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

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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. 
@article{SJVM_2013_16_4_a1,
     author = {A. I. Zadorin and N. A. Zadorin},
     title = {An analogue of {Newton--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},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2013_16_4_a1/}
}
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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/