Applied and Computational Mathematics

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A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept “Strongly Equivalent”

Received: 30 April 2017    Accepted: 02 May 2017    Published: 13 May 2017
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Abstract

We introduce the concept “strongly equivalent” for integral algebraic equations (IAEs). This definition and its corresponding theorems construct powerful tools for the classifying and analyzing of IAEs (especially numerical analysis). The related theorems with short proofs provide powerful techniques for the complete convergence analysis of discretised collocation methods on discontinuous piecewise polynomial spaces.

DOI 10.11648/j.acm.s.2018070101.12
Published in Applied and Computational Mathematics (Volume 7, Issue 1-1, January 2018)

This article belongs to the Special Issue Singular Integral Equations and Fractional Differential Equations

Page(s) 12-17
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Volterra Integral Equation, Volterra Equation, Integral equation, Discontinuous Piecewise Polynomial Spaces, Collocation Methods

References
[1] H. Brunner, “Collocation Methods for Volterra Integral and Related Functional Equations,” Cambridge university press, 2004.
[2] M. V. Bulatov, “Transformations of differential-algebraic systems of equations,” hurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 1996.
[3] V. F. Chistyakov, “Algebro-Differential Operators with Finite-Dimensional Core,” Novosibirsk: Naukka, Siberian Publishing Company RAS., 1996.
[4] C. W. Gear, “Differential algebraic equations, indices and integral algebraic equations,” SIAM J. Numer. Anal., 1990, 27(6), 1527-1534.
[5] F. Ghoreishi, M. Hadizadeh and S. Pishbin, “On the convergence analysis of the spline collocation method for system of integral algebraic equations of index-2,” Int. J. Comput. Methods, 2012, 9(4), 131-148.
[6] M. Hadizadeh, F. Ghoreishi and S. Pishbin, “Jacobi spectral solution for integral algebraic equations of index-2,” Applied Numerical Mathematics, 2011, 61(1), 131-148.
[7] J. P. Kauthen, “The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods,” Math. Comp., 2001, 70(236), 1503-1514.
[8] P. Kunkel and Mehrmann, “Differential-algebraic equations: analysis and numerical solution,” European Mathematical Society, 2006.
[9] P. K. Lamm, “A survey of regularization methods for first-kind Volterra equations,” Springer Vienna. 2000.
[10] H. Liang, and H. Brunner, “Integral-Algebraic Equations: Theory of Collocation Methods I,” SIAM Journal on Numerical Analysis, 2013, 51(4), 2238-2259.
[11] S. Pishbin, “Optimal convergence results of piecewise polynomial collocation solutions for integral–algebraic equations of index-3,” J. Comput. Appl. Math, 2015, 279(1), 209-224.
[12] B. Shiri, “Numerical solution of higher index nonlinear integral algebraic equations of Hessenberg type using discontinuous collocation methods,” Mathematical Modelling and Analysis, 2014, 19(1), 99-117.
[13] B. Shiri, S. Shahmorad and G. Hojjati, “Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of Hessenberg type,” AMCS, 2013, 23(2), 341-355.
Author Information
  • Shahid Sattari Aeronautical University of Science and Technology, South Mehrabad, Tehran, Iran

  • Shahid Sattari Aeronautical University of Science and Technology, South Mehrabad, Tehran, Iran

  • Atatürk University Faculty of Science, Department of Mathematics, Erzurum, Turkey

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    Gholamreza Karamali, Babak Shiri, Elham Sefidgar. (2017). A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept “Strongly Equivalent”. Applied and Computational Mathematics, 7(1-1), 12-17. https://doi.org/10.11648/j.acm.s.2018070101.12

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    ACS Style

    Gholamreza Karamali; Babak Shiri; Elham Sefidgar. A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept “Strongly Equivalent”. Appl. Comput. Math. 2017, 7(1-1), 12-17. doi: 10.11648/j.acm.s.2018070101.12

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    AMA Style

    Gholamreza Karamali, Babak Shiri, Elham Sefidgar. A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept “Strongly Equivalent”. Appl Comput Math. 2017;7(1-1):12-17. doi: 10.11648/j.acm.s.2018070101.12

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  • @article{10.11648/j.acm.s.2018070101.12,
      author = {Gholamreza Karamali and Babak Shiri and Elham Sefidgar},
      title = {A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept “Strongly Equivalent”},
      journal = {Applied and Computational Mathematics},
      volume = {7},
      number = {1-1},
      pages = {12-17},
      doi = {10.11648/j.acm.s.2018070101.12},
      url = {https://doi.org/10.11648/j.acm.s.2018070101.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.s.2018070101.12},
      abstract = {We introduce the concept “strongly equivalent” for integral algebraic equations (IAEs). This definition and its corresponding theorems construct powerful tools for the classifying and analyzing of IAEs (especially numerical analysis). The related theorems with short proofs provide powerful techniques for the complete convergence analysis of discretised collocation methods on discontinuous piecewise polynomial spaces.},
     year = {2017}
    }
    

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