Optimization of nodes of composite quadrature formulas in the presence of a boundary layer

The problem of numerical integration of a function of one variable with large gradients in the region of the exponential boundary layer is studied. The problem is that the use of composite quadrature formulas on a uniform grid with decreasing of the small parameter value leads to significant errors, regardless of the number of nodes of the basic quadrature formula. In the paper it is proposed to choose nodes based on the composite quadrature formula error minimizing. Basic quadrature formula applied between grid nodes, takes into account the cases of the Newton-Cotes and Gauss formulas. It is proved that the minimum error is achieved on the Bakhvalov mesh, and the error of the quadrature formula becomes uniform in a small parameter.