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Non-standard Lagrange–Burman methods for the numerical integration of differential equations

We present new non-standard methods based on Mickens' ideas for constructing numerical schemes for differential equations. By analysing his schemes, we were able to give generalized versions by means of Lagrange–Burman expansions. Our generalization provides a theoretical frame to understand non-standard finite difference methods and also to analyse the errors in the approximations. We apply the new non-standard methods to several examples.