A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept “Strongly Equivalent”
Applied and Computational Mathematics
Volume 7, Issue 1-1, January 2018, Pages: 12-17
Received: Apr. 30, 2017; Accepted: May 2, 2017; Published: May 13, 2017
Views 2299      Downloads 154
Gholamreza Karamali, Shahid Sattari Aeronautical University of Science and Technology, South Mehrabad, Tehran, Iran
Babak Shiri, Shahid Sattari Aeronautical University of Science and Technology, South Mehrabad, Tehran, Iran
Elham Sefidgar, Atatürk University Faculty of Science, Department of Mathematics, Erzurum, Turkey
Article Tools
Follow on us
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.
Volterra Integral Equation, Volterra Equation, Integral equation, Discontinuous Piecewise Polynomial Spaces, Collocation Methods
To cite this article
Gholamreza Karamali, Babak Shiri, Elham Sefidgar, A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept “Strongly Equivalent”, Applied and Computational Mathematics. Special Issue: Singular Integral Equations and Fractional Differential Equations. Vol. 7, No. 1-1, 2018, pp. 12-17. doi: 10.11648/j.acm.s.2018070101.12
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
H. Brunner, “Collocation Methods for Volterra Integral and Related Functional Equations,” Cambridge university press, 2004.
M. V. Bulatov, “Transformations of differential-algebraic systems of equations,” hurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 1996.
V. F. Chistyakov, “Algebro-Differential Operators with Finite-Dimensional Core,” Novosibirsk: Naukka, Siberian Publishing Company RAS., 1996.
C. W. Gear, “Differential algebraic equations, indices and integral algebraic equations,” SIAM J. Numer. Anal., 1990, 27(6), 1527-1534.
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.
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.
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.
P. Kunkel and Mehrmann, “Differential-algebraic equations: analysis and numerical solution,” European Mathematical Society, 2006.
P. K. Lamm, “A survey of regularization methods for first-kind Volterra equations,” Springer Vienna. 2000.
H. Liang, and H. Brunner, “Integral-Algebraic Equations: Theory of Collocation Methods I,” SIAM Journal on Numerical Analysis, 2013, 51(4), 2238-2259.
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.
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.
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.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186