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This article introduces a novel methodology for tackling stochastic Itô Volterra-Fredholm integral equations through a pseudo-Spectral Galerkin method utilizing shifted Vieta-Fibonacci polynomials. The method is very efficient in solving integral and integro-differential equations, which often arise in many scientific and engineering fields. This approach is widely applicable in many domains, including signal processing, electromagnetic theory, population dynamics, finance, and related areas. By employing this method, the intricate task of solving such equations is simplified into a set of linear algebraic equations that are efficiently solvable via the Gauss elimination method for numerical solutions. The article also presents the existence, uniqueness, and stability of the solution. The convergence of the proposed method is rigorously established, ensuring its reliability. To validate its effectiveness, two numerical examples are provided. In addition to the examples, a comparison study is conducted between the orthonormal Bernoulli polynomials-based, second-kind shifted Chebyshev polyunomials-based pseudo-Spectral Galerkin approach and existing results based on improved hat functions. Numerical results demonstrate the superiority of the suggested approach in terms of accuracy and computational efficiency, showcasing its applicability across various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

Typically, the parameters of a practical problem are considered to be deterministic. Nevertheless, there may be possibility that these parameters are incomplete owing to observational or experimental errors. Fuzzy sets are capable of handling such problems. In addition, the majority of studies on fuzzy fractional heat equations have employed two spatial variables and fuzzy sets with deterministic membership. In some instances, however, there may be more than two spatial variables, and the membership grade associated with an uncertain parameter may also contain imprecise information. Consequently, dealing with such a situation may be fascinating. In this regard, the main objective of this research is to develop alternative methods for solving the n + 1 -dimensional fractional order heat equation with an external source term in an uncertain environment. The problem is addressed by using the Elzaki transformed homotopy perturbation method and the fractional reduced differential transformation method in the fuzzy environment. Caputo fuzzy fractional derivatives are considered here. In this study, triangular fuzzy numbers and Gaussian fuzzy numbers are utilised to handle type-1 fuzzy uncertainty, whereas triangularly perfect quasi type-2 fuzzy numbers and newly introduced Gaussian-triangular type-2 fuzzy numbers are used to handle type-2 fuzzy uncertainty. Numerical examples with graphical representations are provided to demonstrate the applicability of the approaches. Convergence plots are also provided. [ABSTRACT FROM AUTHOR]

Recently, numerous demonstrative investigations on the mathematical modelling and analysis of Covid-19 have been performed. In this regard, a mathematical model is introduced to demonstrate the effects of vaccination on Covid-19. This study presents the five compartmental groups of the model, namely susceptible, infected, exposed, recovered groups, and vaccinated people. To make the model more realistic, the existing integer-order system has been modified using the Mittag–Leffler kernel. The formation of a fractional-order model shows the validity of this generalization with memory. In addition to the positivity solution, existence, uniqueness, and biologically feasible region are considered. The Ulam–Hyers stability is established through fixed-point theory and nonlinear analysis. The basic reproduction number, which means the number of people predicted to be secondarily infected among the susceptible population, is calculated. The considered model is assessed by using the Atangana–Baleanu fractional operator, and the fractional Adams–Bashforth numerical technique based on Lagrange polynomial interpolation is adopted to solve the model. Further, the error analysis of the present method has also been included. The behaviour of the solution is demonstrated through numerical simulations with the help of graphical representations. [ABSTRACT FROM AUTHOR]

The multidimensional poverty index of poverty is the most widely used method for estimating the poverty rate. It uses preassigned weights for poverty indicator that plays a critical role in determining an individual's poverty status. But the question immediately arises: how do we justify these weights of poverty indicators? In order to overcome this issue, we have formulated composite multidimensional poverty index and composite headcount ratio using principal component analysis. Further, the self-organizing maps clustering algorithm, the K-means clustering algorithm, and spatial analysis are used to determine the spatial and temporal disparity of different poverty levels. In this respect, an Indian state viz. Odisha has been considered as a case study. These methods cluster poverty data into different poverty levels by analysing the poverty indicators only. This study concluded that multidimensional poverty in the considered state is widespread and severe, requiring prompt government action. In comparison to other methods, this one is proven to be more robust, all-encompassing, and indicator-neutral. [ABSTRACT FROM AUTHOR]

In this article, we accumulate theories and computing procedures to find eigenvalue bounds for interval matrices. Also, we have pointed out the computational complexity and applications of interval eigenvalue problems. Computing exact eigenvalue for the general interval matrices is an NP-hard problem. Various eigenvalue bounds for symmetric, non-symmetric, and complex interval matrices are included here. The article attempts to present some of the important methods proposed by various authors in one place, analyze them, and then final observations are made. In this respect, each of the methods has been addressed by taking simple example problems of interval matrices so that readers may understand the philosophy of those methods. Further, the increase of tightness of the solutions is also discussed. The interlinks of their development and gaps among them are highlighted. [ABSTRACT FROM AUTHOR]

Much of the time, geological data are not accurately known. Instead, geoscientists work with a range of possible data. These data can have a range since they can vary spatially and temporally. In this context, we demonstrate how a well-established structural geological model of velocity profile of ductile simple shear, expressed as an equation, can be furthered by fuzzy set concept. The implementation of fuzzy set concepts in terms of fuzzification in (geo)scientific models takes into consideration the uncertainties of magnitudes of the parameters that define the model. The present study shows that the fuzzified velocity profile of a ductile simple shear zone with parallel boundaries is effectively independent of the dip of the shear zone. This article demonstrates how models of (structural) geological processes can be modelled by fuzzy set approach capable of incorporating a range of magnitudes of parameters. [ABSTRACT FROM AUTHOR]

This article deals with the solution of a fractionally-damped generalized Bagley–Torvik (BT) equation whose damping characteristics are well-defined by means of the fractional derivative (FD) of the Riemann–Liouville and the Liouville–Caputo types. The Ho-motopy Analysis Method (HAM) is implemented for computing the dynamic response (DR) analysis. Two external forces or loads (namely, the unit step function and the unit impulse function) are considered for the analysis presented here. The FD is first defined and then used here in the Riemann–Liouville sense and the Liouville–Caputo sense. In order to show the efficiency, powerfulness, and validations of the present analysis, the obtained results are compared with the solutions derived earlier by Suarez and Shokooh (1997) who used the eigenfunction expansion method and by Podlubny (1999) who used fractional-order Green's function. [ABSTRACT FROM AUTHOR]

Risk and uncertainties plays a major role in stock market investments. It is a pedagogical practice to deduce probability distributions for analysing stock market returns using theoretical models of investor behaviour. Generally, economists estimate probability distributions for stock market returns that are observed from the history of past returns. Besides this, there are impreciseness involved in various factors affecting market investment and returns. As such, we need to model a more reliable strategy that will quantify the uncertainty with better confidence. Here, we have presented a computational method to solve fuzzy stochastic Volterra–Fredholm integral equation which is based on the block pulse functions (BPFs) using fuzzy stochastic operational matrix (SOM). The concept of fuzziness has been hybridized with BPFs, and the corresponding stochastic integral equation has been modelled. For illustration, the developed model has been used to investigate an example problem of Black–Scholes fuzzy stochastic differential equation (FSDE), and the results are compared in special cases. [ABSTRACT FROM AUTHOR]

In causal study we are interested in finding the graphical structure in the form of directed acyclic graphs (DAGs). These DAGs describe the directions and connection strength to connecting variables represented by nodes. In this regard, various methods have been developed to estimate the appropriate structure of the causal model and to explain a fair number of its features. Our review aims to provide a complete and systematic analysis of selected articles from past few decades, having powerful methods to infer the area of study. In this article, we categorized all selected articles in three groups, on the basis of techniques these used to construct the causal model. To provide a full comparative study under categories of probabilistic, statistical and algebraic approaches, we discussed underlying difficulties, limitations, merits and disadvantages in applying these techniques. The reader will find it helpful to choose and use the appropriate method for a better implication. [ABSTRACT FROM AUTHOR]

This paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method. [ABSTRACT FROM AUTHOR]

This article uses powerful technique of artificial neural network (ANN) models to simulate and estimate structural response of two-storey shear building by training the model for a particular earthquake. The neural network is first trained for a real earthquake data and the numerically generated responses of different floors of two-storey buildings as the training patterns. Trained ANN architecture is then used to simulate and test the structural response of different floors for various intensity earthquake data and it is found that the predicted responses given by ANN model are good for practical purposes. It is worth mentioning that although the simulation is done with numerically generated response data for particular earthquake, the idea may also be used for actual experimental (response) data. [ABSTRACT FROM AUTHOR]

The accurate long-range forecast of southwest rainfall can have manifold benefits for the country, from disaster mitigation and town planning to crop planning and power generation. In this paper, the rainfall has been modeled using artificial neural network (ANN) with different network configurations. Performance of these networks are compared with some results found in the literature. The networks have also been tested for the data outside the range of the trained data and compared with known results. The present network is found to be better in term of predictions than the previous results by others. Southwest monsoon rainfall over India for 6 years in advance has been predicted. [ABSTRACT FROM AUTHOR]