Hybrid approximations via second-order crossed dynamic derivatives with the ♢α♢α derivative

Second-order crossed dynamic derivatives with the ♢α♢α 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.