The operational calculus model for the nth-order backward difference

An Operational Calculus Model with the (m,n) - Symmetric Difference

“…When , we get in turn the model with the central difference (4) constructed in [17]. In [15] there was presented a model, in which to the backward difference (3) there correspond the following limit conditions (12) and integrals (13) This model is different to the one obtained from Theorem 1 for . The limit conditions (10) are uniquely determined by the initial conditions while the limit conditions (12) depend on Hence, the integrals ( 9) and ( 13) are also different, so…”

“…If we define the objects (1), then we have in mind a representation, or a model of the operational calculus. In particular, representations of the operational calculus (1) are discrete models in which the derivative can be understood as: an -order forward difference [16] (2) an -order backward difference [15] (3) or a 2 -order central difference [17] (4)…”

“…For being a constant sequence, representations of operational calculus with derivatives were discussed in [16], [15], [17].…”

An Operational Calculus Model for the nTH – Order Forward Difference

“…It is not difficult to notice that for we have , while for we can write (12) Finally, it be stated that the property is satisfied.…”

A Generalization of Difference Models of the Bittner Operational Calculus

“…The idea of generalizing the model with the forward difference (2) can be applied to constructing other discrete representations of the Bittner operational calculus. Namely, considering models with the backward difference and the central difference, described in [8] and [10], similarly as above we can determine integrals and limit conditions that correspond to the below difference operations:…”