Existence and approximation of a solution of boundary value problems for delay integro-differential equations

Igor Cherevko; Andrew Dorosh

doi:10.33993/jnaat442-1054

Authors

Igor Cherevko Yuriy Fedkovych Chernivtsi National University, Ukraine
Andrew Dorosh Yuriy Fedkovych Chernivtsi National University, Ukraine

DOI:

https://doi.org/10.33993/jnaat442-1054

Keywords:

boundary value problems, spline functions, delay integro-differential equations, cubic splines

Abstract views: 309

Abstract

The properties of a solution of the boundary value problems for delay integro-differential equations are investigated, the conditions for the existence and uniqueness of their solution are found. An iterative scheme of finding an approximate solution using cubic splines with defect two is suggested. The sufficient conditions for the convergence of the iterative scheme are obtained.

Downloads

Download data is not yet available.

References

Alberg, J., Nilson, E., Walsh, J. The theory of splines and their applications. Academic, New York, 1967.

Andreeva, E., Kolmanovsky, V., Shayhet, D. Control of systems with aftereffect. Nauka, Moscow, 1992.

Athanasiadou, E.S. On the existence and uniqueness of solutions of boundary problems for second order functional differential equations. Mathematica Moravica, 17 (2013) 1, pp.51-57. DOI: https://doi.org/10.5937/MatMor1301051A

Biga, A., Gaber, R. Existence, uniqueness and approximation for the solution of a second order neutral differential equation with delay in Banach spaces. Mathematica, 49 (2007) 2, pp.117-130.

Dorosh, A., Cherevko, I. Application of spline functions for approximating solutions of linear boundary value problems with delay. Mathematical and computer modelling. Series: Physical and mathematical sciences, 10 (2014), pp.80-88.

Cherevko, I.M., Yakimov, I.V. Numerical method of solving boundary value problems for integro-differential equations with deviating argument. Ukrainian Mathematical Journal, 41 (1989) 6, pp.854-860, http://doi.org/10.1007/BF01060583 DOI: https://doi.org/10.1007/BF01060583

Grim, L.J., Schmitt, K. Boundary value problems for delay differential equations. Bull. Amer. Math. Soc, 74 (1968) 5, pp.997-1000, http://doi.org/10.1090/S0002-9904-1968-12114-7 DOI: https://doi.org/10.1090/S0002-9904-1968-12114-7

Hartman, F. Ordinary differential equations. The Society for Industrial and Applied Mathematics, Philadelphia, 2002. DOI: https://doi.org/10.1137/1.9780898719222

Kamensky, G., Myshkis, A. Boundary value problems for nonlinear differential equations with deviating argument of neutral type. Differential equations, 8 (1972) 12, pp.2171-2179.

Nastasyeva, N., Cherevko, I. Cubic splines with defect two and their applications to boundary value problems. Bulletin of Kyiv University. Physics and mathematics, 1 (1999), pp.~69--73.

Nikolova, T.S., Bainov, D.D. Application of spline-functions for the construction of an approximate solution of boundary problems for a class of functional-differential equations. Yokohama Math. J, 29 (1981) 1, pp.108-122.