General derivative approximations for finite different schemes

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Abstract
A general expression for finite difference approximations for all derivatives up to the (N − 1)th order on a finite difference mesh with N‐nodes is established. The mesh may be non‐uniform and the derivative evaluated at any of the N‐nodes. An estimate for the truncation error is obtained and shown to depend on the nonlinearity of the mesh as well as on the mesh diameter. Specific formulae for first‐ and second‐order derivatives are given as examples.

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