Discrete Fractional Sumudu Transform by Inverse Fractional Difference Operator

In this research work, the aim is to develop the fractional proportional delta operator and present the generalized discrete Laplace transform and its convolution with the newly introduced fractional proportional delta operator. Moreover, this transform is a connection between Sumudu and Laplace transforms, which yields several applications in pure and applied science. The research work also investigates the fractional proportional differences and its sum on Riemann–Liouville and Mittag–Leffler functions. As an application of this research is to find new results and properties of fractional Laplace transform, the comparison of the existing results with this research work is also done. Moreover, we used the two types of solutions, namely, closed and summation forms in Laplace transform and verified with numerical results.

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“…In view [1], (10) and (11) are the discrete versions of the following versions of integration by parts. Definition 4 (integration by parts).…”

Section: The Delta Fractional Sums and Differencesmentioning

confidence: 99%

“…Recently, Bastos et al developed the theory on h-sum and h-di erence operators in discrete fractional calculus. For more recent applications of fractional Laplace transform, one can refer [8][9][10][11]. ese concepts are very well applied in discrete fractional transforms in the field of fractional calculus [12].…”

Section: Introductionmentioning

confidence: 99%

The Generalized Fractional Proportional Delta Operator and New Generalized Transforms in Discrete Fractional Calculus

Amalraj¹,

Manuel²,

Meganathan³

et al. 2022

Mathematical Problems in Engineering

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“…In [2], the authors introduced fractional differential equations and fractional calculus, whereas [10][11][12] examined the delta operators and their properties in fractional sums. Applications of discrete and continuous fractional calculus are provided in books [13][14][15], and for more details one can also refer to articles [16][17][18][19][20][21] and references therein.…”

Section: Introductionmentioning

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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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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“…Inverse Laplace transforms the resulting which expression yields the solutions in the time domain, revealing the transient behavior (initial response) and the steady-state behavior (long-term response) of the system. At recently, the authors [8,9,10], have been obtained and analysed with many applications using Laplace transform with n−tuples.…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis of Discrete Laplace Transform for Higher Order Cosine Function and Applications in Initial Value Problem

2023

Advances in the Theory of Nonlinear Analysis and Its Application

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Discrete Fractional Sumudu Transform by Inverse Fractional Difference Operator

Cited by 3 publications

References 22 publications

The Generalized Fractional Proportional Delta Operator and New Generalized Transforms in Discrete Fractional Calculus

The Generalized Fractional Proportional Delta Operator and New Generalized Transforms in Discrete Fractional Calculus

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

Numerical Analysis of Discrete Laplace Transform for Higher Order Cosine Function and Applications in Initial Value Problem

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