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Machinery components degrade over time due to continuous use. A reliable prognosis framework can improve machinery health by monitoring the behavior of its parts and providing warnings before critical failures occur. Bearings, which are essential components of rotating machinery, help maintenance personnel assess the machine’s condition during continuous wear. In this study, vibration data from roller bearings under various conditions and faults were collected. The Vibration analysis technique was employed to detect and classify different faults in bearings based on the characteristics of the vibration signals generated by the machinery. Faults can be detected, diagnosed, and classified by analyzing bearing vibration signatures using techniques such as frequency analysis, time-domain analysis, spectral analysis, and kurtogram classifiers. This enables appropriate maintenance actions to be taken in time, preventing further damage or failures.

A survey of results on Grüss-type inequalities associated with a variety of fractional integral and differential operators is presented. The fractional differential operators includes, Riemann-Liouville fractional integral operators, Riemann-Liouville fractional integrals of a function with respect to another function, Katugampola fractional integral operators, Hadamard's fractional integral operators, $ k $-fractional integral operators, Raina's fractional integral operators, tempered fractional integral operators, conformable fractional integrals operators, proportional fractional integrals operators, generalized Riemann-Liouville fractional integral operators, Caputo-Fabrizio fractional integrals operators, Saigo fractional integral operators, quantum integral operators, and Hilfer fractional differential operators.

A review on Hermite-Hadamard type inequalities connected with a different classes of convexities and fractional differential operators is presented. In the various classes of convexities it includes, classical convex functions, quasi-convex functions, p-convex functions, strongly-m-convex functions, strongly-(θ,m)-convex functions, (s, m)-convex functions, (θ, h- m)-convex functions, strongly (θ, h- m)-convex functions, (h, m)-convex functions of the second type, m-convex functions, h-convex functions, (h,m)-convex functions, relative-convex functions, exponentially (θ, h- m)-convex functions, harmonically h-convex functions and geometric-arithmetically s-convex functions. In the fractional differential operators it includes, Caputo fractional derivative, k-Caputo fractional derivative and Hilfer fractional derivative.

In this article, we aim to introduce and explore a new class of preinvex functions called $ \mathfrak{n} $-polynomial $ m $-preinvex functions, while also presenting algebraic properties to enhance their numerical significance. We investigate novel variations of Pachpatte and Hermite-Hadamard integral inequalities pertaining to the concept of preinvex functions within the framework of the Caputo-Fabrizio fractional integral operator. By utilizing this direction, we establish a novel fractional integral identity that relates to preinvex functions for differentiable mappings of first-order. Furthermore, we derive some novel refinements for Hermite-Hadamard type inequalities for functions whose first-order derivatives are polynomial preinvex in the Caputo-Fabrizio fractional sense. To demonstrate the practical utility of our findings, we present several inequalities using specific real number means. Overall, our investigation sheds light on convex analysis within the context of fractional calculus.

This paper presents a novel approach to studying divorce dynamics and elimination strategies using nonlinear differential equations. A mathematical model is formulated to capture the key factors influencing divorce rates. The model undergoes a rigorous theoretical analysis, including parameter estimation, solution existence/uniqueness, positivity, boundedness, and invariant regions. A qualitative analysis explores equilibria, stability conditions, and a sensitivity analysis. Numerical simulations and discussions are presented to validate the model and shed light on divorce dynamics. Finally, conclusions and future research directions are outlined. This work offers valuable insights for understanding and potentially mitigating divorce rates through targeted interventions.

The measurements of thermal transport are complex to optimize from high temperatures; this is because increased thermal-management performance have typically been accompanied for the sake of applicability in thermal barriers and heat dissipation. In this manuscript, an investigation for the flow of Brinkman nanofluid is carried out in which shapes of nanoparticles are suspended in ethylene glycol and their thermophysical properties have been emphasized. In order to have an optimized heat and mass transfer, the mathematical modeling is traced out for temperature, concentration and velocity profiles through non-singularized fractional technique. The analytical solutions for temperature, concentration and velocity profiles have been derived by invoking integral transforms. Special analytical solutions have been traced out for Stoke's first and second problems by varying boundary conditions of the problem. The rheological parameters have shown the pertinent effects on the basis of numerical variations and similarities. The results suggested that velocity of ethylene glycol-alumina is slower than ethylene glycol-copper, ethylene glycol-gold and ethylene glycol-silver during non-magnetization process.

In this work, a new adaptive numerical method is proposed for solving nonlinear, singular, and stiff initial value problems often encountered in real life. Starting with a fixed step size, the new method’s performance can be significantly enhanced by introducing an adaptive step-size approach. The qualitative properties of the proposed method have been investigated to determine the efficiency and reliability of the method. The proposed method is of fifth-order accuracy, zero stable, L-stable, and consistent. In addition, the proposed method is convergent, and its stability properties are also shown through its Order Stars. Finally, numerical experiments are conducted to illustrate the performance of the method. The results obtained show that the proposed method compares favourably with existing methods.

A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions, (p,s)-convex functions, modified (p,s)-convex functions, (p,h)-convex functions, tgs-convex functions, η-quasi-convex functions, ϕ-convex functions, (α,m)-convex functions, ϕ-quasi-convex functions, and coordinated convex functions. Quantum H-H type inequalities via preinvex functions and Green functions are also presented. Finally, H-H type inequalities for (p,q)-calculus, h-calculus, and (q−h)-calculus are also included.

Abstract Implicit block approaches are used by a number of numerical analyzers to model mild, medium, and hard differential systems. Their excellent stability characteristics, self-starting nature, quick convergence, and large decrease in computing cost all contribute to their widespread application. With these numerical benefits in mind, a new one-step implicit block method with three intrastep grid points has been created. The major term of the local truncation error is minimized to determine which of these points is optimal. The reformulation of the suggested technique leads to a significant decrease in computing cost while maintaining the same consistency, zero-stability, A $\mathcal{A}$ -stability, and convergence. Several sorts of error are calculated, together with CPU time and efficiency plot, to determine which is superior. Differential models from the fields of heat transfer, population dynamics, and chemical engineering show that the suggested method does a better job than some of the current hybrid block and implicit Radau methods with similar properties.

Machinery components undergo wear and tear over time due to regular usage, necessitating the establishment of a robust prognosis framework to enhance machinery health and avert catastrophic failures. This study focuses on the collection and analysis of vibration data obtained from roller bearings experiencing various fault conditions. By employing a combination of techniques sourced from existing literature, distinct configurations within vibration datasets were examined to pinpoint the primary defects in roller bearings. The significant features identified through this analysis were utilized to formulate optimized stochastic model equations. These models, developed separately for inner and outer race fault features in comparison to healthy bearing features under random conditions, offer valuable insights into machinery prognosis. The application of these models aids in effective maintenance management, optimization of machinery performance, and the minimization of catastrophic failures and downtime, thereby contributing to overall machinery reliability.

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher’s interest even more.

Complex processes of the physical world require novel and sophisticated mathematical notions to get deep insights. In this research analysis, two standard mathematical models for the series RL and RC circuits having time-invariant sources taken from the discipline of electrical engineering have been investigated with the help of differential operators known with the name of truncated M- derivative, Atangana beta-derivative, and the conformable derivative operators. The exact solutions for these two models have been found in terms of the transcendental exponential function of time under the truncated M-derivative, Atangana beta-derivative, and the conformable derivative operators. The numerical simulations carried out via MATLAB ”9.4.0.813654 (R2018a)” have been interpreted to explore new behavior for solutions of the models not possible to obtain through standard classical calculus wherein one is restricted to have integer-order derivatives unlike the differential operators used in the present research study. The models under consideration for the three differential operators have been investigated with varying parameters’ values including the differential orders α and β whereupon the classical case is resumed for α=β=1 and τ=0.

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, (θ,h−m)−p-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, (k−p)-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and (p,q)-calculus are also included.

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m,tgs)-type s-preinvex function, Godunova–Levin (s,m)-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings.

The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of Hermite–Hadamard and Pachpatte-type integral inequalities involving the idea of the preinvex function in the frame of a fractional integral operator, namely the Caputo–Fabrizio fractional operator. By employing our approach, a new fractional integral identity that correlates with preinvex functions for first-order differentiable mappings is presented. Moreover, we derive some refinements of the Hermite–Hadamard-type inequality for mappings, whose first-order derivatives are generalized preinvex functions in the Caputo–Fabrizio fractional sense. From an application viewpoint, to represent the usability of the concerning results, we presented several inequalities by using special means of real numbers. Integral inequalities in association with convexity in the frame of fractional calculus have a strong relationship with symmetry. Our investigation provides a better image of convex analysis in the frame of fractional calculus.

The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, the theory of convexity and inequalities have recently attracted a lot of attention from historians and modern researchers. This article explores the concept of a new group of modified harmonic exponential s-convex functions. Some of its significant algebraic properties are elegantly elaborated to maintain the newly described idea. A new sort of Hermite–Hadamard-type integral inequality using this new concept of the function is investigated. In addition, several new estimates of Hermite–Hadamard inequality are presented to improve the study. These new results illustrate some generalizations of prior findings in the literature.

In this study, we analyzed the inertia effect on the axisymmetric squeeze flow of slightly viscoelastic fluid film between two disks. A system of nonlinear partial differential equations (PDEs) in cylindrical coordinates, along with nonhomogenous boundary conditions, illustrates the phenomenon of fluid flow caused by squeezing with the inertia effect. The Langlois recursive approach was applied to obtain the analytical solution of the system having a stream function, axial and radial velocities, pressure distribution, normal and tangential stresses and normal squeeze force. These flow variables are also portrayed graphically to describe the effects of the Reynolds number and slightly viscoelastic parameter. The results show that by increasing the Reynolds number, the velocity profile decreases, and both the pressure distribution and shear stresses increase. Moreover, there is a small increase in normal squeeze force. When the slightly viscoelastic parameter approaches zero, the obtained solution of flow variables matches with the classical results. This study can be applied to understand the mechanism of load-bearing features in thrust bearings and in arthrodial human joint (knee and hip) diseases.

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α,m)-convex functions, (α,m)-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ,m)-convex functions, m-harmonic harmonically convex functions, (s,r)-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, (α,m)-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable ϕ-convex functions, MT-convex functions, (s,m)-convex functions, p-convex functions, h-convex functions, σ-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, σ,s-convex functions, modified (p,h)-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, (α,h−m)−p-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, (k,s)-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator.

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators.

The main objective of this article is to introduce a new notion of convexity, i.e., modified exponential type convex function, and establish related fractional inequalities. To strengthen the argument of the paper, we introduce two new lemmas as auxiliary results and discuss some algebraic properties of the proposed notion. Considering a generalized fractional integral operator and differentiable mappings, whose initial absolute derivative at a given power is a modified exponential type convex, various improvements of the Hermite–Hadamard inequality are presented. Thanks to the main results, some generalizations about the earlier findings in the literature are recovered.

Fractional derivative and integral operators are often employed to present new generalizations of mathematical inequalities. The introduction of new fractional operators has prompted another direction in different branches of mathematics and applied sciences. First, we investigate and prove new fractional equality. Considering this equality as the auxiliary result, we attain some estimations of a Hermite–Hadamard type inequality involving s-preinvex, s-Godunova–Levin preinvex, and prequasi invex functions. In addition, we investigate a fractional order Hadamard–Fejér inequality and some of its refinements pertaining to h-preinvexity via a non-conformable fractional integral operator. Finally, we present a Pachpatte type inequality for the product of two preinvex functions. The findings as well as the special cases presented in this research are new and applications of our main results.

The subject of convex analysis and integral inequalities represents a comprehensive and absorbing field of research within the field of mathematical interpretation. In recent times, the strategies of convex theory and integral inequalities have become the subject of intensive research at historical and contemporary times because of their applications in various branches of sciences. In this work, we reveal the idea of a new version of generalized harmonic convexity i.e., an m–polynomial p–harmonic s–type convex function. We discuss this new idea by employing some examples and demonstrating some interesting algebraic properties. Furthermore, this work leads us to establish some new generalized Hermite–Hadamard- and generalized Ostrowski-type integral identities. Additionally, employing Hölder’s inequality and the power-mean inequality, we present some refinements of the H–H (Hermite–Hadamard) inequality and Ostrowski inequalities. Finally, we investigate some applications to special means involving the established results. These new results yield us some generalizations of the prior results in the literature. We believe that the methodology and concept examined in this paper will further inspire interested researchers.

Integral inequalities make up a comprehensive and prolific field of research within the field of mathematical interpretations. Integral inequalities in association with convexity have a strong relationship with symmetry. Different disciplines of mathematics and applied sciences have taken a new path as a result of the development of new fractional operators. Different new fractional operators have been used to improve some mathematical inequalities and to bring new ideas in recent years. To take steps forward, we prove various Grüss-type and Chebyshev-type inequalities for integrable functions in the frame of non-conformable fractional integral operators. The key results are proven using definitions of the fractional integrals, well-known classical inequalities, and classical relations.

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity.

The eigenvalue problems arise in a variety of Science and Engineering applications. This paper is significantly concerned with the students’ academic performance using Eigen values and Eigen vectors of covariant matrix .A case study of two departments i.e Electronics and Electrical Engineering for 13 and 14 batches is taken into consideration. It aims at analyzing performance of students on the basis of their results in various subjects. Covariance matrix has been developed from the data and the sample size of each batch has been taken 100 for the better testing hypothesis. From the results it has been analyzed that the performance of 14electrical engineering students is better in the subject of Applied Calculus whereas poor performance of 13 Electrical students in the subject of applied physics has been noticed.

This study has been discussed a trapezoidal iterated method for estimating a single root of nonlinear problems, which arises in many scientific and engineering fields. The proposed iterative method has second order of convergence, and derived from trapezoidal rule. Several numerical examples to illustrate the efficiency of proposed method by Comparison with the famous second order convergence Newton Raphson Method. C++ and EXCELL are used to analyze the results and graphical representations of proposed method. Henceforth, it has been observed from the outcomes and assessment of trapezoidal iterative method with the Newton Raphson Method that the trapezoidal second order method is performance healthier than existing method.

This study develops MHGM (1,1) (Modified Hybrid Grey Model) which is the combination of two models first one is improved GM (1,1), this model consists of optimization of initial and background values and other is concave EDDGM (1,1) (Dynamic Discrete Grey Model) termed, in this model equal division technique is applied to fit the concavity of cumulative sequence and after that created dynamic average value and on the basis of that dynamic average value dynamic discrete GM (1,1) model is established and by the gradual heuristics method or the dichotomy approach the initial equal division number is obtained. We have fixed equal division number ‘n’ between 0 and 1in MHGM (1,1). For forecasting of starting half years we use y(0)(m) as initial condition of model in time restored function and also multiply by a factor e-b 1 to adjust the model. This model has applied without solving by heuristics or dichotomy method. Subscribers of cellular networks increase day by day in Pakistan; cellular industry has total five networks in Pakistan. In this paper data of three cellular networks subscribers that are Mobilink, Ufone and Zong have taken as application of models and it has been proved by using mean absolute percentage error that the forecast accuracy of MHGM (1,1) is better than GM (1,1) (Grey Model) and improved grey model (1,1).

In this paper Modified Heun’s algorithm of Heun’s algorithm is presented with different formulations which are applied on exponential Population growth rate problems. In Heun’s algorithm the average of two formulations is considered as A.M mean, where as in Modified Heun’s algorithm G.M and Modified Heun’s algorithm H.M are also considered as averages which are also applied on exponential population growth rate problems respectively. Comparison between numerical results of both Modified Heun’s algorithm and existing Heun’s algorithm shows that Modified Heun’s algorithm of Heun’s algorithm is more convergent then Heun’s algorithm. Both algorithms will be analyzed by different errors for the convergent purpose.

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function and to present inequalities of the Hermite–Hadamard type in the setting of the generalized s-type m–preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite–Hadamard-type inequality using Hölder’s integral inequality, the improved power-mean integral inequality, and the Hölder-İşcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory.

There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a new nonlinear three-step method with tenth-order convergence while using six functional evaluations (three functions and three first-order derivatives) per iteration. The method has an efficiency index of about 1.4678, which is higher than most optimal methods. Convergence analysis for single and systems of nonlinear equations is also carried out. The same is verified with the approximated computational order of convergence in the absence of an exact solution. To observe the global fractal behavior of the proposed method, different types of complex functions are considered under basins of attraction. When compared with various well-known methods, it is observed that the proposed method achieves prespecified tolerance in the minimum number of iterations while assuming different initial guesses. Nonlinear models include those employed in science and engineering, including chemical, electrical, biochemical, geometrical, and meteorological models.

This work is related to establish the exact solutions of sine hyperbolic and cosine hyperbolic oscillations of Maxwell fluid over the velocity field and shear stress. Under the effects of sine hyperbolic and cosine hyperbolic oscillations, the general solutions are derived for the motions of incompressible Maxwell fluid. For the sack of the general solutions the mathematical techniques of integral transformations (Laplace and Fourier Sine transforms) are applied. We have expressed the obtained solutions under form of theorem of convolutions product and integral notation, satisfying the boundary and initial conditions. The expressions for similar solutions are specialized as a limiting case of Newtonian fluid.

Many encoding schemes are used in OCDMA (Optical Code Division Multiple Access Network) but SAC (Spectral Amplitude Codes) is widely used. It is considered an effective arrangement to eliminate dominant noise called MAI (Multi Access Interference). Various codes are studied for evaluation with respect to their performance against three noises namely shot noise, thermal noise and PIIN (Phase Induced Intensity Noise). Various Mathematical models for SNR (Signal to Noise Ratios) and BER (Bit Error Rates) are discussed where the SNRs are calculated and BERs are computed using Gaussian distribution assumption. After analyzing the results mathematically, it is concluded that ZCC (Zero Cross Correlation Code) performs better than the other selected SAC codes and can serve larger number of active users than the other codes do. At various receiver power levels, analysis points out that RDC (Random Diagonal Code) also performs better than the other codes. For the power interval between -10 and -20 dBm performance of RDC is better ZCC. Their lowest BER values suggest that these codes should be part of an efficient and cost effective OCDM access network in the future.