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This paper proposes an approach based on the collocation technique with fifth-degree B-splines to numerically solve higher-order Korteweg–de-Vries (KdV) equations. The discretization of the temporal part is performed via a finite difference scheme and the quintic B-spline approximates the function and its derivatives. Consequently, the equation leads to an algebraic system being solved using matrices. The stability of the proposed scheme is also discussed and is found to be unconditionally stable. The solved test problems exhibit the accuracy and stability of the technique. The error is computed using the L∞ norm and the L2 norm. When compared to previous results, the observations suggest that the technique is better than many other methods present in the literature.

Entropy is a scientific term that finds applications in various domains, such as the laws of thermodynamics, where it was initially discovered, as well as statistical physics and information theory. We used unified hybrid censored data to investigate some inverse Weibull distribution entropy metrics. Entropy is defined using three measures: Rényi, Shannon, and Tsallis entropy. The classical estimates of the entropy measures were developed using the unified hybrid censored data, which included both point and approximation confidence intervals. The Bayesian method utilized the Markov Chain Monte Carlo sampling technique to develop Bayesian estimations. This was done by employing two loss functions, namely squared error and general entropy loss functions. Additionally, we delved into the investigation of Bayes credible intervals. Monte Carlo simulations were applied to explain how the estimates functioned at different sample sizes and censoring strategies via some accuracy criteria. Several observations were made in light of the simulation results. To provide a clear explanation of the offered methodologies, two applications using mechanical and cancer data sets were investigated.

The primary objective of the present study was to investigate the enhancement of heat transfer in a Jeffery–Hamel hybrid nanofluid through a porous medium, within stretching/shrinking and convergent/divergent channels. The Darcy–Forchheimer (DF) law was employed to model the flow and thermal behavior of the nanofluid. The governing system of equations was derived using appropriate transformations. Numerical computations were performed using the NDSolve method in Mathematica-11. Results are presented through numerical data and graphical representations, illustrating the effects of various physical parameters on the flow profiles. Key findings indicate that increasing the inertia coefficient and nanoparticle volume fraction accelerates the velocity of the nanofluid in both divergent and convergent channels. Furthermore, higher porosity and inertia coefficients lead to increased drag forces exerted by the channel. Jeffery–Hamel hybrid nanofluids are significantly enhanced by increasing nanoparticle volume fraction, inertia coefficient, porosity, and the presence of radiation and heat source parameters, with a notably higher rate observed in the case of an expanding channel compared to a contracting one.

A system's reliability is defined as the likelihood that its strength surpasses its stress, referred to as the stress–strength index. In this work, we introduce a new stress–strength model based on the inverted Chen distribution. By analyzing the failure times of organic white light-emitting diodes and pump motors, we focus on the inferences of the stress–strength index $ \mathfrak{R} = P(Y < X) $, where: (1) the strength $ (X) $ and stress $ (Y) $ are independent random variables following inverted Chen distributions, and (2) the data are acquired using the adaptive progressive type-Ⅱ censoring plan. The inferences are based on two estimation approaches: maximum likelihood and Bayesian. The Bayes estimates are obtained with the Markov Chain Monte Carlo sampling process leveraging the squared error and LINEX loss functions. Furthermore, two approximate confidence intervals and two credible intervals are developed. A simulation study is done to examine the various estimations presented in this work. To assess the effectiveness of different point and interval estimates, some precision metrics are applied, especially root mean square error, interval length, and coverage probability. Finally, two practical problems are examined to demonstrate the significance and applicability of the given estimation approaches. The analysis demonstrates the suitability of the proposed model for examining engineering data and highlights the superiority of the Bayesian estimation approach in estimating the unknown parameters.

Function spaces are significant in the study and application of mathematical inequalities. The objective of this article is to develop several new bounds and refinements for well-known inequalities that involve Hilbert spaces within a tensorial framework. Using self-adjoint operators in tensor Hilbert spaces, we developed Simpson type inequalities by using different types of generalized convex mappings. Our next step involved developing a variety of new variations of the Hermite and Hadamard inequalities using convex mappings with some special means, specifically arithmetic and geometric means. Furthermore, we developed a number of new fractional identities, which are used in our main findings, by using Riemann-Liouville integrals. In addition, we discuss some examples involving log convex functions and their consequences.

One of the most famous classes of differential equations is the pantograph equation, which is a unique kind of functional differential equation with proportional delay. The pantograph equation which can be addressed both numerically and analytically owing to their different practical applications for modeling natural systems and nonlinearity. In linear state equations with time-varying behavior, the relation of the state variables to input signal may vary over time, and delay differential equations are those where a quantity’s rate of change is dependent upon its value at a prior time point. Accordingly, a pantograph-type fractional order differential model with input delay is formulated and investigated in this study where the system’s solution is needed to examine the intended outcome, which is posed as a fixed-point problem employing the Mittag–Leffler function and the Laplace transform. The existence of solution and other dynamical aspects of implicit differential equations have also been conducted in-depth. The underlying model’s controllability is subsequently assessed while taking into account a number of auxiliary conditions on the independent variable and the nonlinear function under consideration. Using various fixed-point theorems, the necessary, as well as sufficient requirements for the newly developed nonlinear fractional-order pantograph-type differential system equipped with input delay, have been investigated. An example is further provided to establish authenticity and validation as an integral feature of different real life related processes where more exact control, precise predictions and robust performance are of pivotal significance in the mathematical modeling of nonlinear problems in natural systems.

This article introduces the concept of m-polar picture fuzzy sets to address the inherent ambiguity in sustainability-related decision-making in Industry 4.0. We present a set of novel aggregation operators specifically developed for m-PPFSs, including the m-polar picture fuzzy weighted averaging operator, which assigns different weights to each element, and the ordered weighted averaging operator, which aggregates elements based on their ranked values and assigned weights. Additionally, the weighted geometric operator applies a geometric mean approach with weighted inputs, and the ordered weighted geometric operator combines the properties of ordered values and geometric aggregation. These operators provide versatile tools for handling m-PPFS data in various decision-making and computational contexts, playing a critical role in enhancing the multi-criteria decision-making process. Our framework further enables the application of blockchain technology for tracking raw material provenance, optimizing resource utilization, automating waste recycling through smart contracts, and supporting circular supply chain practices. In addition, Industry 4.0 technologies such as the Internet of Things and advanced analytics enable real-time monitoring and predictive maintenance, strengthening sustainability efforts. To evaluate the performance of the proposed aggregation operators, we compare them with existing methods in fuzzy set theory. The results demonstrate that our approach offers superior precision, robustness, and adaptability, outperforming traditional methods in addressing sustainability challenges within the manufacturing sector. In this study, we proposed novel aggregation operators that significantly improve efficiency, accuracy, and flexibility over existing methods, offering robust solutions to sustainability-related decision-making challenges.

The stress-strength index measures the likelihood that a system's strength exceeds its stress. This study focuses on deducting the stress-strength index, denoted as $ \mathfrak{R} = P(Y < X) $, where the strength $ (X) $ and stress $ (Y) $ are independent random variables following new extended xgamma distributions. Inferences are made based on progressively first-failure censored samples. Both maximum likelihood and Bayesian estimation approaches, including point and interval estimations, are considered. The estimations take into account the model parameters as well as the reliability index. The Bayes estimates are obtained using the Markov chain Monte Carlo sampling procedure with the squared error loss function. Additionally, the approximate confidence intervals and Bayes credible intervals are developed. A simulation experiment is conducted to assess the different estimates presented in this paper. Precision metrics such as root mean square error, mean relative absolute bias, and interval length are used to evaluate the efficiency of various point and interval estimates. Two insulating fluid data sets are analyzed to demonstrate the relevance and applicability of the proposed estimation methods.

In this note, we introduce the concept of ℏ-Godunova–Levin interval-valued preinvex functions. As a result of these novel notions, we have developed several variants of Hermite–Hadamard and Fejér-type inequalities under inclusion order relations. Furthermore, we demonstrate through suitable substitutions that this type of convexity unifies a variety of other convex classes. To further support our main findings, we formulated several remarks and nontrivial examples.

This research article investigate the dynamics of fluid flow in non-parallel stretchable walls which focusing on the Jeffery-Hamel (JH) flow. This study analyzed activation and dissipation energies, investigating how these factors interact in the framework of differential type fluid flow. The findings provide insights into the behavior of fluid in complex geometries, with potential applications in microfluidics, biomechanics, and industrial processes. The main equations are non-dimensionalized through valid parameters and used NDSolve software numerically. Magnetic field increases as well as skin friction and Nusselt also increases.

This study examined the estimations of Weibull distribution using progressively first-failure censored data, under the assumption that removals follow the beta-binomial distribution. Classical and Bayesian approaches for estimating unknown model parameters have been established. The estimations included scale and shape parameters, reliability and failure rate metrics as well as beta-binomial parameters. Estimations were considered from both point and interval viewpoints. The Bayes estimates were developed by using the squared error loss and generating samples for the posterior distribution through the Markov Chain Monte Carlo technique. Two interval estimation approaches are considered: approximate confidence intervals based on asymptotic normality of likelihood estimates and Bayes credible intervals. To investigate the performance of classical and Bayesian estimations, a simulation study was considered by various kinds of experimental settings. Furthermore, two examples related to real datasets were thoroughly investigated to verify the practical importance of the suggested methodologies.

This paper investigates the effect of nonlinear thermal radiation on SWCNT-TiO2 and MWCNT-CoFe2O4 nanoparticles suspended in a water-based hybrid nanofluid, flowing past rotating disks. The study employs the Cattaneo-Christov heat flux model to capture the influence of non-Fourier heat conduction. The rotational motion of the disks generates the fluid flow, and the governing partial differential equations are transformed into dimensionless forms using similarity variables. These equations are then solved using a New Iterative Technique (NIT) in Mathematica, which is known for its rapid convergence and accuracy. The analysis focuses on the behavior of various parameters, including velocity components (û, vˆ, ŵ), temperature (Tˆ), and thermal conductivity (kˆ), under different heat transfer conditions. Graphical representations illustrate the effects of these parameters, providing insights into the thermal and fluid dynamic performance of the hybrid nanofluid. The study demonstrates that the NIT is highly effective for solving complex fluid dynamics problems, offering precise and swift solutions. NIM provide an efficient and accurate solution for complex nonlinear problems, overcoming the limitations of traditional methods. This approach enhances computational efficiency and solution accuracy in modeling hybrid nanofluid behavior. This research contributes to the understanding of hybrid nanofluids in engineering applications, particularly in optimizing heat transfer in systems involving rotating machinery.

Nanoparticle concentrations have recently attracted attention as a means to increase the effectiveness of parabolic trough solar collectors (PTSC). Computational fluid dynamics (CFD) study has recently gained popularity for evaluating the performance of PTSC with Dimpled Tubes using a nanofluid made of TiO2 nanoparticles distributed in deionized water (DI-H2O). The effect of using Dimpled Tubes with varied turbulent flow rates, from 0.5 kg/minto 2.5 kg/min, is studied via a battery of tests and computational fluid dynamics simulations. TiO2nanofluid performance is compared to water, the “base fluid,” in this study.This study investigates the utilization of TiO2 nanofluids in enhancing the heat transfer performance of parabolic trough solar collector (PTSC) technology, particularly when integrated with dimpled tubes.The importance of this research lies in the quest for improving the efficiency of solar thermal systems, which is crucial for sustainable energy production.Variables such as solar collector efficiency, Reynolds number, and Nusselt number are tracked as the inquiry progresses. When TiO2 nanofluids are added to the base fluid within the dimpled tube, the convective heat transfer coefficient increases by an astonishing 34.25 percent.This improvement is most pronounced at a mass flow rate of 2.5 kg/min and a volume concentration of TiO2 nanofluid of 0.3 %. Parabolic trough solar collectors have recently been the focus of attempts to improve efficiency via nanoparticle concentrations. The paper uses Computational Fluid Dynamics analysis to investigate how well Dimpled Tubes work with TiO2/DI-H2O nanofluid. The significant increase in the convective heat transfer coefficient was clarified by experimental and CFD calculations that considered fluctuations in turbulent flow rates. The improvement is particularly striking at a mass flow rate and a volume concentration of TiO2 nanofluid.Findingsindicate a significant improvement in heat transfer rates when TiO2 nanofluids are used, particularly in conjunction with dimpled tubes. The combined effect leads to a notable thermal efficiency enhancement, representing the potential of nanofluid-based improvements in optimizing PTSC performance.

Fractional calculus and convex inequalities combine to form a comprehensive mathematical framework for understanding and analyzing a variety of problems. This note develops Hermite-Hadamard, Fejér, and Pachpatte type integral inequalities within pseudo left-right order relations by applying fractional operators with non-singular kernels. Recently, results have been developed using classical Riemann integral operators in addition to various other partial order relations that have some defects that we explained in literature in order to demonstrate the unique characteristics of pseudo order relations. To verify the developed results, we constructed several interesting examples and provided a number of remarks that demonstrate that this type of fractional operator generalizes several previously published results when different things are set up. This work can lead to improvements in mathematical theory, computational methods, and applications across a wide range of disciplines.

The present paper deals with the hybrid nano fluid, for this we have used Single and Multi-wall carbon nano tubes along with CuO and Water. The inclined magneto-hydrodynamics is considered along with Hall effect, soret and dufour for the impact on Williamson fluid flowing through porous medium over stretching/shrinking lamina having suction and injection in consideration. Heat and mass convective boundaries are taken to solve the nonlinear problem using shooting method and bvp4c MATLAB solver. The study shows the comparison between nanofluid (Cuo-Water, SWCNT-Water and MWCNT-Water) and found that the hybrid carbon nanotubes possess excellent result in skin friction and local Nusselt number compared to the SWCNT-water and MWCNT-water.Then the effect of different parameters are analysed through graphical representation and data tables.

Delay partial differential equations have significant applications in numerous fields, such as population dynamics, control systems, neuroscience, and epidemiology, where they are required to efficiently model the effects of past states on current system behavior. This work presents an RBF-based localized meshless method for the numerical solution of delay partial differential equations. In the suggested numerical scheme, the localized meshless method is combined with the Laplace transform. The main attractive features of the localized meshless method are its simplicity, adaptability, and ease of implementation for complex problems defined on complex shaped domains. In a localized meshless scheme, a linear system of equations is solved. The Laplace transform, which is one of the most powerful techniques for solving integer- and non-integer-order problems, is used to represent the desired solution as a contour integral in the complex plane, known as the Bromwich integral. However, the analytic inversion of contour integral becomes very laborious in many situations. Therefore, a contour integration method is utilized to numerically approximate the Bromwich integral. The aim of utilizing the Laplace transform is to handle the costly convolution integral associated with the Caputo derivative and to avoid the effects of time-stepping techniques on the stability and accuracy of the numerical solution. We also discuss the convergence and stability of the suggested scheme. Furthermore, the existence and uniqueness of the solution for the considered model are studied. The efficiency, efficacy, and accuracy of the proposed numerical scheme have been demonstrated through numerical experiments on various problems.

Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we achieve significant results for these equations within the context of Caputo–Katugampola derivatives. First, we establish the existence and uniqueness of solutions by employing the contraction mapping principle with a suitably weighted norm and demonstrate that the solutions continuously depend on both the initial values and the fractional exponent. The second part examines the regularity concerning time. Third, we illustrate the results of the averaging principle using techniques involving inequalities and interval translations. We generalize these results in two ways: first, by establishing them in the sense of the Caputo–Katugampola derivative. Applying condition β=1, we derive the results within the framework of the Caputo derivative, while condition β→0+ yields them in the context of the Caputo–Hadamard derivative. Second, we establish them in Lp space, thereby generalizing the case for p=2.

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing ε in front of the drift term and ε in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in Lp space, generalizing the case of p=2. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives.

This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation features a unique time–space coupled derivative while simultaneously incorporating both wave-diffusion and subdiffusion terms. Our proposed method follows three basic steps: (i) The given equation is transformed into a time-independent form using the Laplace transform (LT); (ii) the LMM is then used to solve the transformed equation in the LT domain; (iii) finally, the time domain solution is obtained by inverting the LT. We use the improved Talbot method and the Stehfest method to invert the LT. The LMM is used to circumvent the shape parameter sensitivity and ill-conditioning of interpolation matrices that commonly arise in global mesh-free methods. Traditional time-stepping methods achieve accuracy only with very small time steps, significantly increasing the computational time. To overcome these shortcomings, the LT is used to provide a more powerful alternative by removing the need for fine temporal discretization. Additionally, the Ulam–Hyers stability of the considered model is analyzed. Four numerical examples are presented to illustrate the effectiveness and practical applicability of the method.

The treatment adherence level of tuberculosis patients is critically related to their awareness level about tuberculosis and anti-tuberculosis therapy. Tuberculosis is always spreading if the community has low level of awareness. In this article a mathematical model describing the impact of treatment adherence and difference in awareness level on tuberculosis transmission dynamics is investigated. It is assumed that the detected infectious individuals with low level of awareness after treatment may enter either the recovered compartment due to treatment adherence or the latent compartment due to lack of adherence. We compute the epidemic threshold parameter, Rb, to study its role on the dynamics and stability of the proposed model. Mathematical analyses of this model reveal that treatment nonadherence of individuals with active tuberculosis may lead to relapse. Analysis of the epidemic threshold parameter shows that increase in awareness level, treatment and adherence to medication has a positive effect on tuberculosis control. Whereas, decrease in effective contact rate and endogenous reactivation rate help to reduce tuberculosis infection. The numerical simulations of the model illustrate the feasibility of our analytical conclusions.

This paper extends the cholera human-to-human direct transmission model from a deterministic to a stochastic framework. This is expressed as mixed system of stochastic and deterministic differential equations. A Lyapunov function is created to investigate the global stability of the stochastic cholera epidemic, which shows the existence of global positivity of the solution using the theory of stopping time. We then find the threshold quantity of the extended stochastic cholera epidemic model. We derive a parametric condition $ \widetilde{R}_0 $, and for additive white noise, we establish sufficient conditions for the extinction and the persistence of the cholera infection. Finally, for a suitable choice of the parameter of the system for $ \widetilde{R}_0 $, we perform numerical simulations for both scenarios of extinction and persistence of the dynamic of the cholera infection.

The hierarchical medical care structure is an efficient means of addressing some inadequate medical capabilities. Evaluating the several degrees of disease based on the physician’s diagnosis is a vital step in building a hierarchical healthcare treatment strategy. This study aims to offer different dynamic T-spherical fuzzy aggregation operators (AOs) for addressing multi-period decision-making (MPDM) situations in which decision-makers (DMs) have access to all information in the form of T-spherical fuzzy numbers (T-SFNs) spanning many time periods. AOs play a crucial role in decision-making, particularly when competing considerations are present. Developing suitable solutions for the aggregation process is a crucial challenge when dealing with unclear data. ”T-spherical fuzzy dynamic Einstein weighted averaging (T-SFDEWA) operator and the T-spherical fuzzy dynamic Einstein weighted geometric (T-SFDEWG) operator” are two innovative Einstein AOs. Several appealing properties of these AOs are addressed in depth. In addition, we present a strategy for addressing MPDM difficulties using optimum solutions. A numerical illustration of a hierarchical medical care system is provided to illustrate how the suggested method could be implemented. An authenticity assessment and comparative analysis is also provided to illustrate the effectiveness of the suggested method. In practice, the proposed AOs and decision-making technique are typically useful in multi-stage and dynamic decision analysis.

The fundamental cause of coronary heart disease is thought to be atherosclerosis. Blood flow in arteries under plaque formation is widely studied using both experimental and theoretical methods in literature. This paper presents a comprehensive study on the mathematical modelling and numerical simulation of non-Newtonian blood flow in an idealized stenosed artery. Non-Newtonian behavior of blood flow is assumed by Casson and Quemada fluid models. An analysis is performed for four forms of stenosis, 50%, 60%, 70%, and 80%, which are defined by a narrowing of the artery's diameter. The mathematical model for non-Newtonian fluid transport is presented in the form of highly nonlinear coupled partial differential equations. A finite-volume approach is used to solve the system of equations numerically. The solutions are obtained using a hybrid mesh with hexahedron and tetrahedron control volumes. Numerical solutions are validated first with experimental results from the literature for a Newtonian fluid, and then the results are extended to non-Newtonian fluid models. It can be observed that the non-Newtonian nature of the fluid model influences flow dynamics and is a significant aspect to consider for arteries with small diameter. The velocity and pressure drop rise as the degree of stenosis develops. Wall shear is substantially larger in stenotic passages and severity of stenosis is directly related with this increasing behavior.

Multiple sclerosis (MS) is a common neurological disorder that affects the central nervous system (CNS) and can cause lesions that spread over space and time. Our study proposes a mathematical model that illustrates the progression of the disease and its likelihood of recurrence. We use Caputo fractional-order (FO) derivative operators to represent non-negative solutions and to establish a steady-state point and basic reproductive number. We also employ functional analysis to prove the existence of unique solutions and use the Ulam-Hyres (UH) notion to demonstrate the stability of the solution for the proposed model. Furthermore, we conduct numerical simulations using an Euler-type numerical technique to validate our theoretical results. Our findings are presented through graphs that depict various behaviors of the model for different parameter values.

This note introduces the concept of $ (h_1, h_2) $-convex stochastic processes using interval-valued functions. First we develop Hermite-Hadmard $ (\mathbb{H.H}) $ type inequalities, then we check the results for the product of two convex stochastic process mappings, and finally we develop Ostrowski and Jensen type inequalities for $ (h_1, h_2) $-convex stochastic process. Also, we have shown that this is a more generalized and larger class of convex stochastic processes with some remark. Furthermore, we validate our main findings by providing some non-trivial examples.

The theory of inequalities is greatly influenced by interval-valued concepts, and this contribution is explored from several perspectives and domains. The aim of this note is to develop several mathematical inequalities such as Hermite–Hadamard, Fejér, and the product version based on center radius CR-order relations. Furthermore, we develop several nontrivial examples and remarks to support the main findings.

In this article, the meshfree radial basis function method based on the Gaussian function is proposed for some time-space dependent fractional order partial differential equation (PDE) models. These PDE models have significant applications in chemical engineering and physical science. Some main advantages of the proposed method are that it is easy to implement, and the output response is quick and highly accurate, especially in the higher dimension. In this method, the time-dependent derivative terms are treated by Caputo fractional derivative while space-dependent derivative terms are treated by Riesz, Riemann-Liouville, and Grünwald-Letnikov derivatives. The proposed method is tested on some numerical examples and the accuracy is analyzed by $ \|L\|_\infty $.

A cubic bipolar fuzzy set (CBFS) is by far the most efficient model for handling bipolar fuzziness because it carries both single-valued (SV) and interval-valued (Ⅳ) bipolar fuzzy numbers at the same time. The sine trigonometric function possesses two consequential qualities, namely, periodicity and symmetry, both of which are helpful tools for matching decision makers' conjectures. This article aims to integrate the sine function and cubic bipolar fuzzy data. As a result, sine trigonometric operational laws (STOLs) for cubic bipolar fuzzy numbers (CBFNs) are defined in this article. Premised on these laws, a substantial range of aggregation operators (AOs) are introduced. Certain features of these operators, including monotonicity, idempotency, and boundedness, are explored as well. Using the proffered AOs, a novel multi-criteria group decision-making (MCGDM) strategy is developed. An extensive case study of carbon capture and storage (CCS) technology has been provided to show the viability of the suggested method. A numerical example is provided to manifest the feasibility of the developed approach. Finally, a comparison study is executed to discuss the efficacy of the novel MCGDM framework.

The non-linear time-fractional Korteweg-de Vries and modified Korteweg-de Vries equations are studied with Caputo's fractional derivative. The general higher-order solitary wave solutions are derived using a novel technique called the Aboodh transform decomposition method. To validate the obtained results, two examples of Caputo's fractional derivative with appropriate subsidiary conditions are illustrated. The accuracy and efficiency are confirmed by using numerical simulations and error analysis, where good agreements are obtained. The numerical analysis shows that, in comparison to the time-fractional Korteweg-de Vries solution, the solitary wave solution for the time-fractional modified Korteweg-de Vries equation is less stable against the oscillations. The variations in the temporal variable t enhance the strength of the wave solutions. Moreover, the wave perturbations taper off as t attains large values. The parameter α signifies the fractional derivative influence on the wave dispersion and nonlinearity effects. This affects the amplitude as well as the spatial extension of the solitary waves. With a relatively small value of t, the obtained solutions admit pulse-shaped solitons. Moreover, the wave's solutions suffer from oscillations when the temporal variable attains large values. This effect cannot be noticed in the soliton solutions obtained in the integer order systems.

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems.

Piecewise fractional-order differential operators have received more attention in recent years because they can be used to describe various evolutionary dynamical problems to investigate crossover behaviors. In this manuscript, we use the aforementioned operators to investigate a mathematical model of COVID-19. By utilizing fractional calculus, our approach aims to capture the crossover dynamics of disease spread, considering heterogeneity and transitions between epidemic phases. This research seeks to develop a framework using specialized mathematical techniques, such as the Caputo fractional derivative, with the potential to investigate the crossover dynamical behaviors of the considered epidemic model. The anticipated contribution lies in bridging fractional calculus and epidemiology, offering insights for both theoretical advancements and practical public health interventions. In order to improve our understanding of epidemic dynamics and support, we used MATLAB to simulate numerical results for a visual representation of our findings. For this interpretation, we used various fractional-order values. In addition, we also compare our simulated results with some reported results for infected and death classes to demonstrate the efficiency of our numerical method.

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p=2 to results in the Lp space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish.

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively.

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in Lp space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the Ψ-Caputo derivative. We generalize the two situations of p=2 and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end.

The efficiency, temperature distribution, and temperature at the tip of straight rectangular, growing and decaying moving exponential fins are investigated in this article. The influence of internal heat generation, surface and surrounding temperatures, convection-conduction, Peclet number and radiation-conduction is studied numerically on the efficiency, temperature profile, and temperature at the tip of the fin. Differential transform method is used to investigate the problem. It is revealed that thermal and thermo-geometric characteristics have a significant impact on the performance, temperature distribution, and temperature of the fin's tip.The results show that the temperature distribution of decaying exponential and rectangular fins is approximately 15 and 7% higher than growing exponential and rectangular fins respectively. It is estimated that the temperature distribution of the fin increases by approximately 6% when the porosity parameter is increased from 0.1 to 0.5. It is also observed that the decay exponential fin has better efficiency compared to growing exponential fin which offers significant advantages in mechanical engineering.

In the current manuscript, we deal with the dynamics of a piecewise covid-19 mathematical model with quarantine class and vaccination using SEIQR epidemic model. For this, we discussed the deterministic, stochastic, and fractional forms of the proposed model for different steps. It has a great impact on the infectious disease models and especially for covid-19 because in start the deterministic model played its role but with time due to uncertainty the stochastic model takes place and with long term expansion the use of fractional derivatives are required. The stability of the model is discussed regarding the reproductive number. Using the non-standard finite difference scheme for the numerical solution of the deterministic model and illustrate the obtained results graphically. Further, environmental noises are added to the model for the description of the stochastic model. Then take out the existence and uniqueness of positive solution with extinction for infection. Finally, we utilize a new technique of piecewise differential and integral operators for approximating Caputo-Fabrizio fractional derivative operator for the purpose of constructing of the fractional-order model. Then study the dynamics of the models such as positivity and boundedness of the solutions and local stability analysis. Solved numerically fractional-order model used Newton Polynomial scheme and present the results graphically.

The current paper deals with the transmission of MERS-CoV model between the humans populace and the camels, which are suspected to be the primary source for the infection. The effect of time MERS-CoV disease transmission is explored using a non-linear fractional order in the sense of Caputo operator in this paper. The considered model is analyzed for the qualitative theory, uniqueness of the solution are discussed by using the Banach contraction principle. Stability analysis is investigated by the aid of Ulam-Hyres (UH) and its generalized version. Finally, we show the numerical results with the help of generalized Adams-Bashforth-Moulton Method (GABMM) are used for the proposed model, for supporting our analytical work.

In this article, we propose some novel results on the existence and uniqueness of generalized Caputo-type initial value problems with delay by using fixed point theory. The characteristics of space of continuous and measurable functions are the main basis of our results. The proposed results are very useful to prove the existence of a unique solution for the various types of fractional-order systems defined under the generalized Caputo fractional derivative consisting of delay terms.

In this article, we study moving longitudinal and exponential fins with variable thermal conductivity, thermal emissivity, and heat transfer coefficient. The problem is solved numerically using shooting technique. To compare the performance of trapezoidal fins with exponential fins, we introduce a novel idea of volume adjusted fin heat transfer rate, fin effectiveness, and fin efficiency. The result shows that heat transfer from the exponential fin is higher than from the trapezoidal fin. It is observed that the capability of an exponential fin is higher than the trapezoidal fin. With increasing thermal conductivity and radiation-conduction number as well as convection-conduction number, the fin’s efficiency increases. The results illustrate the effect of heat transfer coefficient, thermal emissivity, radiation-conduction and convection-conduction parameters, dimensionless radiation, and convection sink temperatures on the fin’s performance.

Thin film flows over porous and moving sheets of variable thickness have been reconsidered here. New and multiple dimensions of classical problems of such type flow are analyzed. Here, we categorically emphasized on the nature and kinds of injection (suction) and moving velocities of the sheet, whereas, variable size of the sheet is also taken into account. We formed and investigated different cases and checked different options for kinematics of sheet, variable sizes of thin film and that of sheet. All possible cases of exact similarities are noticed by which the system of partial differential equations and boundary conditions are exactly transformed into ordinary differential equations. The final systems of exact equations are solved by bvp4c technique. The present simulations are exactly matched with the previously published analyses for special choices of functions and parameters. Strict behavioral changes have been observed in the velocity profiles by changing the nature of sheet’s kinematics.

We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor ω considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.

The purpose of this article is to discuss some midpoint type HHF fractional integral inequalities and related results for a class of fractional operators (weighted fractional operators) that refer to harmonic convex functions with respect to an increasing function that contains a positive weighted symmetric function with respect to the harmonic mean of the endpoints of the interval. It can be concluded from all derived inequalities that our study generalizes a large number of well-known inequalities involving both classical and Riemann-Liouville fractional integral inequalities.

This paper is devoted to present a numerical scheme based on operational matrices to compute approximate solutions to fractional-order boundary value problems. For the mentioned operational matrices, we utilize fractional-order Bernoulli polynomials. Since fractional-order problems are usually difficult to treat for their corresponding analytical or exact solutions, therefore, we need sophisticated methods to find their best numerical solutions. The presented numerical scheme has the ability to reduce the proposed problem to the corresponding algebraic equations. The obtained algebraic equations are then solved by using the computational software MATLAB for the corresponding numerical results. The used method has the ability to save much more time and also is reliable to secure the proper amount of memory. Several examples are solved by using the considered method. Also, the solutions are compared with their exact solution graphically. In addition, the absolute errors for different scale values are presented graphically. In addition, we compare our results with the results of the shifted Legendre polynomials spectral method.

Abstract In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results.

The concept of linear Diophantine fuzzy set (LDFS) theory with its control parameters is a strong model for machine learning and data-driven multi-criteria decision making (MCDM). The sine-trigonometric function (STF) has two significant features, periodicity and symmetry about the origin that are very useful tools for information analysis. Keeping in view the characteristics of both STF and LDFS theory, this article introduces the sine-trigonometric operations for linear Diophantine fuzzy numbers (LDFNs). These operational laws lay a foundation for developing new linear Diophantine fuzzy sine-trigonometric aggregation operators (LDFSTAOs). The integration of Industry 4.0 technology into healthcare has the potential to revolutionize patient care. One of the most challenging tasks is the selection of efficient suppliers for the healthcare supply chain (HSC). The traditional suppliers are not efficient in accordance with Industry 4.0, with particular uncertainties. A new MCDM framework is presented based on LDFSTAOs to examine the HSC performance in industry 4.0. A credibility test, sensitivity analysis and comparative analysis are performed to express the novelty, reliability, and efficiency of the proposed methodology.

Abstract In this paper, we establish some dynamic Hilbert-type inequalities in two independent variables on time scales by using the Fenchel–Legendre transform. We also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as particular cases. Our results give more general forms of several previously established inequalities.

In the current manuscript, we are tying to study one of the important class of differential equations known is evolution equations. Here, we considered the problem under controllability term and with proportional delay. Before going to numerical or analytical solution it is important to check the existence and uniqueness of the solution. So, we will consider our problem for qualitative theory using fixed point theorems of Banach's and Krasnoselskii's type. For numerical solution the stability is important, hence the problem is also studied for Ulam-Hyer's type stability. At the end an example is constructed to ensure the establish results.

Abstract This paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method.