Optimization of nodes of Newton-Cotes formulas in the presence of an exponential boundary layer

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