TY  - JOUR
T1  - Piecewise polynomial numerical method for Volterra integral equations of the fourth-kind with constant delay
AU  - Darania, Parviz
AU  - Pishbin, Saeed
PY  - 2024
DA  - February
DO  - 10.15672/hujms.1055681
JF  - Hacettepe Journal of Mathematics and Statistics
PB  - Hacettepe University
WT  - DergiPark
SN  - 2651-477X
SP  - 74
EP  - 87
VL  - 53
IS  - 1
LA  - en
AB  - This work studies the fourth-kind integral equation as a mixed system of first and second-kind  Volterra integral equations (VIEs) with constant delay. Regularity, smoothing properties and uniqueness of the solution of this mixed system are obtained by using theorems which give the relevant conditions related to first and second-kind  VIEs with delays. A numerical collocation algorithm making use of piecewise polynomials is designed and the global convergence of the proposed numerical method is established. Some typical numerical experiments are also performed which confirm our theoretical result.
KW  - fourth-kind integral equations
KW  - Volterra integral equations with constant
delay
KW  - piecewise polynomial collocation method
KW  - convergence analysis
CR  - [1] I. Ali, H. Brunner and T. Tang, Spectral methods for pantograph-type differential and
integral equations with multiple delays Front. Math. China 4, 49-61, 2009.
CR  - [2] J. Belair, Population models with state-dependent delays. In: Arino, O., Axelrod, D.E.,
Kimmel, M. (eds.) Mathematical Population Dynamics, 165-176. Marcel Dekker, New
York, 1991.
CR  - [3] H. Brunner, Iterated collocation methods for Volterra integral equations with delay
arguments. Math. Comput. 62, 581-599, 1994.
CR  - [4] H. Brunner, Collocation and continuous implicit Runge-Kutta methods for a class of
delay Volterra integral equations. J. Comput. Appl. Math. 53, 61-72, 1994.
CR  - [5] H. Brunner, The discretization of neutral functional integro-differential equations by
collocation methods. J. Anal. Appl. 18, 393-406, 1999.
CR  - [6] H. Brunner, Collocation methods for Volterra integral and related functional differential
equations. Cambridge university press, Cambridge, 2004.
CR  - [7] H. Brunner and Y. Yatsenko, Spline collocation methods for nonlinear Volterra integral
equations with unknown delay. J. Comput. Appl. Math. 71, 67-81, 1996.
CR  - [8] M.V. Bulatov and M. N. Machkhina, On a class of integro-algebraic equations with
variable integration limits Zh. Sredn. Mat. Obshch. 12 (2), 40-45, 2010.
CR  - [9] M.V. Bulatov, M. N. Machkhina and V.N. Phat, Existence and uniqueness of solutions
to nonlinear integral-algebraic equations with variable limits of integrations Commun.
Appl. Nonlinear Anal. 21 (1), 65-76, 2014
CR  - [10] F. Calio, E. Marchetti and R. Pavani, About the deficient spline collocation method
for particular differential and integral equations with delay. Rend. Sem. Mat. Univ.
Pol. Torino, 61, 287-300, 2003.
CR  - [11] F. Calio, E. Marchetti, R. Pavani and G. Micula, About some Volterra problems solved
by a particular spline collocation. Studia Univ. Babes Bolyai. 48, 45-52, 2003.
CR  - [12] K.L. Cooke,An epidemic equation with immigration. Math. Biosci. 29, 135-158, 1976.
CR  - [13] S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, 1999.
CR  - [14] 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. 9 (4) , 1250048, 2012.
CR  - [15] E. Hairer, Ch. Lubich and S.P. Nrset, Order of convergence of one-step methods for
Volterra integral equations of the second kind, SIAM. J. Numer. Anal. 20, 569-579,
1983.
CR  - [16] V. Horvat, On collocation methods for Volterra integral equations with delay arguments.
Math. Commun. 4, 93-109, 1999.
CR  - [17] Q. Hu, Multilevel correction for discrete collocation solutions of Volterra integral equations
with delay arguments. Appl. Numer. Math. 31, 159-171, 1999.
CR  - [18] J. P. Kauthen, The numerical solution of integral-algebriac equations of index-1 by
polynomial spline collocation methods, Math. Comp. 236, 1503-1514, 2000.
CR  - [19] M. Khasi, F. Ghoreishi and M. Hadizadeh, Numerical analysis of a high order method
for state-dependent delay integral equations. Numerical Algorithms, 66, 177-201, 2013.
UR  - https://doi.org/10.15672/hujms.1055681
L1  - https://dergipark.org.tr/en/download/article-file/2183486
ER  -