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Abstract The Hermite–Hadamard–Fejér-type inequality is an effective utensil for examining upper and lower estimations of the integrals of convex functions. In this study, the power mean inequality and Hölder inequality are employed. The outcomes are new Hermite–Hadamard–Fejér-type inequalities via Raina fractional integrals for m-convex functions. The findings are extensions of many previous investigations.

In this paper, we present a computational analysis of data-driven solutions of the convection–diffusion–reaction equation using Physics Informed Neural Networks (PINNs). PINNs enforce laws of physics when solving non-linear partial differential equations that govern physical dynamics. The PINN technique for solving boundary value problems in partial differential equations is presented as an alternative to the available numerical techniques. Three model initial-boundary value problems are implemented through MATLAB using the presented technique. The computed numerical solutions of these model problems are compared with the actual solution to observe the accuracy of the numerical implementation. It is noted that the predicted solution in the case of these model problems through PINNs is in strong agreement with the corresponding exact solution. The analysis of the presented algorithm is performed to observe what changes in the accuracy of the solution when the number of neurons and the number of layers working within the neural network structure are altered. Moreover, the impact of the number of training data points and collocation points on the model’s accuracy is also presented to develop a better understanding of the algorithm. It is observed that the presented method is capable of efficiently computing numerical solutions of boundary value problems in partial differential equations and has the potential to solve a large number of related problems that arise in engineering physics.

This research represents a crucial step forward in the design and application of magnetized ternary hybrid nanofluids, paving the way for innovations in thermal management and medical treatment technologies. Optimization analysis of nanoscale heat transport for magnetized ternary hybrid bio nanofluid containing SiO2, TiO2, and Al2O3 nanoparticles over a three-dimensional wedge geometry is made in this article. Heat transport analysis is studied through thermal and joule heating effects and the ternary hybrid composition is chosen to leverage the combined thermal properties of the nanoparticles. Furthermore, detail streamlines of flow pattern are investigated for different physical parameters. Physical system is formulated with the system of Partial differential equation (PDEs) and it is processed with similarity transformation to convert it into system of ordinary differential equation (ODEs). Furthermore, Keller box scheme is applied to fetch its numerical solution and compared with finite element technique for validation and found smooth agreement.Temperature is increased with pressure gradient, shear strain and thermal radiation parameter. Magnitude of drag force is increasing with increasing Weissenberg number. For We = 0.2, 0.3, 0.4, 0.5, 0.6, skin friction increases in ternary nanofluid with 13 %,17 %, 21 %, 23 % and 26 % respectively. For β = 0.3, 0.4, 0.5, 0.6, 0.7, skin friction decreases in ternary nanofluid with 30 %,25 %, 20 %, 15 % and 10 % respectively.

Significance of this study is to explore Arrhenius activation energy, microgravity, chemical reaction and porous medium effects on Darcy nanofluid over a stretching sheet with nonlinear thermal radiations. Purpose of this study is to enhance the lubrication efficiency, excessive heat reduction, surface roughness and tool wear in drilling, milling, grinding and cutting tools of machining procedures. Use of Brownian and thermophoretic nanoparticles in base liquid is very effective to reduce friction in tool-chip and tool-work interfaces, tool life ability and cooling ability through nanofluid minimum quantity lubrication in machining operations. A refined mathematical model is solved using dimensionless variables; oscillatory stokes conditions and primitive variables. The implicit form of finite difference method is applied for numerical results using Gaussian elimination approach. The numerical and physical outputs are secured using Gaussian elimination methodology in FORTRAN tool and TecPlot-360. Graphs are displayed by using numerical data in Tecplot-360 program. Impact of activation energy (AE), solar radiation (Rd), microgravity (RG), porosity (Ω), thermophoresis (NT), chemical reaction (Kr), Prandtl (Pr) and Schmidt factor (Sc) on oscillating frequency of heat and mass transport is depicted. Amplitude in fluid velocity, surface temperature and volume of concentration is enhanced as radiation, microgravity and activation energy increases. It is noted that the increasing amplitude in fluid velocity, temperature and fluid concentration is found with activation energy, radiation and buoyancy force effects. Steady behavior of skin friction and mass transport decreases as Darcy porous medium and reaction rate decreases. Frequency of oscillating skin friction and oscillating heat transfer increases as Schmidt and Prandtl coefficient increases. Fluctuations and amplitude in the frequency of heat and mass transport enhances as thermophoretic nanoparticle enhances.

We consider the positivity of the discrete sequential fractional operators $ \left(^{\rm RL}_{a_{0}+1}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ defined on the set $ \mathscr{D}_{1} $ (see (1.1) and Figure 1) and $ \left(^{\rm RL}_{a_{0}+2}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ of mixed order defined on the set $ \mathscr{D}_{2} $ (see (1.2) and Figure 2) for $ \tau\in\mathbb{N}_{a_{0}} $. By analysing the first sequential operator, we reach that $ \bigl(\nabla {f}\bigr)(\tau)\geqq 0, $ for each $ \tau\in{\mathbb{N}}_{a_{0}+1} $. Besides, we obtain $ \bigl(\nabla {f}\bigr)(3)\geqq 0 $ by analysing the second sequential operator. Furthermore, some conditions to obtain the proposed monotonicity results are summarized. Finally, two practical applications are provided to illustrate the efficiency of the main theorems.

In this study, we consider positivity and other related concepts such as α−convexity and α−monotonicity for discrete fractional operators with exponential kernel. Namely, we consider discrete Δ fractional operators in the Caputo sense and we apply efficient initial conditions to obtain our conclusions. Note positivity results are an important factor for obtaining the composite of double discrete fractional operators having different orders.

The aim of this research is to combine the concept of inequalities with fractional integral operators, which are the focus of attention due to their properties and frequency of usage. By using a novel fractional integral operator that has an exponential function in its kernel, we establish a new Hermite-Hadamard type integral inequality for an LR-convex interval-valued function. We also prove new fractional-order variants of the Fejér type inequalities and the Pachpatte type inequalities in the setting of pseudo-order relations. By showing several numerical examples, we further validate the accuracy of the results that we have derived in this study. We believe that the results, presented in this article are novel and that they will be beneficial in encouraging future research in this field.

This work deals with the construction and analysis of convexity and nabla positivity for discrete fractional models that includes singular (exponential) kernel. The discrete fractional differences are considered in the sense of Riemann and Liouville, and the υ1-monotonicity formula is employed as our initial result to obtain the mixed order and composite results. The nabla positivity is discussed in detail for increasing discrete operators. Moreover, two examples with the specific values of the orders and starting points are considered to demonstrate the applicability and accuracy of our main results.

Due to the highly competitive marketing economy, for different kinds of products’ related suppliers confer various incentives to their respective retailers with certain terms and conditions. Even though suppliers may require advance payments before delivering their products, they incentivize the scheme by offering instant price-discounts, maybe with some other additional benefits and we have constructed an inventory-model consisting of two-warehouses mathematically with deteriorating products. In this proposed model, the suppliers offer some price-discounts for advance payments made by their retailers. As advance payments put a constraint on the capital position of the retailers, the retailers meanwhile, enjoy some delay in the final payment of the rest amount which acts as a booster for their business. A partially backlogged shortage is allowed and its rate is considered to be dependent on the duration of waiting time from the replenishment of a lot to the arrival of the next lot. Finally, a numerical example is performed for validation of the suggested model.

Security of personal information has become a major concern due to the increasing use of the Internet by individuals in the digital world. The main purpose here is to prevent an unauthorized person from gaining access to confidential information. The solution to such a problem is by authentication of users. Authentication has a very important role in achieving security. Mutually orthogonal graph squares (MOGS) are considered the generalization of mutually orthogonal Latin squares (MOLS). Also, MOGS are generated from edge decompositions of complete bipartite graphs by isomorphic graphs. Graph-orthogonal arrays can be constructed by MOGS. In this paper, graph-orthogonal arrays are used for constructing authentication codes. These arrays are the encoding matrices of authentication tags. We introduce the concepts and basic theorems of MOGS, graph-orthogonal arrays, and authentication codes. After constructing graph-orthogonal arrays by MOGS, then there is an established mapping between graph-orthogonal arrays and message set. This manages us to construct perfect non-splitting and splitting Cartesian authentication codes. In both cases, we calculate the probabilities of successful impersonation attacks and substitution attacks. Besides that, the performance of constructed non-splitting and splitting authentication codes is analyzed. In the end, optimal authentication codes and secure authentication codes are constructed.

Abstract A number of mathematical methods have been developed to determine the complex rheological behavior of fluid’s models. Such mathematical models are investigated using statistical, empirical, analytical, and iterative (numerical) methods. Due to this fact, this manuscript proposes an analytical analysis and comparison between Sumudu and Laplace transforms for the prediction of unsteady convective flow of magnetized second grade fluid. The mathematical model, say, unsteady convective flow of magnetized second grade fluid, is based on nonfractional approach consisting of ramped conditions. In order to investigate the heat transfer and velocity field profile, we invoked Sumudu and Laplace transforms for finding the hidden aspects of unsteady convective flow of magnetized second grade fluid. For the sake of the comparative analysis, the graphical illustration is depicted that reflects effective results for the first time in the open literature. In short, the obtained profiles of temperature and velocity fields with Laplace and Sumudu transforms are in good agreement on the basis of numerical simulations.

In this study, we introduce and study new fuzzy-interval integral is known as fuzzy-interval double integral, where the integrand is fuzzy-interval-valued functions (FIVFs). Also, some fundamental properties are also investigated. Moreover, we present a new class of convex fuzzy-interval-valued functions is known as coordinated convex fuzzy-interval-valued functions (coordinated convex FIVFs) through fuzzy order relation (FOR). The FOR (≼) and fuzzy inclusion relation (⊇) are two different concepts. With the help of fuzzy-interval double integral and FOR, we have proved that coordinated convex fuzzy-IVF establish a strong relationship between Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) inequalities. With the support of this relation, we also derive some related HH-inequalities for the product of coordinated convex FIVFs. Some special cases are also discussed. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

Among various improvements in fuzzy set theory, a progressive development has been in process to investigate fuzzy analogues of fixed point theorems of the classical fixed point results. In this direction, taking the ideas of θ-contractions as well as Feng-Liu's approach into account, some new fuzzy fixed point results for nonlinear fuzzy set-valued θ-contractions in the framework of metric-like spaces are introduced in this paper without using the usual Pompeiu-Hausorff distance function. Our established concepts complement, unify and generalize a few important fuzzy and classical fixed point theorems in the corresponding literature. A handful of these special cases of our notions are pointed and analyzed. Some of the main results herein are further applied to derive their analogues in metric-like spaces endowed with partial ordering and binary relations. Comparisons and nontrivial examples are given to authenticate the hypotheses and significance of the obtained ideas.

This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional (GPF) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-GPF operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations (FVFIEs) via generalized fuzzy Hilfer-GPF Hukuhara differentiability (HD) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy FVFIEs which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.

The aim of this paper is to initiate a new concept of nonlinear contraction under the name r-hybrid ψ-contraction and establish some fixed point results for such mappings in the setting of complete metric spaces. The presented ideas herein unify and extend a number of well-known results in the corresponding literature. A few of these special cases are pointed out and analysed. From application point of view, we investigate the existence and uniqueness criteria of solutions to certain functional equation arising in dynamic programming and integral equation of Volterra type. Nontrivial illustrative examples are provided to show the generality and validity of our obtained results.

Abstract In this article, we are interested in some well-known dynamic inequalities on time scales. For this reason, we will prove some new Hermite–Hadamard (H-H) and Opial dynamic inequalities on time scales. The main results here will be derived via the dynamic integration by parts and chain rule formulas on time scales. In addition, we will extend and unify the inequalities for the convex functions.

In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function Ψ, we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and Grüss type inequalities with respect to increasing, positive and monotone functions Ψ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on ω and Ψ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

Efficient thermal transportation in compact heat density gadgets is a prevailing issue to be addressed. The flow of a mono nanofluid (SiO2/Kerosene oil) and hybrid nanofluid (TiO2 + SiO2/Kerosene oil) is studied in context of Riga wedge. The basic purpose of this work pertains to improve thermal conductivity of base liquid with inclusions of nano-entities. The hybrid nanofluid flow over Riga wedge is new aspect of this work. The concentration of new species is assumed to constitute the base liquid to be non-Newtonian. The fundamental formulation of the concentration laws of mass, momentum and energy involve partial derivatives. The associated boundary conditions are taken in to account. Similarity variables are utilized to transform the leading set of equations into ordinary differential form. Shooting procedure combined with Runge-Kutta method is harnessed to attain numerical outcomes. The computational process is run in matlab script. It is seen that the velocity component f′(η) goes upward with exceeding inputs of modified Hartmann number Mh and it slows down when non-dimensional material parameter αh takes large values. Also, Nusselt number − θ′(0) is enhanced with developing values of Eckert number Ec and Biot number Bi.

Oncolytic virotherapy is an efficacious chemotherapeutic agent that addresses and eliminates cancerous tissues by employing recombinant infections. M1 is a spontaneously produced oncolytic alphavirus with exceptional specificity and powerful activity in individual malignancies. The objective of this paper is to develop and assess a novel fractional differential equation (FDEs)-based mathematical formalism that captures the mechanisms of oncogenic M1 immunotherapy. The aforesaid framework is demonstrated with the aid of persistence, originality, non-negativity, and stability of systems. Additionally, we also examine all conceivable steady states and the requirements that must exist for them to occur. We also investigate the global stability of these equilibria and the characteristics that induce them to be unstable. Furthermore, the Atangana–Baleanu fractional-order derivative is employed to generalize a treatment of the cancer model. This novel type of derivative furnishes us with vital understanding regarding parameters that are widely used in intricate mechanisms. The Picard–Lindelof approach is implemented to investigate the existence and uniqueness of solutions for the fractional cancer treatment system, and Picard’s stability approach is used to address governing equations. The findings reveal that the system is more accurate when the fractional derivative is implemented, demonstrating that the behaviour of the cancer treatment can be interpreted when non-local phenomena are included in the system. Furthermore, numerical results for various configurations of the system are provided to exemplify the established simulation.

In this paper, we introduced response analysis and controlling the nonlinear vibration of van der pol duffing oscillator subject to parametric and external excitations by connecting the nonlinear integral positive position feedback (NIPPF). The controller combines the characteristics of the positive position feedback controller (PPF) and integral resonant control (IRC) and these controllers are viewed in a blocked scheme for the closed-loop system (system under control). The analytical solution and all resonance cases are obtained by applying the multiple time scales perturbation method (MSPM). The stability analysis is studied using the frequency response equations at the supposed worst resonance case. For numerical simulation, the system is examined before and after connecting the NIPPF controller using Matlab software program (package ode45). The influences of different parameters are examined numerically for the vibrating system and controller. The approximate analysis is confirmed with numerical simulation outcomes. Finally, making a comparison with previously related published work.

Intravenous substance consumption is on the upswing all over the globe, especially in Europe and Asia. It is extremely harmful to society; excessive substance consumption is the leading cause of death. Beyond all prohibited narcotics, heroin is a narcotic that has a substantial negative impact on society and the world at large. In this paper, a heroin epidemic model is developed via an Atangana–Baleanu fractional-order derivative in the Caputo sense describe accurately real world problems, equipped with recovery and persistent immunity. Meanwhile, we have established a globally asymptotically stable equilibrium for both the drug-free and drug-addiction equilibriums. Additionally, we apply a novel scheme that is mingled with the two-step Lagrange polynomial and the basic principle of fractional calculus. The simulation results for various fractional values indicate that as the fractional order decreases from 1, the growth of the epidemic diminishes. The modelling data demonstrates that the suggested containment technique is effective in minimizing the incidence of instances in various categories. Furthermore, modelling the ideal configuration indicated that lowering the fractional-order from 1 necessitates a swift commencement of the implementation of the suggested regulatory technique at the maximum rate and sustaining it throughout a significant proportion of the pandemic time frame.

The Pythagorean fuzzy soft set (PFSS) is the most proficient and manipulative leeway of the Pythagorean fuzzy set (PFS), which contracts with parameterized values of the alternatives. It is a generalized form of the intuitionistic fuzzy soft set (IFSS), which provides healthier and more accurate evaluations through decision-making (DM). The main determination of this research is to prolong the idea of Einstein’s aggregation operators for PFSS. We introduce the Einstein operational laws for Pythagorean fuzzy soft numbers (PFSNs). Based on Einstein operational laws, we construct two novel aggregation operators (AOs) such as Pythagorean fuzzy soft Einstein-weighted averaging (PFSEWA) and Pythagorean fuzzy soft Einstein-weighted geometric (PFSEWG) operators. In addition, important possessions of proposed operators, such as idempotency, boundedness, and homogeneity, are discussed. Furthermore, to validate the practicability of the anticipated operators, a multiple attribute group decision-making (MAGDM) method is developed. We intend innovative AOs considering the Einstein norms for PFSS to elect the most subtle business. Pythagorean fuzzy soft numbers (PFSNs) support us to signify unclear data in real-world perception. Furthermore, a numerical description is planned to certify the efficacy and usability of the projected method in the DM practice. The recent approach’s pragmatism, usefulness, and tractability are validated through comparative exploration with the support of some prevalent studies.

A new extended direct algebraic method is applied to extract new traveling wave solutions for non-linear directional couplers with the optical meta-materials. This model studies light distribution from the main fiber into other branch fibers, which can be one or more than one. So, for this reason, twin couplers and multiple core couplers are studied in this paper. Further two cases in multiple core couplers namely multiple core couplers near the neighbor and multiple core couplers with a complete neighbor are exhaustively discussed. The non-linearity type that is considered here is Kerr Law. The obtained solutions set is more generalized than the existing solution as it is transformed into fractional-order derivative because it gives the solutions of problems with heavy tails or infinity fluctuations. Furthermore, the obtained solutions contain almost all types of solitons such as dark, bright, dark-bright, rational, periodic, breather, and singular soliton solutions, etc. Moreover, the obtained solutions are fractional solitons with Atangana-Baleanu and modified Riemann–Liouville fractional operator. These non-linear directional couplers also act as limiters in intensity-dependent switches. The propagation constants of the modes of a system are changed by non-linearity. In order to discuss the physical nature of couplers, 2D and 3D figures are also shown.

The purpose and novelty of our study include the scrutinization of the unsteady flow and heat characteristics of the unsteady Sutterby nano-fluid flow across an elongated cone using slip boundary conditions. The bioconvection of gyrotactic micro-organisms, Cattaneo–Christov, and thermal radiative fluxes with magnetic fields are significant physical aspects of the study. Anisotropic constraints on the cone surface are taken into account. The leading formulation is transmuted into ordinary differential formate via similarity functions. Five coupled equations with nonlinear terms are resolved numerically through the utilization of a MATLAB code for the Runge–Kutta procedure. The parameters of buoyancy ratio, the porosity of medium, and bioconvection Rayleigh number decrease x-direction velocity. The slip parameter retard y-direction velocity. The temperature for Sutterby fluids is at a hotter level, but its velocity is vividly slower compared to those of nanofluids. The temperature profile improves directly with thermophoresis, v-velocity slip, and random motion of nanoentities.

Recently, cases of influenza virus resistance are being observed. Resistance is deadly and can cause many pandemics in future. That is why, here we investigate the situation on which the strains exist side – by – side and the difference in their mode of transmission. This paper studies two strain (resistance and non - resistance) flu model. The non-resistant strain mutates to give the resistant strain. These strains are differentiated by their incidence rates which are; bilinear and saturated for the non-resistant and saturated resistant strain respectively. This will help in studying the difference in the mode of transmission of the two strains. Equilibrium solutions are computed and Lyapunov functions are used to show their global stability. It is clear from the analysis that disease – free equilibrium is globally asymptotically stable if maxRR,RN1. Any strain with biggest basic reproduction ratio out performs the other. In order to support the analytic results some numerical simulations are carried out.

Positivity analysis is used with some basic conditions to analyse monotonicity across all discrete fractional disciplines. This article addresses the monotonicity of the discrete nabla fractional differences of the Riemann–Liouville type by considering the positivity of ∇b0RLθg(z) combined with a condition on g(b0+2), g(b0+3) and g(b0+4), successively. The article ends with a relationship between the discrete nabla fractional and integer differences of the Riemann–Liouville type, which serves to show the monotonicity of the discrete fractional difference ∇b0RLθg(z).

The discrete fractional operators of Riemann–Liouville and Liouville–Caputo are omnipresent due to the singularity of the kernels. Therefore, convexity analysis of discrete fractional differences of these types plays a vital role in maintaining the safe operation of kernels and symmetry of discrete delta and nabla distribution. In their discrete version, the generalized or modified forms of various operators of fractional calculus are becoming increasingly important from the viewpoints of both pure and applied mathematical sciences. In this paper, we present the discrete version of the recently modified fractional calculus operator with the Mittag-Leffler-type kernel. Here, in this article, the expressions of both the discrete nabla derivative and its counterpart nabla integral are obtained. Some applications and illustrative examples are given to support the theoretical results.

In this paper, we find the solution of the fractional-order Kaup–Kupershmidt (KK) equation by implementing the natural decomposition method with the aid of two different fractional derivatives, namely the Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF). When investigating capillary gravity waves and nonlinear dispersive waves, the KK equation is extremely important. To demonstrate the accuracy and efficiency of the proposed technique, we study the nonlinear fractional KK equation in three distinct cases. The results are given in the form of a series, which converges quickly. The numerical simulations are presented through tables to illustrate the validity of the suggested technique. Numerical simulations in terms of absolute error are performed to ensure that the proposed methodologies are trustworthy and accurate. The resulting solutions are graphically shown to ensure the applicability and validity of the algorithms under consideration. The results that we obtain confirm that the proposed method is the best tool for handling any nonlinear problems arising in science and technology.

The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave (MRLW) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L2 and L∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.

In this study, the solutions of 2+1-dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation are characterized, which can be used in mathematical physics to model water waves with low surface tension and long wavelengths. The integration scheme, namely, the extended direct algebraic method, is used to extract complex trigonometric, rational and hyperbolic functions. The complex-valued solutions represent traveling waves in different structures, such as bell-, V-, and W-shaped multiwaves. The results obtained in this article are novel and more general than those contained in the literature (Wang et al., 2014, Yuan et al., 2017, Pu and Hu 2019, Singh and Gupta 2018). Furthermore, the mechanical features and dynamical characteristics of the obtained solutions are demonstrated by three-dimensional graphics.

This article is aimed at studying novel generalizations of Hermite-Mercer-type inequalities within the Riemann-Liouville k-fractional integral operators by employing s-convex functions. Two new auxiliary results are derived to govern the novel fractional variants of Hadamard-Mercer-type inequalities for differentiable mapping Ψ whose derivatives in the absolute values are convex. Moreover, the results also indicate new lemmas for Ψ′, Ψ′′, and Ψ′′′ and new bounds for the Hadamard-Mercer-type inequalities via the well-known Hölder’s inequality. As an application viewpoint, certain estimates in respect of special functions and special means of real numbers are also illustrated to demonstrate the applicability and effectiveness of the suggested scheme.

There is a strong correlation between convexity and symmetry concepts. In this study, we investigated the new generic class of functions called the (n,m)–generalized convex and studied its basic algebraic properties. The Hermite–Hadamard inequality for the (n,m)–generalized convex function, for the products of two functions and of this type, were proven. Moreover, this class of functions was applied to several known identities; midpoint-type inequalities of Ostrowski and Simpson were derived. Our results are extensions of many previous contributions related to integral inequalities via different convexities.

Many scholars have recently become interested in establishing integral inequalities using various known fractional operators. Fractional calculus has grown in popularity as a result of its capacity to quickly solve real-world problems. First, we establish new fractional inequalities of the Hadamard–Mercer, Pachpatte–Mercer, and Dragomir–Agarwal–Mercer types containing an exponential kernel. In this regard, the inequality proved by Jensen and Mercer plays a major role in our main results. Integral inequalities involving convexity have a wide range of applications in several domains of mathematics where symmetry is important. Both convexity and symmetry are closely linked with each other; when working on one of the topics, you can apply what you have learned to the other. We consider a new identity for differentiable mappings and present its companion bound for the Dragomir–Agarwal–Mercer type inequality employing a convex function. Applications involving matrices are presented. Finally, we conclude our article and discuss its future scope.

A cubic-position negative-velocity (CPNV) feedback controller is proposed in this research in order to suppress the nontrivial oscillations of the 1/3 order subharmonic resonance of a mass-damper-spring model. Based on the Krylov–Bogoliubov (KB) averaging method, the model’s equation of motion is approximately solved and tested for stability. The nontrivial solutions region is plotted to determine where these solutions occur and try to quench them. The controller parameters can play crucial roles in eliminating such regions, keeping only the trivial solutions, and improving the transient response of the car’s oscillations. Different response curves and relations are included in this study to provide the reader a wide overview of the control process.

The purpose of this study is to prove the existence of fractional integral inclusions that are connected to the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for χ-pre-invex fuzzy-interval-valued functions. Some of the related fractional integral inequalities are also proved via Riemann–Liouville fractional integral operator, where integrands are fuzzy-interval-valued functions. To prove the validity of our main results, some of the nontrivial examples are also provided. As specific situations, our findings can provide a variety of new and well-known outcomes which can be viewed as applications of our main results. The results in this paper can be seen as refinements and improvements to previously published findings.

The aim of this paper is not only to propose a new extreme distribution, but also to show that the new extreme model can be used as an alternative to well-known distributions in the literature to model various kinds of datasets in different fields. Several of its statistical properties are explored. It is found that the new extreme model can be utilized for modeling both asymmetric and symmetric datasets, which suffer from over- and under-dispersed phenomena. Moreover, the hazard rate function can be constant, increasing, increasing–constant, or unimodal shaped. The maximum likelihood method is used to estimate the model parameters based on complete and censored samples. Finally, a significant amount of simulations was conducted along with real data applications to illustrate the use of the new extreme distribution.

The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results confirm the generality of our attempt.