The goal of this paper is to develop and present a precise theory for integer and fractional order \(\ell\)-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 \(\ell\)-nabla operator, we used the factorial-coefficient method. For developing the theory of fractional order \(\ell\)-nabla operator and its integration, we introduce a function called \(N_{\nu}\)-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.

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 $\nu$th-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 $\nu$th-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.

This paper begins by analyzing the key mathematical properties of diffusive vaccinated models, including existence, uniqueness, positivity, and boundedness. Equilibria are identified, and the basic reproductive number is calculated. The Banach contraction mapping principle is applied to rigorously establish the solution existence and uniqueness. In order to understand the disease’s time transmission, it is important to examine the global stability of the equilibrium points. Disease‐free equilibrium and endemic equilibrium are the two equilibria in this model. Here, we demonstrate that the endemic equilibrium is worldwide asymptotic stable when the basic reproductive number is greater than 1, and the disease‐free equilibrium is globally asymptotic stable whenever the basic reproductive number is less than 1. Moreover, based on the Caputo fractional derivative of order and the implicit Euler’s approximation, we offered an unconditionally stable numerical solution for the resultant system. This work explores the solution of some significant population models of noninteger order using an approach known as the iterative Laplace transform. The proposed methodology is developed by effectively combining Laplace transformation with an iterative procedure. A series form solution that exhibits some convergent behavior towards the precise solution can be attained. It is noted that there is a close contact between the obtained and precise solutions. Moreover, the suggested method can handle a variety of fractional order derivative problems because it involves minimal computations. This information will be helpful in further studies to determine the ideal strategy of action for preventing or stopping the spread disease transmission.