Geodesic
An analogue of Newton–Cotes formula with four nodes for a function with a boundary-layer component
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 4, pp. 313-323 
A. I. Zadorin; N. A. Zadorin. An analogue of Newton–Cotes formula with four nodes for a function with a boundary-layer component. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 16 (2013) no. 4, pp. 313-323. http://geodesic.mathdoc.fr/item/SJVM_2013_16_4_a1/
@article{SJVM_2013_16_4_a1,
     author = {A. I. Zadorin and N. A. Zadorin},
     title = {An analogue of {Newton{\textendash}Cotes} formula with four nodes for a~function with a~boundary-layer component},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {313--323},
     year = {2013},
     volume = {16},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2013_16_4_a1/}
}
TY  - JOUR
AU  - A. I. Zadorin
AU  - N. A. Zadorin
TI  - An analogue of Newton–Cotes formula with four nodes for a function with a boundary-layer component
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2013
SP  - 313
EP  - 323
VL  - 16
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SJVM_2013_16_4_a1/
LA  - ru
ID  - SJVM_2013_16_4_a1
ER  - 
%0 Journal Article
%A A. I. Zadorin
%A N. A. Zadorin
%T An analogue of Newton–Cotes formula with four nodes for a function with a boundary-layer component
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2013
%P 313-323
%V 16
%N 4
%U http://geodesic.mathdoc.fr/item/SJVM_2013_16_4_a1/
%G ru
%F SJVM_2013_16_4_a1

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

The construction of the Newton–Cotes formulas is based on approximating an integrand by the Lagrange polynomial. The error of such quadrature formulas can be serious for a function with a boundary-layer component. In this paper, an analogue to the Newton–Cotes rule with four nodes is constructed. The construction is based on using non-polynomial interpolation that is accurate for a boundary layer component. Estimates of the accuracy of the quadrature rule, uniform on gradients of the boundary layer component, are obtained. Numerical experiments have been performed.

[1] Berezin I. S., Zhidkov N. P., Metody vychislenii, Nauka, M., 1966

[2] Bakhvalov N. S., Chislennye metody, Nauka, M., 1975

[3] Zadorin A. I., “Metod interpolyatsii dlya zadachi s pogranichnym sloem”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 10:3 (2007), 267–275

[4] Zadorin A. I., Zadorin N. A., “Splain-interpolyatsiya na ravnomernoi setke funktsii s pogransloinoi sostavlyayuschei”, Zhurn. vychisl. matem. i mat. fiziki, 50:2 (2010), 221–233 | MR | Zbl

[5] Zadorin A. I., “Spline interpolation of functions with a boundary layer component”, Int. J. of Num. Analysis and Modeling. Series B, 2:2–3 (2011), 562–579 | MR

[6] Zadorin A. I., Zadorin N. A., “Kvadraturnye formuly dlya funktsii s pogransloinoi sostavlyayuschei”, Zhurn. vychisl. matem. i mat. fiziki, 51:11 (2011), 1952–1962 | MR | Zbl

[7] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, UrO RAN, Ekaterinburg, 1992

[8] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996 | MR

[9] Kellogg R. B., Tsan A., “Analysis of some difference approximations for a singular perturbation problems without turning points”, Math. Comput., 32 (1978), 1025–1039 | DOI | MR | Zbl

[10] Miln V. E., Chislennyi analiz, IL, M., 1951

[11] Nikolskii S. M., Kvadraturnye formuly, Nauka, M., 1974 | MR

[12] Dulan E., Miller D., Shilders U., Ravnomernye chislennye metody resheniya zadach s pogranichnym sloem, Mir, M., 1983 | MR