In this article, the exact travelling wave solutions for the nonlinear time-fractional Clannish Random Walker’s Parabolic equation are discussed. This study establishes the new extended direct algebraic method in which periodic, bright, multiple U-shaped bright and kink-type wave solitons are obtained with exact solutions offered by the mixed hyperbolic and trigonometry solutions, mixed periodic and periodic solutions, plane solutions, shock solutions, mixed trigonometric solutions, mixed singular solutions, mixed shock single solutions, complex solitary singular solutions, shock solutions and shock wave solutions. The obtained solutions of the non-linear time-fractional Clannish Random Walker’s Parabolic equation model are graphically presented for different values of the involved parameters using Wolfram Mathematica software. The propagating behaviours are visualised through 3D, contour, and 2D surface plots to illustrate the influence of key parameters on the solution profiles. The time-fractional derivative introduces a memory effect into the model, making it more suitable for describing real-world physical processes that involve hereditary and nonlocal behaviour. The presence of novel soliton structures, such as multiple U-shaped solitons and bright-shaped solitons, further highlights the novelty and complexity of the model’s dynamics. In this study, a planar dynamical system is constructed from the proposed model to carry out a sensitivity analysis of initial conditions. This transformation enables the investigation of how small variations in the initial values influence the system’s long-term behaviour. The proposed method proves to be efficient, reliable, and broadly applicable for generating new analytical solutions to both integer and non-integer-order differential equations arising in mathematical physics and engineering.

This study advances the application of fuzzy hypersoft sets (FHSS) in machine learning (ML) decision-making by introducing a novel energy metric to quantify multi-sub-attribute uncertainty. Building on fuzzy set theory, soft sets, and hypersoft sets, FHSS integrates fuzzy membership with multi-sub-attribute parameterization, addressing limitations of traditional uncertainty models in handling complex, high-dimensional datasets. Inspired by spectral graph theory, the proposed energy metric — the sum of singular values of the FHSS matrix — quantifies systemic significance and enables a robust ranking of alternatives. An algorithm leveraging this metric has been developed and validated through applications in healthcare (heart risk profiling) and energy systems, achieving 90.83% accuracy and an F1-score of 0.8706 in a dataset of 500 participants. A comparative analysis demonstrates the superiority of FHSS energy over fuzzy soft and hypersoft sets, particularly in capturing attribute interdependencies. Despite the computational challenges posed by large matrices, the framework provides interpretable and scalable solutions for ML-driven decision-making under uncertainty. Future work will optimize computational efficiency and extend applications to domains such as financial risk analysis, further reinforcing FHSS energy as a transformative tool for precise, uncertainty-aware decision-making.

Carotid atherosclerosis is a major contributor to cardiovascular morbidity and mortality, primarily due to plaque-induced arterial steno-sis. This study develops a patient-specific computational model of Casson blood flow in symmetrically stenosed carotid arteries using a finite volume method. Realistic artery geometries are reconstructed from MRI and CT imaging. The model indicates a significant pressure drop (from approxi-mately 917 Pa to -411 Pa) and an increase in blood velocity, peaking at 1.5 m/s within the stenosed region. Temperature gradients are noted along ar-terial walls due to frictional heating. The model is quantitatively validated with reference data and clinically confirmed by MRI diagnostics. The main findings of computational results further show that wall shear stress increases with stenosis severity, and recirculation zones intensify near the stenosis throat, potentially promoting plaque buildup. The Casson fluid model captures the non-Newtonian nature of blood effectively, while the finite volume scheme ensures stable and accurate numerical predictions. These insights advance the understanding of hemodynamic behavior in stenosed arteries and may aid in personalized treatment planning.

In this research we employ first model of f(R,T) gravitational theory f(R, T) = R + 2f(T) to explore the type III Bianchi Universe featuring domain wall. It was assumed that the expansion scalar is directly related to the shear scalar to find solution of the field equation. This results the metric potentials connection Z = kY n and the consideration of two scenarios for the integrating constant D. We observed that the domain wall existed early in the universe’s history and eventually vanished. A few of the model’s physical parameters are also covered.

WedemonstratedHermite-Hadamardtyperesultsviar−convex function class. The proofs of results are based on the identities which are given in [16] and [3], established by Set et al. and Chun and Qi respectively.

In this article, we present the generalized ρ dependent polynomials for the calculations of eccentricity, distance, total distance, and degree-based topological indices of the identity graph of Zρ. This is a thorough work in which we present many topological indices and coindices as ρ dependent polynomials. The polynomials presented here can play a key role in the further development of the theory of topological indies for commutative rings. This paper presents a brand new approach to generalizing the topological indices because instead of the traditional way. A set-theoretic method is introduced here that can be very helpful and game-changing in the field of algebraic graph theory first of all sets of vertices for the identity graph of the commutative ring Zρ are partitioned into various sets, which makes it easier to generalize the degrees, distances, and eccentricities of this graph. This paper presents various results that make it easier to generalize the topologicalindices of the identity graph of Zρ.

In this article, the translation hypersurfaces in Euclidean 4- space are defined as the sum of three curves with distinct parameters with unit speed, and non-planar. These curves are called the generator curves of the hypersurface. Utilizing the hypersurface theory in Euclidean 4-space, unit normal vector field, shape (Weingarten) operator matrix, fundamental forms, Gaussian curvature and mean curvature have been expressed for the translation hypersurfaces. Finally, the computational example is stated to efficiency of the theoretical results.

The purpose of the work is to generate the ordinary differential equations and their solutions for the probability density function, quantile rate function, survival rate function, inverse survival rate function, haz-ard rate function and reversed hazard rate function of the Length Biased Exponential Distribution. The ordinary differential equations and their so-lutions are obtained using the Math of differentiation and integration as a tool together with their boundary conditions. The boundaries and para-meters that portrayed the distribution unavoidably decide the nature, presence, uniqueness, arrangement and the various possible solutions of these ordinary differential equations are better approaches to understand these characteristics. The work will be helpful to analyze the lifetime growth or risk and is of great significance in the field of ecological studies. The method can be very useful for the other probability distributions and can serve as a substitute for the approximation study.

This research paper has developed a numerical iterative method by using Simpson 3/8 rule for solving non-linear equations, which equations are studied in different sciences and engineering fields. The developed technique has quadratic converge. This paper reflects the idea and more better results from the work of the authors of [7-8]. Examples are given to show that the proposed iterative method is better than compared methods. Our developed method is compared with the Newton Raphson method and the Trapezoidal method. C++/MATLAB was used to compute the numerical results. It can be observed from the results that the simpson iterative method is better than the Newton Raphson method, and the Trapezoidal method in terms of iteration and accuracy perception.

Segmenting natural and outdoor images are challenging for most of the latest variational segmentation models. For this purpose we employ derived image data (DID) and propose a robust variational model. The DID rely on three images by utilizing image local and global statistics as well as filter image which is obtained through our design high pass filtering techniques. Then these derived image data are incorporated into our proposed energy functional which can robustly segment images having inhomogeneity, mix backgrounds and multi-regions. Furthermore, the results of DID are compared with other well known methods with finding Jaccard similarity index to proof the efficient and qualitative performance of proposed model over the traditional methods. Finally, the proposed DID based model is tested on real world 3D images to ensure that it also preserve its performance in vector valued images as well.