We first introduce a novel modeling framework, called linear coupled component automata (LCCA), to facilitate the modeling of discrete-continuous dynamical systems with piecewise constant derivatives. Second, we provide a procedure for transforming models in this framework to mixed-integer linear programming (MILP) constraints. Traditionally, such systems have been modeled directly with MILP co...

We prove a more general version of the Grüss inequality by using the recent theory of combined dynamic derivatives on time scales and the more general notions of diamond-α derivative and integral. For the particular case when α = 1, one gets a delta-integral Grüss inequality on time scales; for α = 0 a nabla-integral Grüss inequality. If we further restrict ourselves by fixing the time scale to...

Fix an integer n > 0. For a multivariate function defined on a (not necessarily rectangular) lattice, an extension is constructed to have, ∀k ≤ n, derivatives of total degree k that are bounded by the function’s tensor product divided differences of total degree k times a constant independent of the lattice and the function. The extension is locally constructed, can have any prescribed smoothne...

Using an exponential model for the variation in energy with respect to the number of electrons it is shown that, within the model, the hardness, softness, electrophilicity and other global parameters connected to higher order derivatives follow an equalization principle after a molecule is formed from two separated species. Two generalizations of the model are also discussed, one of which prese...

In this paper, we present a numerical computational approach for solving Caputo type fractional differential equations. This method is based on approximation of Caputo derivative in terms of integer order derivatives and waveform relaxation method. The utility of the method is shown by applying it to several examples. A comparative study indicates that our approach is more efficient and accurat...

and integral calculus from integer orders n to the entire complex plane. Methods are presented for using this generalized calculus with Laplace transforms of complex-order derivatives to solve analytically many differential equations in physics, facilitate numerical computations, and generate new infinite-series representations of functions. As examples, new exact analytic solutions of differen...

The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial are used to represent physical processes, which then solved analytically or numerically examine the dynamical behaviour system. new iterative approach Homotopy perturbation method this article solve fractional order Fokker–Planck equation numerically. Caputo sense is cha...

In order to investigate the honeybee population dynamics, many differential equation models were proposed. Fractional derivatives incorporate history of dynamics. We numerically study inverse problem parameter identification in with Caputo and Caputo–Fabrizio operators. use a gradient method minimizing quadratic cost functional. analyze compare results for integer (classic) fractional models. T...

The concept of a fractional derivative is not at all intuitive, starting with having clear geometrical interpretation. Many different definitions have appeared, to the point that need for order has arisen in field. diversity potential applications even more overwhelming. When modeling problem, one must think carefully about what introduction derivatives model can provide was already adequately ...