The two-energy neutron diffusion model in slab reactors characterizes neutron behavior across two energy groups: fast and thermal. Fast neutrons, generated by fission, decelerate through collisions, transitioning into thermal neutrons. This model employs diffusion equations to compute neutron flux distributions and reactor parameters, thereby optimizing reactor design and safety to ensure efficient neutron utilization and stable, sustained nuclear reactions. The primary objective of this research is to explore both analytical and numerical solutions to the two-energy neutron diffusion model in slab reactors. Specifically, we will utilize the Laplace transform method for an analytical solution of the two-energy neutron diffusion model. Subsequently, employing the Caputo differentiator, we transform the original neutron diffusion model into its fractional-order equivalents, yielding the fractional-order two-energy group neutron diffusion model in slab reactors. To address the resulting fractional-order system, we develop a novel approach aimed at reducing the 2β-order system to a β-order system, where β ∈ (0, 1]. This transformed system is then solved using the Modified Fractional Euler Method (MFEM), an advanced variation of the fractional Euler method. Finally, we present numerical simulations that validate our results and demonstrate their applicability.

This paper presents the so-called shifted Jacobi method, an efficient numerical technique to solve second-order periodic boundary value problems with finitely many singularities involving nonlinear systems of two points. The method relies on the Jacobi polynomials used as natural basis functions in the conformable sense of fractional derivative. A study is carried out to compare the outcomes of the shifted Jacobi approach with those of other methods that are currently in use. In the same vein, a theoretical result for establishing a bound of the error generated from the proposed approximate solution is proved accordingly. The efficiency and effectiveness of the shifted Jacobi technique with conformable fractional derivative are discussed numerically.

We introduce the quantum indeterminate set (QIS), a novel mathematical framework that integrates complex-valued membership functions and phase-based interference into classical and generalized set theory. Unlike complex fuzzy sets, QIS formally encodes both amplitude and phase interactions, allowing constructive and destructive interference to emerge naturally in reasoning. This feature distinguishes QIS as a bridge between fuzzy logic and quantum probability. We define its structure, explore its algebraic properties, and demonstrate its practical capability through a real-world decision-making case study in energy-system evaluation. The model generalizes fuzzy, intuitionistic, and neutrosophic sets while introducing a quantum-inspired interference mechanism that enables more nuanced reasoning under indeterminacy. This paper lays the theoretical foundation for future work in quantum decision theory, quantum-inspired soft computing, and uncertainty modeling in artificial intelligence.

The Neutrosophic Soft Set (NSS) is an advanced and highly effective expansion of soft sets, specifically designed to handle parameterized values of alternatives. As an enhanced version of fuzzy soft sets, it provides a novel mathematical framework that offers significant advantages in dealing with uncertain information. This model is created by merging soft sets and neutrosophic sets, providing a robust approach to uncertainty management. Various algorithms have been proposed for making neutrosophic decisions using NSSs. However, these algorithms neglect external effective that influence the Decision-Making (DM) process, focusing solely on parameters. To address this issue, the article introduces the concept of Effective Neutrosophic Soft Sets (ENSSs). Additionally, we extend and generalize the innovative concept of Effective Fuzzy Soft Sets (EFSSs) to accommodate three independent membership criteria, aiming to enhance effectiveness and realism. We also introduce operations on ENSSs, including subset, complement, union, intersection, AND, and OR, which are defined along with illustrative examples. Furthermore, we examine some of its properties. Moreover, we present applications of this concept in DM problems and Medical Diagnosis (MD)

This study addresses the critical need for accurate neutron diffusion modeling in cylindrical reactors, focusing on the two-energy groups neutron diffusion system. Such modeling is essential for optimizing reactor design and safety in nuclear engineering. The research primarily aims to enhance computational methods by transitioning from a traditional integer-order model to a more sophisticated fractional-order model, which can capture complex physical phenomena with greater precision. The study employs the Laplace Transform Method (LTM) to first solve the integer-order system and then extends this approach to a fractional-order system using the Caputo derivative, a method well-suited for systems with memory effects. To efficiently solve the resulting fractional-order model, we introduce the Modified Fractional Euler Method (MFEM), designed to improve numerical accuracy and stability. The effectiveness of this approach is demonstrated through specific numerical applications, such as simulating neutron flux distributions, which validate the model’s accuracy and its potential impact on advancing reactor physics. These applications showcase the practical relevance of the proposed methods and their contribution to improving nuclear reactor simulations.

The neutrosophic soft set emerges as a highly valuable and efficient adaptation of soft sets, specifically addressing parameterized values of alternatives. However, numerous decision-making algorithms rooted in neutrosophic soft sets often neglect the external factors impacting their effectiveness. This paper introduces the innovative concept of an effective neutrosophic soft expert set, meticulously crafted to encapsulate external influences on both neutrosophic soft sets and expert opinions within a unified model. This eliminates the necessity for additional operations. Notably, our groundbreaking approach seamlessly amalgamates the strengths of the neutrosophic soft expert set and the effective set, resulting in heightened efficiency and realism in this domain. The article comprehensively explores the fundamental operations of an effective neutrosophic soft expert set, elucidating these processes through apt examples. Finally, the paper showcases the practical application of this concept in decision-making problems, providing algorithms and illustrative examples to underscore its efficacy.

Shadow soft set is a new concept defined as a new tool with uncertainty where the values of membership taken from 0, 1 and [0,1]. In this thesis as a generalization of shadow soft set we introduce a new concept which is an Intuitionistic possibility shadow soft set and study its properties. Furthermore, some examples of Intuitionistic possibility shadow soft set and its properties are presented. We also introduce some of the operations of this concept and give some results related to these operations. Finally, the definitions of (AND) and (OR) operations are given and the properties of these operations in their application to decision-making problems are shown.

The most useful extension for fuzzy soft groups is the effective fuzzy soft group, which explains the effect of external efficacy on soft groups and is called effective fuzzy soft set (EFSS). This extension is the first extension of fuzzy soft set which considers the external effect on fuzzy soft set. Another extension of the fuzzy soft set was the time fuzzy soft set (TFSS) introduced by Ayman A. Hazaymeh in his Ph.D. thesis. In this paper we define the concept of effective time fuzzy soft set with respect time (ETFSS), which is the combination between effective fuzzy soft set and time fuzzy soft set. Also, we introduce basic operations and some properties of (ETFSS) with suitable examples.

This article aims to study the existence and uniqueness of a weak solution for a singular and degenerate nonlinear parabolic equation with a generalized nonlinear integral condition of the second type. The proof of the existence and uniqueness of the weak solution to such a problem will be proceeded with in three steps. In the same regard, the solvability of the linear case of the problem at hand will be handled with the use of the Faedo-Galerkin method, a priori estimate, and by imposing some nonlinear conditions of the second kind.

Parkinson's disease (PD) must be identified early to provide prompt treatment and better patient outcomes. This study looks into using machine learning methods in conjunction with vocal biomarkers to identify Parkinson's disease early. Vocal data was analyzed using four techniques: Support Vector Machine (SVM), K-Nearest Neighbor (KNN), Random Forest, and Logistic Regression algorithms. The third strategy, which used the Synthetic Minority Over-sampling Technique (SMOTE) for data balance, performed exceptionally well and stood out among the others. Utilizing SMOTE for dataset balancing, the Random Forest algorithm produced an exceptional 98.3 % accuracy. The study highlights the effectiveness of voice biomarkers, examines the various approaches used, and draws comparisons between them.

In this work, we suggest a new numerical scheme called the fractional higher order Taylor method (FHOTM) to solve fractional differential equations (FDEs). Using the generalized Taylor’s theorem is the fundamental concept of this approach. Then, the local truncation error generated by the suggested FHOTM is estimated by proving suitable theoretical results. At last, several numerical applications are given to demonstrate the applicability of the suggested approach in relation to their exact solutions.

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.

In this study, a thorough methodology is used to present a unique way for improving skin cancer prediction accuracy. The research uses sophisticated preprocessing methods, such as the Frost filter for noise reduction and histogram equalization for contrast enhancement, to boost contrast on dermoscopic pictures from various sources, using an ISIC 2020 dataset. These actions greatly raise the dermoscopic pictures’ overall quality and usefulness for diagnosis. Utilizing labeled data for training, we offer a Fuzzy-based C-means clustering technique based on Neutrosophic Logic during the segmentation phase. In order to overcome ambiguities in skin lesion segmentation, the neutrosophic set—a groundbreaking idea in philosophy—is used. The suggested model enhances the accuracy of segmentation by modifying the neutrosophic set functions. For precise prediction, the approach combines Support Vector Machine (SVM) classification with Histogram of Oriented Gradient (HOG) feature extraction. While SVM, a supervised learning algorithm, diagnoses skin lesions based on the collected features, HOG features capture gradient information. To improve object recognition and classification, the HOG-SVM architecture is made to methodically collect and quantify essential information using dermoscopic pictures. The use of Neutrosophic Fuzzy Logic, which combines the benefits of fuzzy clustering with neutrosophic sets to produce more precise and nuanced predictions, sets the suggested method apart. The integration of different approaches into a holistic solution for skin cancer prediction is what makes the proposed study innovative. Findings and performance analysis show of the HOG-SVM method exhibits an outstanding accuracy of 98.69%, outperforming LR, KNN, and GNB methods. Python software is used to accomplish the suggested approach. This discovery opens up a possible path for better skin cancer diagnosis and advances the rapidly developing fields of dermatology and medical image processing.

Parkinson's disease presents challenges in assessing severity, especially in older adults with dementia. This study introduces a novel approach using vision-based systems for gait analysis [1]. 2D human pose tracking in video data from a dementia unit extracted gait features. Correlation between these features and parkinsonism severity, assessed by UPDRS, was observed [6]. Five machine learning algorithms were evaluated, with KNN demonstrating superior performance [9]. Random Forest and Extra Trees also performed well [10], while Logistic Regression faced challenges [11], [12]. XGBoost showed exceptional predictive ability [8]. The study underscores the effectiveness of these algorithms in non-invasively identifying gait abnormalities, particularly highlighting KNN's adeptness with high-dimensional features [9]. This research advances healthcare diagnostics by showcasing machine learning's potential in early detection and ongoing monitoring of neurodegenerative disorders, promising improved medical outcomes.

Fuzzy soft set is the most powerful and effective extension of soft sets which deals with parameterized values of the alternative. It is an extended model of soft set and a new mathematical tool that has great advantages in dealing with uncertain information and is proposed by combining soft sets and fuzzy sets. Many fuzzy decision making algorithms based on fuzzy soft sets were given. However, these do not consider the external effective on the decision it depends on the parameters without considering any external effective. In order to solve these problems, in this paper, we introduce the concept of effective fuzzy soft set and its operation and study some of its properties. We also give an application of this concept in decision making (DM) problem. Finally, we give an application of this theory to medical diagnosis (MD) and exhibit the technique with a hypothetical case study.

We introduce the concept of an effective neutrosophic soft set, which aims to capture the influence on three independent membership functions representing degrees of truth (T), indeterminacy (I) and falsity (F). We go further by presenting a generalization of the effective neutrosophic soft set, which includes the incorporation of a degree to signify the potential for an approximate value-set. This enhancement contributes to improved efficiency and realism in the concept. Notably, this innovative approach leverages the strengths of both the generalized neutrosophic set and the effective neutrosophic soft set. The subsequent sections delve into fundamental operations on the generalized effective neutrosophic soft set, providing clarity through illustrative examples and propositions. Furthermore, we demonstrate the practical application of the generalized effective neutrosophic soft set in addressing decision-making problems and medical diagnoses.

Data could be uncertain, and the levels of precision of data are intuitively different. Neutrosophic set expressions are considered an alternative to represent imprecise data in such cases. In this paper, a general definition of neutrosophic conditional probability is introduced as a generalization of the classical conditional probability. Additionally, the properties of this neutrosophic conditional probability are presented. The concepts of joint distribution function, regular conditional probabilities, marginal density function, expected value, and joint density function in the classical type are generalized to a neutrosophic type with two discrete and continuous neutrosophic random variables. Various properties and examples are presented to demonstrate the significance of this study.