# Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

Babak Shiri; Sedaghat Shahmorad; Gholamreza Hojjati

International Journal of Applied Mathematics and Computer Science (2013)

- Volume: 23, Issue: 2, page 341-355
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topBabak Shiri, Sedaghat Shahmorad, and Gholamreza Hojjati. "Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type." International Journal of Applied Mathematics and Computer Science 23.2 (2013): 341-355. <http://eudml.org/doc/256713>.

@article{BabakShiri2013,

abstract = {In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results.},

author = {Babak Shiri, Sedaghat Shahmorad, Gholamreza Hojjati},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {piecewise continuous collocations methods; Volterra integral equations; integral algebraic equations; piecewise continuous collocation methods; high index Volterra integral algebraic equations; convergence; system of first kind Volterra linear integral equations; Newton's iterative method; numerical examples},

language = {eng},

number = {2},

pages = {341-355},

title = {Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type},

url = {http://eudml.org/doc/256713},

volume = {23},

year = {2013},

}

TY - JOUR

AU - Babak Shiri

AU - Sedaghat Shahmorad

AU - Gholamreza Hojjati

TI - Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

JO - International Journal of Applied Mathematics and Computer Science

PY - 2013

VL - 23

IS - 2

SP - 341

EP - 355

AB - In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results.

LA - eng

KW - piecewise continuous collocations methods; Volterra integral equations; integral algebraic equations; piecewise continuous collocation methods; high index Volterra integral algebraic equations; convergence; system of first kind Volterra linear integral equations; Newton's iterative method; numerical examples

UR - http://eudml.org/doc/256713

ER -

## References

top- Atkinson, K. (2001). Theoretical Numerical Analysis: A Functional Analysis Framework, Springer-Verlag, New York, NY. Zbl0966.65001
- Bandrowski, B., Karczewska, A. and Rozmej, P. (2010). Numerical solutions to integral equations equivalent to differential equations with fractional time, International Journal of Applied Mathematics and Computer Science 20(2): 261-269, DOI: 10.2478/v10006-010-0019-1. Zbl1201.35020
- Brunner, H. (1977). Discretization of Volterra integral equations of the first kind, Mathematics of Computation 31(139): 708-716. Zbl0372.65052
- Brunner, H. (1978). Discretization of Volterra integral equations of the first kind (II), Numerische Mathematik 30(2): 117-136. Zbl0366.65058
- Brunner, H. (2004). Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, New York, NY. Zbl1059.65122
- Bulatov, M.V. (1994). Transformations of differential-algebraic systems of equations, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 34(3): 360-372. Zbl0818.34002
- Bulatov, M.V. (2002). Regularization of degenerate integro-differential equations, Computational Mathematics and Mathematical Physics 42(11): 1602-1608.
- Chistyakov, V.F. (1987). On Singular Systems of Ordinary Differential Equations. Lyapunov Functions and Their Applications, Siberian Publishing House NAUKA, Novosibirsk, pp. 231-239.
- Chistyakov, V.F. (1996). Algebro-Differential Operators With Finite-Dimensional Core, Siberian Publishing House NAUKA, Novosibirsk. Zbl0999.34002
- De Hoog, F.R. and Weiss, R. (1973a). High order methods for Volterra integral equations of the first kind, SIAM Journal on Numerical Analysis 10(4): 647-664. Zbl0261.65086
- De Hoog, F.R. and Weiss, R. (1973b). On the solution of Volterra integral equations of the first kind, Numerische Mathematik 21(1): 22-32. Zbl0262.65078
- Gear, C.W. (1990). Differential algebraic equations indices and integral algebraic equations, SIAM Journal on Numerical Analysis 27(6): 1527-1534. Zbl0732.65061
- Hadizadeh, M., Ghoreishi, F. and Pishbin, S. (2011). Jacobi spectral solution for integral algebraic equations of index-2, Applied Numerical Mathematics 61(1): 131-148. Zbl1206.65260
- Hochstadt, H. (1973). Integral Equations, John Wiley, New York, NY. Zbl0259.45001
- Kauthen, J.P. (1997). The numerical solution of Volterra integral-algebraic equations by collocation methods, Proceedings of the 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Berlin, Germany, Vol. 2, pp. 451-456.
- Kauthen, J.P. (2001). The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods, Mathematics of Computation 70(236): 1503-1514. Zbl0979.65122
- Kauthen, J.-P. and Brunner, H. (1997). Continuous collocation approximations to solutions of first kind Volterra equations, Mathematics of Computation 66(220): 1441-1459. Zbl0890.65138
- Lamm, P.K. (2005). Full convergence of sequential local regularization methods for Volterra inverse problems, Inverse Problems 21(3): 785-803. Zbl1085.65123
- Lamm, P.K. and Scofield, T.L. (2000). Sequential predictorcorrector methods for the variable regularization of Volterra inverse problems, Inverse Problems 16(2): 373-399. Zbl0972.65121
- Saeedi, H., Mollahasani, N., Mohseni Moghadam, M. and Chuev, G.N. (2011). An operational Haar wavelet method for solving fractional Volterra integral equations, International Journal of Applied Mathematics and Computer Science 21(3): 535-547, DOI: 10.2478/v10006-011-0042-x. Zbl1233.65100
- Weiss, R. (1972). Numerical Procedures for Volterra Integral Equations, Ph.D. thesis, Australian National University, Canberra.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.