Geodesic
Numerical analytical method of studying some linear functional differential equations
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 3, pp. 275-285 
V. B. Cherepennikov. Numerical analytical method of studying some linear functional differential equations. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 3, pp. 275-285. http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a7/
@article{SJVM_2013_16_3_a7,
     author = {V. B. Cherepennikov},
     title = {Numerical analytical method of studying some linear functional differential equations},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {275--285},
     year = {2013},
     volume = {16},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a7/}
}
TY  - JOUR
AU  - V. B. Cherepennikov
TI  - Numerical analytical method of studying some linear functional differential equations
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2013
SP  - 275
EP  - 285
VL  - 16
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a7/
LA  - ru
ID  - SJVM_2013_16_3_a7
ER  - 
%0 Journal Article
%A V. B. Cherepennikov
%T Numerical analytical method of studying some linear functional differential equations
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2013
%P 275-285
%V 16
%N 3
%U http://geodesic.mathdoc.fr/item/SJVM_2013_16_3_a7/
%G ru
%F SJVM_2013_16_3_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper presents the results of studies of the scalar linear functional-differential equation of a delay type $\dot x(t)=a(t)x(t-1)+b(t)x(t/q)+f(t)$, $q>1$. The main attention is being given to the original problem with the initial point, when the initial condition is specified at the initial point, and the classical solution, whose substitution into the original equation transforms it into the identity, is sought for. The method of polynomial quasi-solution, based on representation of an unknown function $x(t)$ as polynomial of degree $N$ is applied as the method of investigation. Substitution of this function in the original equation results in the residual $\Delta(t)=O(t^N)$, for which an accurate analytical representation is obtained. In this case, the polynomial quasi-solution is understood as exact solution in the form of polynomial of degree $N$, disturbed because of the residual of the original initial problem. The theorems of existence of polynomial quasi-solutions for the considered linear functional-differential equation and exact polynomial solutions have been proved. The results of the numerical experiment are presented.

[1] Bernoulli J., “Meditations de chordis vibrantibus, cum pondusculis aequali a intervallo a se invicem dissitis ubi nimirum ex principio virium vivarum quaqeritur numerus vibrationum chordia pro una oscillatione penduli datae longitudinis D”, Commentarial Academia Scientiarum Imperialis Petropolitanae, Collected Work, v. 3, 1728, 198–221

[2] Myshkis A. D., Lineinye differentsialnye uravneniya s zapazdyvayuschim argumentom, Gostekhizdat, M.–L., 1951

[3] Bellman R., Kuk K., Differentsialno-raznostnye uravneniya, Mir, M., 1967 | MR | Zbl

[4] Azbelev N. V., Maksimov V. P., Rakhmatullina L. F., Vvedenie v teoriyu funktsionalno-differentsialnykh uravnenii, Nauka, M., 1991 | MR | Zbl

[5] Cherepennikov V. B., “Polinomialnye kvaziresheniya lineinykh sistem differentsialno-raznostnykh uravnenii”, Izv. VUZov, ser. Matematika, 1999, no. 10, 49–58 | MR | Zbl

[6] Cherepennikov V. B., Ermolaeva P. G., “Polynomial quasisolutions of linear differential difference equations”, Opuscula Mathematica, 26:3 (2006), 431–443 | MR | Zbl

[7] Cherepennikov V. B., Ermolaeva P. G., “Gladkie resheniya nachalnoi zadachi dlya nekotorykh differentsialno-raznostnykh uravnenii”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 13:2 (2010), 213–226 | Zbl

[8] Cherepennikov V. B., “Ob analiticheskikh resheniyakh nekotorykh sistem funktsionalno-differentsialnykh uravnenii”, Diff. uravneniya, 26:6 (1990), 1094–1095