Discrete Mathematics and Mathematical Modelling in the Digital Era

This manuscript investigates the approximate controllability results for a class of second-order Hilfer fractional integro-differential equations (HFIDEs). A new set of appropriate mild solutions has been derived. Further, the existence of mild solutions for the proposed system has been verified using Mönch fixed point theorem (MFPT) and derived sufficient condition utilizing the measure of non-compactness (MNC). The approximate controllability results of the proposed system have been established by presuming that the associated linear system is an approximate control system. Finally, for the understanding, a numerical example is included.

Extracting the human brain tissues from MRI is a vital pre-processing step required for many automated medical image-processing pipelines. We propose two methods NDTM and NDTME based on the Novel Distance Transform (NDT) technique to identify the brain boundary in a 2D MRI and separate it. These methods utilize the largest connected component analysis and mathematical morphology. Experiments are done by applying these two methods on thirtyeight sets of MRI head scans (3500 images) collected from the publicly available Internet Brain Segmentation Repository (IBSR) image database. Achievement of these methods is assessed by estimating the Jaccard and Dice similarity indices. It was found that our method NDTME produced higher precision values than BSE, BET, and a few other algorithms. The suggested algorithms are simple, noise tolerant, and have lower computational complexity than the existing methods.

This study focuses on systems that provide services in batches instead of personalized one-on-one assistance. In order to acquire the probability and features of the queueing system, it first introduces controllable arrival rates and interdependency in the service and arrival processes of such a system. Such models are helpful in various circumstances where arrivals are regulated by non-manual robotic systems. Consider that two parallel servers are homogeneous. The input is assumed to be Poisson (during each epoch of the Poisson occurrence, a single arrival) and controllable according to the speed of arrivals. For assistance, each server is equally likely to take a batch when both servers are accessible. Although there have been studies on bulk service in queuing theory, this novel method aims to provide a link between bulk service and interdependency in the arrival and service processes, as well as controlled arrival rates. M/M(a, b)/2/K is the notation for the system. Steady-state solutions and characteristics of the model are derived and analysed. The outcome of analysis is also numerically shown for various parameter values and levels.

In this manuscript, we focus on the development of a fuzzy fault-tolerant control strategy for switched nonlinear systems (SNSs) utilizing the mode-dependent average dwell time (MDADT) technique. Distinguished from previous studies, our approach introduces an innovative concept, the Lyapunov–Krasovskii functional (LKF), which is dependent on membership functions for nonlinear systems. This pioneering technique incorporates membership function derivatives through a switching mechanism, leading to the formulation of a fuzzy fault-tolerant controller. This controller is designed to establish stabilization conditions for SNSs, utilizing the proposed LKF scheme to ensure globally uniform exponential stability (GUES) through the application of linear matrix inequalities (LMI). To demonstrate the practical applicability and effectiveness of our proposed controller scheme, we provide a numerical example for validation purposes.

The delay differential equation encapsulates nuanced temporal variations within solution conditions. This study delves into scrutinizing the convergence of an iterative approach across a spectrum of fuzzy number domains, aimed at elucidating proximity and eliminating ambiguity within the solutions. The resolution of Fuzzy Delay Differential Equations (FDDE) leverages the Runge-Kutta-Fehlberg method (RKF). Numerical outcomes are scrutinized to underscore the efficacy of the trapezoidal fuzzy number scheme.

Articles describing the limited capacity of malfunctioning and operating services exist, but this article aims to connect the rates of controllable arrivals and the interdependence of the arrival-and-service procedure. The study of parallel servers with breakdown and controlled arrival rates and the clients approach the system individually in a Poisson procedure. The arrival pattern follows a Poisson procedure, and service follows an exponential. In real-life situations, if we go to a toll plaza, there is a parallel server available for every client. The server furnishes the service individually, first-come, first-served (FCFS). At that time, if the machine breaks down, no service will occur until it is repaired, so the count of clients waiting in queue increases. These models are used to analyze practical situations such as import and export ships, taxi stands, computer-related applications for time-sharing, system architecture, and so on. Numerical examples are calculated using MATLAB software, and graphical analysis is provided for better comprehension. According to this article, this method has a finite population of parallel servers. Future study of the proposed model is included in this study.

Deduction of edges is an important operation in image processing. Among various existing edge detection techniques in recent days fractional derivatives have been utilized widely. In this paper, the image edges are identified by using fractional derivative of order \(v \in (0,1)\). The order of the fractional derivative has been optimized for each image by converting each edge image obtained for various orders between 0 to 1 into fuzzy images. From these fuzzy images, a single image is considered whose divergence measure is minimum. A fuzzy image with minimum divergence is defuzzified to get an edge image. The results of the proposed technique have been compared with the existing edge detection method. Quantitatively metric values have been evaluated which explains the performance of the proposed technique.

Objectives The essential persistence of this study is to enhancing on slip boundary conditions and the stream of the two-dimensional, steady, incompressible flow of a hybrid nanofluid over a moving thin needle has been explored. Design/Methodology/Approach Similarity transformations can be used to transform a system of Partial Differential Equations into a scheme of Ordinary Differential Equations and it is solved by a numerical technique based on Runge-Kutta fourth order method. In addition, the momentum equation considers an inclined magnetic field, and the energy equation considers the stuff of thermal radiation and viscous dissipation. Findings It is initiated that slip conditions (velocity and thermal) exist for a range of the forced convection boundary layer flow. In the hybrid nanofluid flow, which consists of \(Al_2 O_3\) and \(Fe_3O_4\) are nanoparticles, \(H_2O\)–\(C_2H_6O_2\) (50 : 50) are considered as the base fluid. The findings show how important variables like the magnet variable, Eckert number, nanoparticles of solid volume fractions, inclined angle parameter, Prandtl numbers, and radiation parameter have a significant impact on velocity and temperature outlines. The arithmetic values of multiple variables, which are presented in a table, are obtained through the skin friction and heat transfer rates. These analyses contest the idea that enhancing the magnetic parameter leads to the temperature and hybrid nanofluid velocity to go down. Novelty This study proposes that in hybrid nanofluid using slip conditions, the significances of linear thermal radiation, and viscous dissipation are significant factors that can significantly affect the flow characteristics of Magnetohydrodynamics (MHD) boundary layer flow over a poignant tinny needle. These findings can be useful in the design of thermal systems that involve in engineering applications.

The Primal endeavor of this manuscript is to demonstrate the theorems of fixed point in asymmetric strong b-Metric Space. Using Modified almost contractions, Existence and Uniqueness of fixed points are proved. Furthermore, we generalize the theorem of Nadler’s Multi-Valued mapping in the environment of asymmetric strong b-metric space. The results were obtained by the use of simple processes.

This article deals with the examination of Strong Altering JS-metric space and certain peculiarities of this general space. Also, it analyses fixed point results for the famous Kannan contraction and Reich type contraction in the context of complete Strong Altering JS-metric space.

In this research, we have suggested a novel technique to handle fuzzy transportation issues utilizing the congruence modulo approach, which is also illustrated by numerical examples. The obtained findings demonstrate that this is one of the quick and easy ways to discover the best solution to the fuzzy transportation problem. The fuzzy optimum transportation issue may be solved using a bipartite graph-based method that we will describe in this paper. For generalized hexagonal fuzzy numbers, it would undoubtedly generate a ranking algorithm. The fuzzy transportation problem has been converted into crisp values. Applying TORA statistical software, the data analysis discovered that the crisp model becomes linear when the membership function is applied.

Let \(\gamma _{CD}(H)\) denote the corona domination number (CD-number) of the graph H. The idea of CD-number was first coined by G. Mahadevan et al. in 2021. The CD-number of a graph is the minimum cardinality of a dominating set S with the condition that every vertex in the graph induced by S is either a pendent vertex or support vertices. In this paper, we carry over the study of CD-number to identify the exact value of \(\gamma _{CD}(P_r \otimes P_s)\), \(\gamma _{CD}(C_r \otimes C_s)\), \(\gamma _{CD}(P_r^c \otimes P_s)\), etc.

“Graph theoretical analysis” [6] is a significant field of biological networks. Nowadays it gives a new direction to worldwide research. A Pt-graph of a protein or peptide is introduced by us rooted in some properties (physical and chemical) and amino acids’ adjacency equivalent to each protein or peptide. Here we analyze the Pt-graph of Endomorphin. We have some observations relating to their relations among the properties (physical and chemical). The Pt-graph of Endomorphin is a connected graph. The average degree of the Pt-graph is 1.5. The hydrophobic amino acids like Proline (P) and Tryptophan (W) have received the highest value of all measures of centrality. The hydrophobic and aromatic amino acids, Tyrosine (Y) and Phenylalanin (F) received the lowest values of all measures of centrality. It will help in the study of drug design and the evolution of proteins or peptides.

Let \(\mathscr {Q}=(\mathbb {R},\psi ,\varrho )\) be a fuzzy graph. A subset B of \(\mathbb {R}\) is termed as fuzzy equitable dominating set if every \(w \in \mathbb {R}\backslash B ~\exists \) a vertex \(s \in B\ni s w \in \mathscr {E}(\mathscr {Q})\) and \(|d_\mathscr {Q}(s)-d_\mathscr {Q}(w)|\) \(\le 1\), where \(\mathscr {E}(\mathscr {Q})\) indicates the edge set of \(\mathscr {Q}\) and \(d_\mathscr {Q}(s)\), \(d_\mathscr {Q}(w)\) indicate the degree of vertices s and w, respectively, and \(\varrho \) (sw) \(\le \psi (s) \wedge \psi (w)\). A fuzzy equitable dominating sets minimal cardinality indicated by the symbol \(\gamma _{fe}\). Fuzzy graphs are often utilized in mathematical models of problems due to their simplicity of representation of the relationships between topics. Numerous other fields of study, such as computer science, intelligent systems, psychology, and the medical sciences, have benefited from the dominance of fuzzy graphs. In the overall structure of social networks, an equitable domination has some fascinating applications. Hence, in this research, we have presented creative ideas of divisor 2-equitable dominating set (d2EDS) and exact divisor 2-equitable dominating set (exact d2EDS) in fuzzy graph. The characterizations of minimal divisor 2-equitable domination sets and exact divisor 2-equitable domination sets are discussed.

In this paper, initially Mahadevan et al. [4] studied CTATD number. A set \(S\subseteq V\) is called a complementary triple connected at most twin dominating set CTATD(G) if every vertex \(v\in V-S, ~1\le |N(v)\cap S|\le 2\) and \(<V-S>\) is triple connected. The smallest cardinality of CTATD-sets is called the CTATD number and is denoted by CTATD(G). In this work, in this article, we examine the CTATD number for the tadpole graph and some particular power graphs.

Let us consider an undirected and a finite graph \(\mathcal {G}=\{\mathcal {V(G),E(G)}\} \) with no loops or parallel edges. Here \(\mathcal {V(G)}\) and \(\mathcal {E(G)}\) represent the collection of vertices and edges respectively. A Dominator Coloring of a graph \(\mathcal {G}\) is a coloring that is proper in such a way that every vertex in \(\mathcal {G}\) is in the neighborhood of all the vertices of at least a single color class. A Total Dominator Coloring of a graph \(\mathcal {G}\) is also a proper coloring in which each vertex dominates at least a color class other than its own. The Dominator coloring has already been employed for the m-shadow graph of paths. In this article, we investigate the Dominator and Total Dominator coloring for m-Shadow graphs of Cycle, Complete graph and Wheel graph and also determine the respective dominator chromatic number and compare the same.