integral equations via Bell polynomials. Comput Methods Differ Equ 5(2):88–102

    MathSciNet  MATH  Google Scholar 

  • Mirzaee F, Hadadiyan E (2012) Approximate solutions for mixed nonlinear Volterra–Fredholm type integral equations via modified block-pulse functions. J Assoc Arab Univ Basic Appl Sci 12(1):65–73

    MATH  Google Scholar 

  • Mirzaee F, Hoseini SF (2014) A fibonacci collocation method for solving a class of fredholm–volterra integral equations in two-dimensional spaces. Beni-Suef Univ J Basic Appl Sci 3(2):157–163

    Google Scholar 

  • Mirzaee F, Samadyar N (2018) Convergence of 2D-orthonormal Bernstein collocation method for solving 2D-mixed Volterra-Fredholm integral equations. Trans A Razmadze Math Inst 172(3):631–641

    MathSciNet  MATH  Google Scholar 

  • Mirzaee F, Samadyar N (2020) Explicit representation of orthonormal Bernoulli polynomials and its application for solving Volterra–Fredholm–Hammerstein integral equations. SeMA J 77(1):81–96

    Article  MathSciNet  Google Scholar 

  • Moradi S, Mohammadi Anjedani M, Analoei E (2015) On existence and uniqueness of solutions of a nonlinear Volterra–Fredholm integral equation. Int J Nonlinear Anal Appl 6(1):62–68

    MATH  Google Scholar 

  • Ordokhani Y, Razzaghi M (2008) Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via a collocation method and rationalized Haar functions. Appl Math Lett 21(1):4–9

    MathSciNet  MATH  Google Scholar 

  • Pachpatte B (2008) On a nonlinear Volterra–Fredholm integral equation. Sarajevo J Math 16:61–71

    MathSciNet  MATH  Google Scholar 

  • Pandit SG (1997) Quadratically converging iterative schemes for nonlinear Volterra integral equations and an application. Int J Stoch Anal 10(2):169–178

    MathSciNet  MATH  Google Scholar 

  • Rashidinia J, Mohammadi R (2011) Tension spline solution of nonlinear sine-Gordon equation. Numer Algorithms 56(1):129–142

    MathSciNet  MATH  Google Scholar 

  • Stoer J, Bulirsch R (2013) Introduction to numerical analysis, vol 12. Springer, Berlin

    MATH  Google Scholar 

  • Wazwaz AM (2002) A reliable treatment for mixed volterra–fredholm integral equations. Appl Math Comput 127(2–3):405–414

    MathSciNet  MATH  Google Scholar 

  • Yalçinbaş S (2002) Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations. Appl Math Comput 127(2–3):195–206

    MathSciNet  MATH  Google Scholar 

  • Yousefi S, Razzaghi M (2005) Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Math Comput Simul 70(1):1–8

    MathSciNet  MATH  Google Scholar 

  • Zarebnia M (2013) A numerical solution of nonlinear Volterra–Fredholm integral equations. J Appl Anal Comput 3(1):95–104

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous reviewers and editors very much for their constructive suggestions and valuable comments, which greatly helped us to improve this work.

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Authors and Affiliations

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

    Khosrow Maleknejad & Hamed Shahi Kalalagh

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  1. Khosrow Maleknejad
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  2. Hamed Shahi Kalalagh
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Correspondence to Khosrow Maleknejad.

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Maleknejad, K., Shahi Kalalagh, H. An Iterative Approach for Solving Nonlinear Volterra–Fredholm Integral Equation Using Tension Spline. Iran J Sci Technol Trans Sci 44, 1531–1539 (2020). https://doi.org/10.1007/s40995-020-00963-8

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  • DOI: https://doi.org/10.1007/s40995-020-00963-8

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