On the solutions of second order generalized difference equations

Manuel, M.M.S.; Kılıçman, Adem; Xavier, G. Britto Antony; Pugalarasu, R.; Dilip, D. S.

doi:10.1186/1687-1847-2012-105

Cited by 5 publications

(6 citation statements)

References 6 publications

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“…The objective of this research is to construct a thorough and exact theory for integerand fractional-order delta integration, as well as its fundamental theorem. By setting h = 1, one can refer to the difference and anti-difference notions from [22][23][24][25].…”

Section: Methodology and Contribution Of The Workmentioning

confidence: 99%

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A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions

Al-Shamiri,

Rexma Sherine,

Britto Antony Xavier

et al. 2023

Mathematics

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This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The νth-fractional sum of a function f has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to nth-sum for a class of delta integrable functions, that is, functions with both discrete integration and nth-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing ∞-order delta integrable functions, the discrete integration related to the νth-fractional sum of f can be obtained by applying Newton’s formula. In addition, this concept is extended to h-delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions.

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Section: Methodology and Contribution Of The Workmentioning

confidence: 99%

“…Since ∆ f n = f n−1 , by replacing f by f n−1 and f 1 by f n in (23), the first term of the right side of Equation ( 25) takes the form…”

Section: Integer-order Delta Integration and Its Summentioning

confidence: 99%

A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions

Al-Shamiri,

Rexma Sherine,

Britto Antony Xavier

et al. 2023

Mathematics

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“….}. For more details on the applications of discrete fractional calculus, one can refer to [5][6][7][8][9] and for more details on other related fields refer [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. For Example, in [1] (see equation 2.1), the ν th order fractional sum of given function f based at a is defined as…”

Section: Introductionmentioning

confidence: 99%

Discrete version of fundamental theorems of fractional order integration for nabla operator

Byeon,

Abisha,

Sherine

et al. 2024

J. Math. Computer Sci.

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The goal of this paper is to develop and present a precise theory for integer and fractional order -nabla integration and its fundamental theorems. In our research work, we take two forms of higher order difference equation such as closed form and summation form. But most of the authors are focused only on the summation part only. Instead of finding the solution for the summation part, finding the solution for the closed gives the exact solution. To find the closed form solution for the integer order using the -nabla operator, we used the factorial-coefficient method. For developing the theory of fractional order -nabla operator and its integration, we introduce a function called N ν -type function. If the summation series is huge, this approach can help us to find the solution quickly. Suitable examples are provided for verification. Finally, we provide the application for detecting viral transmission using the nabla operator.

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“…But recently, when we took up the definition of ∆ as given in (2) we developed the theory of difference equations in a different direction ([8]- [9]). For convenience, we labelled the operator ∆ defined by (2) as ∆ ℓ and by defining its inverse ∆ −1 ℓ , many interesting results and applications in number theory were established (see [8], [10], [11], [12], [13]). …”

Section: Introductionmentioning

confidence: 99%