### Zadorin A. Blatov I. Kitaeva E.

## Approximation of derivatives of functions with large gradients based on spline interpolation

### Reporter: Zadorin A.

The problem of approximation the derivatives of functions with large gradients in the exponential boundary layer is investigated. The problem is that the application of classical polynomial formulas of numerical differentiation to functions with large gradients can lead to significant errors. The problem of approximation of the derivatives with respect to the values of the function at grid nodes on the basis of spline interpolation is considered. This makes it possible to approximate the derivatives by smooth functions of their argument. Two approaches are explored: the use of a cubic spline on a Shishkin mesh and the use of an exponential spline of the same smoothness on the uniform grid. Estimates of the error in calculating the first and second derivatives, taking into account the uniformity with respect to a small parameter, are obtained. The results of numerical experiments are discussed.

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