In this paper, the uncertain static responses of structures under fuzzy external loadings are investigated using the fuzzy finite element approach. External loads are represented as fuzzy values in this study, although material and geometric properties are regarded as precise. Accordingly, the obtained governing static equilibrium equations is a fuzzy system of linear equations with crisp coefficients, fuzzy unknowns, and a fuzzy right-hand side vector. To solve this fuzzy system, a novel method based on the concept of convex combinations is proposed. The method leverages convex combinations after converting the fuzzy system into an analogous interval form in order to produce a crisp system representation. To obtain the ultimate solution, the resulting crisp systems are solved for two distinct parametric values of the convex combination together. The work examines a mathematical example in addition to actual examples such as an 8-bar truss, a fixed-fixed beam, and a uniform rectangular sheet subjected to external loads. The results are validated by comparing the obtained results with existing techniques.

The bending analysis of nonhomogeneous nanobeams is vital for ensuring the structural reliability and functional efficiency of nanoscale devices such as sensors, actuators, and nano resonators, where even minor deformations can significantly influence performance. These nanobeams, when subjected to exponentially varying loads, exhibit complex mechanical behaviour due to spatially dependent material properties and pronounced nanoscale effects. The assumption of exponentially varying flexural rigidity along the axial direction further influences the deformation characteristics. In this work, we develop a physics-informed kernel-based scientific machine learning framework employing the Physics-Informed Least Squares Support Vector Machine (PI-LSSVM) for the bending analysis of nonhomogeneous nanobeams subjected to exponentially varying loads. The PI-LSSVM approach integrates the governing physical laws and boundary conditions directly into the machine learning architecture, thereby ensuring physically consistent and data-efficient predictions. This hybrid paradigm combines the strengths of physics-based modelling and data-driven learning to approximate displacement fields, stress distributions, and deflection profiles with high accuracy. Comparative analysis demonstrates the high prediction accuracy and strong physical consistency of the proposed model. The study highlights the potential of physics-informed machine learning in addressing complex nanoscale structural mechanics problems and contributes to the intelligent design and optimization of next-generation nanostructures.

This study introduces a model capable of investigating the vibrational response of nanobeams that simultaneously incorporate structural perforations and axially functionally graded materials while being supported by elastic foundations. By using Euler–Bernoulli beam theory with the nonlocal elasticity framework, the model captures nanoscale phenomena often neglected in classical analysis and governing equation of motion is derived. The objective of this study is to examine the coupled effects of axial material gradation, periodic perforations, and elastic foundation parameters on the vibrational behavior of nanobeams, highlighting how variations in perforation geometry and material gradation jointly influence the mode shape and frequency response. The modified equivalent modeling strategy is employed to account for the periodic perforations. The Galerkin method is utilized to handle and extract natural frequencies and mode shapes. The proposed numerical scheme exhibits high accuracy, as verified by comparison with existing results from the literature, with the maximum deviation limited to just 0.081%. Perforation and axial material gradation significantly impact the vibrational characteristics, governed by the nonlocal effects and geometric configuration. Numerical findings highlight the sensitivity of dynamic response to filling ratio, perforation geometry, and foundation interaction across different boundary conditions.

This paper introduces the competition component to Sprott's nonlinear love triangle model and observes that the new system likewise exhibits chaotic behaviour. In order to make the model more realistic, we have modified the existing fractional-order system using exponential decay memory. The occurrence of fractional systems proves the validity of this generalization with memory, since memory impacts romantic relationships over time. The considered model is assessed by using Caputo-Fabrizio fractional derivatives, and the fractional Adams-Bashforth numerical approach is used to solve the system based on Lagrange polynomial interpolation. Existence, uniqueness, and Ulam–Hyers stability are established through fixed-point theory and nonlinear analysis. Further, the error analysis of the current method has also been incorporated. Finally, numerical simulations are used to demonstrate the behaviour of the solution using graphical representations.

In advanced studies of bionanoscience, magnetic nanomaterials serve as therapeutic transporters for treating vascular disorders, such as carotid and peripheral artery diseases, along with other biomedical applications. This study explores the theoretical behavior of hybrid nanoparticles (Cu-Fe2O3) in two-dimensional peristaltic blood flow through an inclined, catheterized artery, accounting for outer wall slip in an uncertain environment. The non-Newtonian Jeffrey nanofluid model is employed, incorporating nonlinear thermal radiation and an externally induced magnetic field to capture novel aspects of nanofluid behavior. However, uncertainty in velocity and temperature patterns may arise due to variations in nanoparticle volume fraction, which cannot be ignored. To address this, these distributions are analyzed within a fuzzy framework, treating them as triangular fuzzy numbers (TFNs). Within this framework, the dimensionless nonlinear flow equations are converted into fuzzy differential equations by introducing symmetrical TFNs, where the nanoparticle volume fractions serve as fuzzy parameters. The Homotopy Perturbation Method (HPM) is then applied to derive fuzzy semi analytical solutions for temperature and velocity profiles using a double parametric approach for fuzzy numbers. Additionally, a comprehensive graphical analysis is presented, incorporating triangular fuzzy representations in both two-dimensional (2D) and three-dimensional (3D) frameworks for the fuzzy solutions of temperature and velocity profiles. The obtained fuzzy solutions are validated by comparing a special case of the present solution with existing precise solutions. An in-depth analysis of key flow characteristics such as wall shear stress, the Nusselt number, and the skin friction coefficient is conducted for the special case under various emerging parameters. It is observed that as the Darcy parameter increases, both the upper and lower bounds of fuzzy velocity improve. Meanwhile, an increase in the thermal radiation parameter leads to a significant drop in the fuzzy temperature profile due to enhanced heat dissipation through radiation.

Poverty is a complex dynamic challenge that cannot be adequately captured using predefined differential equations. Nowadays, artificial machine learning (ML) methods have demonstrated significant potential in modelling real-world dynamical systems. Among these, Neural Ordinary Differential Equations (Neural ODEs) have emerged as a powerful, data-driven approach for learning continuous-time dynamics directly from observations. This chapter applies the Neural ODE framework to analyze poverty dynamics in the Indian state of Odisha. Specifically, we utilize time-series data from 2007 to 2020 on the key indicators of economic development and poverty reduction. Within the Neural ODE architecture, the temporal gradient of the system is represented by a multi-layer perceptron (MLP). The obtained neural dynamical system is integrated using a numerical ODE solver to obtain the trajectory of over time. In backpropagation, the adjoint sensitivity method is utilized for gradient computation during training to facilitate effective backpropagation through the ODE solver. The trained Neural ODE model reproduces the observed data with high accuracy. This demonstrates the capability of Neural ODE to capture the dynamics of the poverty indicator of concrete-structured households. The obtained results show that ML methods, such as Neural ODEs, can serve as effective tools for modeling socioeconomic transitions. It can provide policymakers with reliable projections, supporting more informed and effective decision-making for poverty alleviation.

This study proposes a novel method for solving Fully Fuzzy Linear Programming (FFLP) problems, with coefficients represented as Symmetric Trapezoidal Fuzzy Numbers (STrFNs). Unlike traditional methods that reduce fuzzy constraints to crisp equivalents, this approach preserves the complexity of uncertainty by employing a double-parametric representation using $\alpha$ and $\beta$ parameters. The method reformulates the FFLP problem into a parametric framework via Double parametric form (DPF) or Bi-parametric form (BPF) and solves it using the simplex algorithm for varying ($\alpha, \beta$) combinations. The suggested method has been evaluated on representative FFLP situations, demonstrating enhanced accuracy and adaptability, while it requires greater computer resources than conventional crisp-based methods. The proposed method demonstrates enhanced accuracy and adaptability in numerical experiments, making it highly suitable for applications in logistics, finance, and operations, where managing inherent uncertainty is crucial.

This paper presents a novel dynamic model of a dual-cable-strapped single-link flexible manipulator under an uncertain workspace. The material of the link of the manipulator is considered to be non- homogeneous with inherent non-linearity, which is crucial in modelling. In this regard, linear and quadratic variations in the density of the material of the flexible link with space coordinates are considered. Uncertainty is introduced into its parameters in order to make the real-time dynamics of the flexible manipulator more challenging. Fuzzy parameters, in support of uncertainty, are invoked in the spring constants of the cables for modelling using Triangular Fuzzy Numbers (TFN). In particular, Interval Type-2 Triangular Fuzzy Numbers (IT2TFN) are considered in this investigation since higher-order fuzzy numbers can be beneficial, as the presence of uncertainty in the membership grade may also be unavoidable. The non-homogeneous flexible link is modelled as an Euler-Bernoulli beam, and the non-linear equations of motion are derived using Hamilton's principle and two boundary conditions. It may be difficult to obtain the exact solution of the governing differential equation because of the non-homogeneity in the parameters associated with the link. In this regard, the Adomian Decomposition Method (ADM) is implemented in the crisp as well as uncertain scenario to determine the natural frequencies of the single-link flexible manipulator. The existence and uniqueness of the solution is also illustrated. Tabular and graphical results are presented for validation and better visualisation.

This article investigates the traveling wave solution for a geophysical Boussinesq-type equation, which serves as a model for equatorial tsunami waves. By employing a combination of traveling wave transformation and the Sine-Gordon expansion method, the study introduces several new traveling wave solutions, including hyperbolic, trigonometric types, and bell-shaped forms, for this nonlinear model. Through appropriate parameter selection, the research utilizes two-dimensional (2D), three-dimensional (3D), and contour plots to efficiently depict the characteristics of specific solutions. These visual representations serve as valuable tools for grasping the essence of these solutions. Importantly, all the derived solutions completely agree with the governing equations when inserted into them. This highlights the reliability and accuracy of the methodologies applied, and the results show the effectiveness, clarity, and efficiency of the techniques employed in this work. Additionally, by transforming the considered equation into a planar dynamical structure, a thorough examination of all possible phase portraits of the dynamical system is conducted using bifurcation theory. Furthermore, periodic, quasi-periodic, and chaotic behaviors are observed in 2D, 3D, and time series analyses when an external force is applied to the dynamical system. These approaches could be useful in analyzing more intricate models that are prevalent in contemporary science and engineering.

Polynomial interval eigenvalue problems in a bounded uncertain environment are studied in this article. Here, the polynomial function has the complex Hermitian interval matrix coefficients. We have computed the eigenvalue enclosures (as new theorems) for the polynomial eigenvalue problem with complex Hermitian interval matrices. We have discussed the computational complexity of the proposed method and analyzed the tightness of different eigenvalue enclosures. Demonstrations are made for the present problem in the system of ordinary differential equations that appears in a dynamical system with bounded uncertainty.

In order to model anomalous diffusion in various physical systems, this article presents a collocation technique for solving intricate, multidimensional stochastic integral equation that include fractional Brownian motion. The method utilizes two categories of orthogonal polynomials: second-kind shifted Chebyshev polynomials and shifted Legendre polynomials. It uses Newton-Cotes nodes as collocation points, and the Itô approximation to handle the stochastic integral of the equation. This approach transforms the equation into solvable algebraic systems, which may be either linear or nonlinear. This process effectively decreases computing complexity while preserving accuracy. Furthermore, a thorough convergence study, with meticulous proof, verifies the accuracy of the process. Additionally, four examples demonstrate the effectiveness of the method. A comparison test reveals that results from shifted Chebyshev polynomials are closer to the exact solutions than results from shifted Legendre polynomials. Moreover, a 99

In this study, we have introduced a new method to develop the Wealth Index (WI). In this regard, primary survey data, along with secondary data obtained from the National Family Health Survey-4 (NFHS-4) and NFHS-5 datasets, is used to construct the WI for the Koraput district in the Indian state of Odisha. Furthermore, we compared the WI obtained from the primary data with the WI obtained from the NFHS-4 and NFHS-5 data sets for Koraput and Odisha. Furthermore, to analyze wealth disparities at a finer scale, we computed and compared sub-indices in a similar manner to the WI. To develop the WI, we adopted a novel approach based on principal component analysis (PCA) with orthogonal rotation of factors whose eigenvalue is greater than 1 to utilize the maximum variance of data. The results derived from the WI and its sub-indices indicate that, when contrasted with Odisha as a whole, Koraput exhibits a lower level of wealth. The findings also reveal that, over time, there has been some improvement in wealth conditions, but they remain a cause for serious concern. Overall, the WI for the said district presents critical results, underscoring the urgent need for government or NGO intervention.

This manuscript introduces a semi-analytical and numerical solutions for the Benjamin Bona Mahony equation (BBME) in the form of convergent series. The BBME holds significance in diverse scientific and engineering applications. Especially to study issues on shallow water waves, solitons and their importance in modern physics. The Fuzzy Homotopy Perturbation Transform Method (FHPTM) and Differential Quadrature Method (DQM) are utilized to obtain the solutions for the BBME. In DQM, grid point based on Shifted Legendre Polynomials (SLP) have been used to solve the BBME. Additionally, we address the uncertainty in the initial condition by representing it in terms of an interval. The interval BBME (iBBME) is subsequently tackled using the FHPTM providing both lower and upper interval solutions. The convergence of the interval solution is validated considering crisp case. The outcomes obtained through FHPTM for BBME are compared with the exact solution and results obtained in this study are exhibiting good agreement. The numerical outcomes by FHPTM are compared with results obtained by DQM. Additionally, we presented the time fractional BBME and developed a fuzzy model for it, accounting for uncertainties in the coefficients associated with wave velocity. To analyze the behavior of the fuzzy time fractional BBME, and examined various numerical results using a double parametric approach.