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boundary value problems; functional differential equations; difference method; consistency; convergence; methods of arbitrary order
Algorithms for finding an approximate solution of boundary value problems for systems of functional ordinary differential equations are studied. Sufficient conditions for consistency and convergence of these methods are given. In the last section, a construction of methods of arbitrary order is presented.
[1] R. P. Agarwal: Difference Equations and Inequalities, Theory, Methods, and Applications. Marcel Dekker, Inc., New York, 1992. MR 1155840 | Zbl 0925.39001
[2] U. M. Ascher, R. M. M. Mattheij and R. D. Russell: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice Hall, New Jersey, 1988. MR 1000177
[3] S. Atdaev: On a method of two-sided approximations for a boundary value problem. Izv. Acad. Nauk. Turkm. Ser. Fiz.-Mat. Tekhn. Khim. Geol. Nauk, No. 1 (1985), 8–10. (Russian) MR 0798362 | Zbl 0649.34024
[4] P. B. Bailey, L. F. Shampine and P. E. Waltman: Nonlinear Two Point Boundary Value Problems. Academic Press, New York and London, 1968. MR 0230967
[5] V. L. Bakke, Z. Jackiewicz: The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments. Appl. Math. 34 (1989), 1–17. MR 0982339
[6] A. Bellen: A Runge-Kutta-Nyström method for delay differential equations. In: Numerical Boundary Value ODEs, U. M. Ascher, R. D. Russell (eds.), Birkhäuser, Boston, Basel, Stuttgard, 1985, pp. 271–285. MR 0832900 | Zbl 0566.65066
[7] A. Bellen, M. Zennaro: A collocation method for boundary value problems of differential equations with functional arguments. Computing 32 (1984), 307–318. DOI 10.1007/BF02243775 | MR 0748933
[8] S. R. Bernfeld, V. Lakshmikantham: An Introduction to Nonlinear Boundary Value Problems. Academic Press, New York, London, 1974. MR 0445048
[9] B. A. Chartres, R. S. Stepleman: Convergence of difference methods for initial and boundary value problems with discontinuous data. Math. Comp. 25 (1971), 729–732. DOI 10.1090/S0025-5718-1971-0303739-1 | MR 0303739
[10] B. A. Chartres, R. S. Stepleman: Convergence of linear multistep methods for differential equations with discontinuities. Numer. Math. 27 (1976), 1–10. DOI 10.1007/BF01399080 | MR 0455409
[11] P. Chocholaty, L. Slahor: A numerical method to boundary value problems for second order delay-differential equations. Numer. Math. 33 (1979), 69–75. DOI 10.1007/BF01396496 | MR 0545743
[12] J. W. Daniel, R. E. Moore: Computation and Theory in Ordinary Differential Equations. W. H. Freeman and Company, San Francisco, 1970. MR 0267765
[13] L. J. Grimm, K. Schmitt: Boundary value problems for delay-differential equations. Bull. Amer. Math. Soc. 74 (1968), 997–1000. DOI 10.1090/S0002-9904-1968-12114-7 | MR 0228785
[14] L. J. Grimm, K. Schmitt: Boundary value problems for differential equations with deviating arguments. Aequationes Math. 4 (1970), 176–190. DOI 10.1007/BF01817758 | MR 0262632
[15] Z. Jackiewicz, M. Kwapisz: On the convergence of multistep methods for the Cauchy problem for ordinary differential equations. Computing 20 (1978), 351–361. DOI 10.1007/BF02252383 | MR 0619909
[16] T. Jankowski: Convergence of difference methods for boundary value problems of ODE’s with discontinuities. Demonstratio Math. 22 (1989), 51–65. MR 1024200 | Zbl 0708.65060
[17] T. Jankowski: Difference methods for boundary value problems of deviated differential equations with discontinuities. Zeszyty Nauk. Politech. Gdańsk. Mat. 15 (1991), 55–66. Zbl 0744.34066
[18] V. Lakshmikantham, D. Trigiante: Theory of Difference Equations. Academic Press, Inc., Toronto, 1988. MR 0939611
[19] K.  de Nevers, K. Schmitt: An application of shooting method to boundary value problems for second order delay equations. J.  Math. Anal. Appl. 36 (1971), 588–597. DOI 10.1016/0022-247X(71)90041-2 | MR 0298166
[20] F. Zh. Sadyrbaev: Solutions of a boundary value problem for a second-order ordinary differential equations. Latv. Mat. Ezhegodnik 31 (1988), 87–90. (Russian) MR 0942118
[21] S. Saito, M. Yamamoto: Boundary value problems of quasilinear ordinary differential systems on a finite interval. Math. Japon. 34 (1989), 447–458. MR 1003933
[22] J. Stoer, R. Bulirsch: Introduction to Numerical Analysis. Springer-Verlag, New York-Heidelberg-Berlin, 1993. MR 1295246
[23] Ya. Virzhbitskii: Necessary and sufficient conditions for the solvability of a two-point boundary value problem. Boundary Value Problems for Ordinary Differential Equations, Latv. Gos. Univ. Riga, 1987, pp. 53–68. (Russian) MR 0901975
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