A note on iterative solutions of an iterative functional differential equation

H. Y.  Zhao

doi:10.37863/umzh.v72i11.6034

H. Y. Zhao School Math., Chongqing Normal Univ., China

DOI: https://doi.org/10.37863/umzh.v72i11.6034

Abstract

UDC 517.9

We propose an iterative method for solving the iterative functional differential equation
$$x\prime \prime (t) = \lambda_1x(t) + \lambda_2x^{[2]}(t) + . . . + \lambda_nx^{[n]}(t) + f(t).$$

References

R. Bellman, K. Cooke, Differential-difference equations, New York: Acadmic Press (1963). DOI: https://doi.org/10.1063/1.3050672

C. Chicone, Ordinary Differential Equations with Applications, Springer, New York (1999).

M. Farkas, Periodic Motions, Applied Mathematical Sciences, vol. 104, Springer-Verlag (1994), https://doi.org/10.1007/978-1-4757-4211-4 DOI: https://doi.org/10.1007/978-1-4757-4211-4

M. Gadella, L. P. Lara, G. P. Pronko, Iterative solution of some nonlinear differential equations, Appl. Math. Comput., 217, no. 22, 9480 – 9487 (2011), https://doi.org/10.1016/j.amc.2011.04.058 DOI: https://doi.org/10.1016/j.amc.2011.04.058

J. Hale, Theory of functional differential equations, Springer-Verlag, New York (1977). DOI: https://doi.org/10.1007/978-1-4612-9892-2

W. J. Kim, N. C. Perkins, Harmonic balance/Galerkin method for non-smooth dynamic systems, J. Sound and Vibrations, 261, no. 2, 213 – 224 (2003), https://doi.org/10.1016/S0022-460X(02)00949-5 DOI: https://doi.org/10.1016/S0022-460X(02)00949-5

R. E. Mickens, Iteration procedure for determining approximate solutions to non-linear oscillator equations, J. Soundand Vibration, 116, no. 1, 185 – 187 (1987), https://doi.org/10.1016/S0022-460X(87)81330-5 DOI: https://doi.org/10.1016/S0022-460X(87)81330-5

R. Mickens, Oscillations in planar dynamics systems, Series on Advances in Mathematics for Applied Sciences, vol. 37, World Scientific (1996), https://doi.org/10.1142/2778 DOI: https://doi.org/10.1142/2778

V. R. Petahov, On a boundary value problem, Trudy Sem. Teor Different. Uravnenii Otklon. Argument, Univ. Druzby Narodov Patrisa Lumumby, 3, 252 – 255 (1965).

I. Sendanovic, Y. Fan, Some advances of the harmonic balance method, J. Sound and Vibration, 191, no. 2, 295 – 307 (1996), https://doi.org/10.1006/jsvi.1996.0123 DOI: https://doi.org/10.1006/jsvi.1996.0123

J. G. Si, X. P. Wang, Analytic solutions of a second-order iterative functional differential equation, J. Comput. Appl. Math., 126, no. 1-2, 277 – 285 (2000), https://doi.org/10.1016/S0377-0427(99)00359-3 DOI: https://doi.org/10.1016/S0377-0427(99)00359-3

J. G. Si, S. S. Cheng, Smooth solutions of a nonhomogeneous iterative functional differential equation, P. Roy. Soc. Edinb., 128(A), no. 4, 821 – 831 (1998), https://doi.org/10.1017/S0308210500021806 DOI: https://doi.org/10.1017/S0308210500021806

J. G. Si, X. P. Wang, Analytic Solutions of an Iterative Functional Differential Equation, J. Math. Anal. Appl., 262, no. 42, 490 – 498 (2001), https://doi.org/10.1006/jmaa.2001.7527 DOI: https://doi.org/10.1006/jmaa.2001.7527

J. G. Si, W. N. Zhang, Analytic solutions of a second-order nonautonomous iterative functional differential equation, J. Math. Anal. Appl., 306, no. 2, 398 – 412 (2005), https://doi.org/10.1016/j.jmaa.2005.01.005 DOI: https://doi.org/10.1016/j.jmaa.2005.01.005

J. G. Si, X. P. Wang, Analytic solutions of a second-order functional differential equation with state dependent delay,Results Math., 39, no. 3-4, 345 – 352 (2001), https://doi.org/10.1007/BF03322694 DOI: https://doi.org/10.1007/BF03322694

J. G. Si, X. P. Wang, Analytic solutions of a second-order functional differential equation with a stste derivative dependent delay, Colloquium Math., 79, no. 2, 273 – 281 (1999), https://doi.org/10.4064/cm-79-2-273-281 DOI: https://doi.org/10.4064/cm-79-2-273-281