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Hybrid approximations via second-order crossed dynamic derivatives with the $\diamondsuit _\alpha$ derivative. (English) Zbl 1134.39302
Summary: Second-order crossed dynamic derivatives with the $\diamondsuit _\alpha$ dynamic derivative on time scales will be used for approximating the conventional second derivative in the study. We will discuss the conditions under which the crossed dynamic derivatives provide consistent approximations of the conventional derivative; the chances in which the dynamic derivatives approximate the derivative only via proper modifications of their formulae; and the situations in which the dynamic derivatives can never fulfill any consistent approximations even with structure modifications. Valuable error estimates will be developed via asymptotic expansions for potential computational and modeling applications.

##### MSC:
 39A13 Difference equations, scaling ($q$-differences) 74H15 Numerical approximation of solutions for dynamical problems in solid mechanics 74S20 Finite difference methods in solid mechanics 39A10 Additive difference equations 39A12 Discrete version of topics in analysis
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