Discrete-Time Fractional Difference Calculus: Origins, Evolutions, and New Formalisms
نویسندگان
چکیده
Differences are introduced as outputs of linear systems called differencers, being considered two classes: shift and scale-invariant. Several types presented, namely: nabla delta, bilateral, tempered, bilinear, stretching, shrinking. Both continuous discrete-time differences described. ARMA-type based on differencers exemplified. In passing, the incorrectness usual delta difference is shown.
منابع مشابه
Some New Gronwall-Bellmann Type Discrete Fractional Inequalities Arising in the Theory of Discrete Fractional Calculus
In this paper, we present some new GronwallBellmann type discrete fractional sum inequalities, and based on them present some Volterra-Fredholm type discrete inequalities. These inequalities are of new forms compared with the existing results in the literature, and can be used in the research of boundedness and continuous dependence on the initial value for solutions of fractional difference eq...
متن کاملNabla discrete fractional calculus and nabla inequalities
Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincaré and Sobolev type inequalities .
متن کاملA Note on Convexity, Concavity, and Growth Conditions in Discrete Fractional Calculus with Delta Difference
We demonstrate that some recent results regarding the connection between the convexity of the map t → f (t) and the sign of Δa f (t) , with 2 < ν < 3 , can be improved. In particular, by utilizing a recent inequality due to Jia, Erbe, and Peterson, we are able to improve some of the existing results in the literature. As part of this study we illustrate the improvements that our results afford ...
متن کاملThe new implicit finite difference method for the solution of time fractional advection-dispersion equation
In this paper, a numerical solution of time fractional advection-dispersion equations are presented.The new implicit nite dierence methods for solving these equations are studied. We examinepractical numerical methods to solve a class of initial-boundary value fractional partial dierentialequations with variable coecients on a nite domain. Stability, consistency, and (therefore) convergenceof t...
متن کاملSome New Discrete Fractional Inequalities and Their Applications in Fractional Difference Equations
In this paper, some new Gronwall-Bellman type discrete fractional difference inequalities and fractional sum inequalities are established. Based on the theory of discrete fractional calculus, explicit bounds for unknown functions concerned are presented. These inequalities can be used as a handy tool in the qualitative analysis of solutions of discrete fractional difference equations. As for ap...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Fractal and fractional
سال: 2023
ISSN: ['2504-3110']
DOI: https://doi.org/10.3390/fractalfract7070502