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The present research deals with a novel three-dimensional fractional difference neural network model within undamped oscillations. Both the frequency and the amplitude of movements in equilibrium are subsequently estimated mathematically for such structures. According to the stability assessment, the thresholds of the fractional order were determined where bifurcations happen, and an assortment of fluctuations bifurcate within an insignificant equilibrium state. For such discrete fractional-order connections, the parameterized spectrum of undamped resonances is also predicted, and the periodicity and strength of variations are calculated computationally and numerically. Several qualitative techniques, including the Lyapunov exponent, phase depictions, bifurcation illustrations, the $ 0-1 $ analysis and the approximate entropy technique, have been presented with the rigorous analysis. These outcomes indicate that the suggested discrete fractional neural network model has crucial as well as complicated dynamic features that have been affected by the model's variability, both in commensurate and incommensurate cases. Furthermore, the approximation entropy verification and $ \mathbb{C}_{0} $ procedure are used to assess variability and confirm the emergence of chaos. Ultimately, irregular controllers for preserving and synchronizing the suggested framework are highlighted.

Both human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type Ⅰ (HTLV-Ⅰ) are retroviruses that afflict CD4 T cells. In this article, the codynamics of within-host HIV-1 and HTLV-Ⅰ are presented via piecewise fractional differential equations by employing a stochastic system with an influential strategy for biological research. It is demonstrated that the scheme is mathematically and biologically feasible by illustrating that the framework has positive and bounded global findings. The necessary requirements are deduced, ensuring the virus's extinction. In addition, the structure is evaluated for the occurrence of an ergodic stationary distribution and sufficient requirements are developed. A deterministic-stochastic mechanism for simulation studies is constructed and executed in MATLAB to reveal the model's long-term behavior. Utilizing rigorous analysis, we predict that the aforesaid model is an improvement of the existing virus-to-cell and cell-to-cell interactions by investigating an assortment of behaviour patterns that include cross-over to unpredictability processes. Besides that, the piecewise differential formulations, which can be consolidated with integer-order, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic processes, have been declared to be exciting opportunities for researchers in a spectrum of disciplines by enabling them to incorporate distinctive features in various temporal intervals. As a result, by applying these formulations to difficult problems, researchers can achieve improved consequences in reporting realities with white noise. White noise in fractional HIV-1/HTLV-Ⅰ codynamics plays an extremely important function in preventing the proliferation of an outbreak when the proposed flow is constant and disease extermination is directly proportional to the magnitude of the white noise.

The aerodynamics analysis has grown in relevance for wind energy projects; this mechanism is focused on elucidating aerodynamic characteristics to maximize accuracy and practicability via the modelling of chaos in a wind turbine system's permanent magnet synchronous generator using short-memory methodologies. Fractional derivatives have memory impacts and are widely used in numerous practical contexts. Even so, they also require a significant amount of storage capacity and have inefficient operations. We suggested a novel approach to investigating the fractional-order operator's Lyapunov candidate that would do away with the challenging task of determining the indication of the Lyapunov first derivative. Next, a short-memory fractional modelling strategy is presented, followed by short-memory fractional derivatives. Meanwhile, we demonstrate the dynamics of chaotic systems using the Lyapunov function. Predictor-corrector methods are used to provide analytical results. It is suggested to use system dynamics to reduce chaotic behaviour and stabilize operation; the benefit of such a framework is that it can only be used for one state of the hybrid power system. The key variables and characteristics, i.e., the modulation index, pitch angle, drag coefficients, power coefficient, air density, rotor angular speed and short-memory fractional differential equations are also evaluated via numerical simulations to enhance signal strength.

Divorce is the dissolution of two parties' marriage. Separation and divorce are the major obstacles to the viability of a stable family dynamic. In this research, we employ a basic incidence functional as the source of interpersonal disagreement to build an epidemiological framework of divorce outbreaks via the fractal-fractional technique in the Atangana-Baleanu perspective. The research utilized Lyapunov processes to determine whether the two steady states (divorce-free and endemic steady state point) are globally asymptotically robust. Local stability and eigenvalues methodologies were used to examine local stability. The next-generation matrix approach also provides the fundamental reproducing quantity $ \bar{\mathbb{R}_{0}} $. The existence and stability of the equilibrium point can be assessed using $ \bar{\mathbb{R}}_0 $, demonstrating that counseling services for the separated are beneficial to the individuals' well-being and, as a result, the population. The fractal-fractional Atangana-Baleanu operator is applied to the divorce epidemic model, and an innovative technique is used to illustrate its mathematical interpretation. We compare the fractal-fractional model to a framework accommodating different fractal-dimensions and fractional-orders and deduce that the fractal-fractional data fits the stabilized marriages significantly and cannot break again.

Dengue viruses have distinct viral regularities due to the their serotypes. Dengue can be aggravated from a simple fever in an acute infection to a presumably fatal secondary pathogen. This article investigates a deterministic-stochastic secondary dengue viral infection (SDVI) model including logistic growth and a nonlinear incidence rate through the use of piecewise fractional differential equations. This framework accounts for the fact that the dengue virus can penetrate various kinds of specific receptors. Because of the supplementary infection, the system comprises both heterologous and homologous antibody. For the deterministic case, we determine the invariant region and threshold for the aforesaid model. Besides that, we demonstrate that the suggested stochastic SDVI model yields a global and non-negative solution. Taking into consideration effective Lyapunov candidates, the sufficient requirements for the presence of an ergodic stationary distribution of the solution to the stochastic SDVI model are generated. This report basically utilizes a novel idea of piecewise differentiation and integration. This method aids in the acquisition of mechanisms, including crossover impacts. Graphical illustrations of piecewise modeling techniques for chaos challenges are demonstrated. A piecewise numerical scheme is addressed. For various cases, numerical simulations are presented.

In this research, we investigate the direct interaction acquisition method to create a stochastic computational formula of cholera infection evolution via the fractional calculus theory. Susceptible people, infected individuals, medicated individuals, and restored individuals are all included in the framework. Besides that, we transformed the mathematical approach into a stochastic model since it neglected the randomization mechanism and external influences. The descriptive behaviours of systems are then investigated, including the global positivity of the solution, ergodicity and stationary distribution are carried out. Furthermore, the stochastic reproductive number for the system is determined while for the case $ \mathbb{R}_{0}^{s} > 1, $ some sufficient condition for the existence of stationary distribution is obtained. To test the complexity of the proposed scheme, various fractional derivative operators such as power law, exponential decay law and the generalized Mittag-Leffler kernel were used. We included a stochastic factor in every case and employed linear growth and Lipschitz criteria to illustrate the existence and uniqueness of solutions. So every case was numerically investigated, utilizing the newest numerical technique. According to simulation data, the main significant aspects of eradicating cholera infection from society are reduced interaction incidence, improved therapeutic rate, and hygiene facilities.

Recent Ebola virus disease infections have been limited to human-to-human contact as well as the intricate linkages between the habitat, people and socioeconomic variables. The mechanisms of infection propagation can also occur as a consequence of variations in individual actions brought on by dread. This work studies the evolution of the Ebola virus disease by combining fear and environmental spread using a compartmental framework considering stochastic manipulation and a newly defined non-local fractal-fractional (F-F) derivative depending on the generalized Mittag-Leffler kernel. To determine the incidence of infection and person-to-person dissemination, we developed a fear-dependent interaction rate function. We begin by outlining several fundamental characteristics of the system, such as its fundamental reproducing value and equilibrium. Moreover, we examine the existence-uniqueness of non-negative solutions for the given randomized process. The ergodicity and stationary distribution of the infection are then demonstrated, along with the basic criteria for its eradication. Additionally, it has been studied how the suggested framework behaves under the F-F complexities of the Atangana-Baleanu derivative of fractional-order ρ and fractal-dimension τ. The developed scheme has also undergone phenomenological research in addition to the combination of nonlinear characterization by using the fixed point concept. The projected findings are demonstrated through numerical simulations. This research is anticipated to substantially increase the scientific underpinnings for understanding the patterns of infectious illnesses across the globe.

Oncolytic virotherapy is a viable chemotherapeutic agent that identifies and kills tumor cells using replication-competent pathogens. Oncolytic alphavirus M1 is a naturally existing disease that has been shown to have rising specificity and potency in cancer progression. The objective of this research is to introduce and analyze an oncolytic M1 virotherapy framework with spatial variability and anti-tumor immune function via piecewise fractional differential operator techniques. To begin, we potentially demonstrate that the stochastic system's solution is non-negative and global by formulating innovative stochastic Lyapunov candidates. Then, we derive the existence-uniqueness of an ergodic stationary distribution of the stochastic framework and we establish a sufficient assumption $ \mathbb{R}_{0}^{p} < 1 $ extermination of tumor cells and oncolytic M1 virus. Using meticulous interpretation, this model allows us to analyze and anticipate the procedure from the start to the end of the tumor because it allows us to examine a variety of behaviours ranging from crossover to random mechanisms. Furthermore, the piecewise differential operators, which can be assembled with operators including classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu, and stochastic derivative, have decided to open up innovative avenues for readers in various domains, allowing them to encapsulate distinct characteristics in multiple time intervals. Consequently, by applying these operators to serious challenges, scientists can accomplish better outcomes in documenting facts.

In this research, we investigate the dynamical behaviour of a HPAI epidemic system featuring a half-saturated transmission rate and significant evidence of crossover behaviours. Although simulations have proposed numerous mathematical frameworks to portray these behaviours, it is evident that their mathematical representations cannot adequately describe the crossover behaviours, particularly the change from deterministic reboots to stochastics. Furthermore, we show that the stochastic process has a threshold number $ {\bf R}_{0}^{s} $ that can predict pathogen extermination and mean persistence. Furthermore, we show that if $ {\bf R}_{0}^{s} > 1 $, an ergodic stationary distribution corresponds to the stochastic version of the aforementioned system by constructing a sequence of appropriate Lyapunov candidates. The fractional framework is expanded to the piecewise approach, and a simulation tool for interactive representation is provided. We present several illustrated findings for the system that demonstrate the utility of the piecewise estimation technique. The acquired findings offer no uncertainty that this notion is a revolutionary viewpoint that will assist mankind in identifying nature.

This article adopts a class of nonlinear fractional differential equation associating Hilfer generalized proportional fractional (GPF) derivative with having boundary conditions, which amalgamates the Riemann-Liouville (RL) and Caputo-GPF derivative. Taking into consideration the weighted space continuous mappings, we first derive a corresponding integral for the specified boundary value problem. Also, we investigate the existence consequences for a certain problem with a new unified formulation considering the minimal suppositions on nonlinear mapping. Detailed developments hold in the analysis and are dependent on diverse tools involving Schauder's, Schaefer's and Kransnoselskii's fixed point theorems. Finally, we deliver two examples to check the efficiency of the proposed scheme.

Very recently, several novel conceptions of fractional derivatives have been proposed and employed to develop numerical simulations for a wide range of real-world configurations with memory, background, or non-local effects via an uncertainty parameter [0,1] as a confidence degree of belief. Under the complexities of the uncertainty parameter, the major goal of this paper is to develop and examine the Atangana-Baleanu derivative in the Caputo sense for a convoluted glucose-insulin regulating mechanism that possesses a memory and enables one to recall all foreknowledge. However, as compared to other existing derivatives, this is a vitally important point, and the convenience of employing this derivative lessens the intricacy of numerical findings. The Atangana-Baleanu derivative in the Caputo sense of fuzzy valued functions (FVF) in parameterized interval representation is established initially in this study. Then, it is leveraged to demonstrate that the existence and uniqueness of solutions were verified using the theorem suggesting the Banach fixed point and Lipschitz conditions under generalized Hukuhara differentiability. In order to explore the regulation of plasma glucose in diabetic patients with impulsive insulin injections and by monitoring the glucose level that returns to normal in a finite amount of time, we propose an impulsive differential equation model. It is a deterministic mathematical framework that is connected to diabetes mellitus and fractional derivatives. The framework for this research and simulations was numerically solved using a numerical approach based on the Adams-Bashforth-Moulton technique. The findings of this case study indicate that the fractional-order model's plasma glucose management is a suitable choice.

In this manuscript, a novel general family of contraction, called hybrid-interpolative Reich-Istr$ \breve{a}ţ $escu-type $ (G $-$ \alpha $-$ \mu) $-contraction is introduced and some fixed point results in generalised metric space that are not deducible from their akin in metric spaces are obtained. The preeminence of this class of contraction is that its contractive inequality can be extended in a variety of manners, depending on the given parameters. Consequently, a number of corollaries that reduce our result to other well-known results in the literature are highlighted and analysed. Substantial examples are constructed to validate the assumptions of our obtained theorems and to show their distinction from corresponding results. Additionally, one of our obtained corollaries is applied to set up unprecedented existence conditions for solution of a family of integral equations.

The main objective of the investigation is to broaden the description of Caputo fractional derivatives (in short, CFDs) (of order $ 0 < \alpha < r $) considering all relevant permutations of entities involving $ t_{1} $ equal to $ 1 $ and $ t_{2} $ (the others) equal to $ 2 $ via fuzzifications. Under $ {g\mathcal{H}} $-differentiability, we also construct fuzzy Elzaki transforms for CFDs for the generic fractional order $ \alpha\in(r-1, r) $. Furthermore, a novel decomposition method for obtaining the solutions to nonlinear fuzzy fractional partial differential equations (PDEs) via the fuzzy Elzaki transform is constructed. The aforesaid scheme is a novel correlation of the fuzzy Elzaki transform and the Adomian decomposition method. In terms of CFD, several new results for the general fractional order are obtained via $ g\mathcal{H} $-differentiability. By considering the triangular fuzzy numbers of a nonlinear fuzzy fractional PDE, the correctness and capabilities of the proposed algorithm are demonstrated. In the domain of fractional sense, the schematic representation and tabulated outcomes indicate that the algorithm technique is precise and straightforward. Subsequently, future directions and concluding remarks are acted upon with the most focused use of references.

Swift-Hohenberg equations are frequently used to model the biological, physical and chemical processes that lead to pattern generation, and they can realistically represent the findings. This study evaluates the Elzaki Adomian decomposition method (EADM), which integrates a semi-analytical approach using a novel hybridized fuzzy integral transform and the Adomian decomposition method. Moreover, we employ this strategy to address the fractional-order Swift-Hohenberg model (SHM) assuming gH-differentiability by utilizing different initial requirements. The Elzaki transform is used to illustrate certain characteristics of the fuzzy Atangana-Baleanu operator in the Caputo framework. Furthermore, we determined the generic framework and analytical solutions by successfully testing cases in the series form of the systems under consideration. Using the synthesized strategy, we construct the approximate outcomes of the SHM with visualizations of the initial value issues by incorporating the fuzzy factor ϖ∈[0,1] which encompasses the varying fractional values. Finally, the EADM is predicted to be effective and precise in generating the analytical results for dynamical fuzzy fractional partial differential equations that emerge in scientific disciplines.

In this research, the $ \bar{\mathbf{q}} $-homotopy analysis transform method ($ \bar{\mathbf{q}} $-HATM) is employed to identify fractional-order Whitham–Broer–Kaup equation (WBKE) solutions. The WBKE is extensively employed to examine tsunami waves. With the aid of Caputo and Atangana-Baleanu fractional derivative operators, to obtain the analytical findings of WBKE, the predicted algorithm employs a combination of $ \bar{\mathbf{q}} $-HAM and the Aboodh transform. The fractional operators are applied in this work to show how important they are in generalizing the frameworks connected with kernels of singularity and non-singularity. To demonstrate the applicability of the suggested methodology, various relevant problems are solved. Graphical and tabular results are used to display and assess the findings of the suggested approach. In addition, the findings of our recommended approach were analyzed in relation to existing methods. The projected approach has fewer processing requirements and a better accuracy rate. Ultimately, the obtained results reveal that the improved strategy is both trustworthy and meticulous when it comes to assessing the influence of nonlinear systems of both integer and fractional order.

Over time, the interpolative approach in fixed point theory (FPT) has been investigated only in the setting of crisp mathematics, thereby dropping-off a significant amount of useful results. As an attempt to fill up the aforementioned gaps, this paper initiates certain hybrid concepts under the names of interpolative Hardy-Rogers-type (IHRT) and interpolative Reich-Rus-Ciric type (IRRCT) intuitionistic fuzzy contractions in the frame of metric space (MS). Adequate criteria for the existence of intuitionistic fuzzy fixed point (FP) for such contractions are examined. On the basis that FP of a single-valued mapping obeying interpolative type contractive inequality is not always unique, and thereby making the ideas more suitable for FP theorems of multi-valued mappings, a few special cases regarding point-to-point and non-fuzzy set-valued mappings which include the conclusions of some well-known results in the corresponding literature are highlighted and discussed. In addition, comparative examples which dwell on the generality of our obtained results are constructed.

We propose a new mathematical framework of generalized fractional-order to investigate the tuberculosis model with treatment. Under the generalized Caputo fractional derivative notion, the system comprises a network of five nonlinear differential equations. Besides that, the equilibrium points, stability and basic reproductive number are calculated. The concerned derivative involves a power-law kernel and, very recently, it has been adapted for various applied problems. The existence findings for the fractional-order tuberculosis model are validated using the Banach and Leray-Schauder nonlinear alternative fixed point postulates. For the developed framework, we have generated various forms of Ulam's stability outcomes. To investigate the estimated response and nonlinear behaviour of the system under investigation, the efficient mathematical formulation known as the ℘-Laplace Adomian decomposition technique algorithm was implemented. It is important to mention that, with the exception of numerous contemporary discussions, spatial coherence was considered throughout the fractionalization procedure of the classical model. Simulation and comparison analysis yield more versatile outcomes than the existing techniques.

This research utilizes the Jafari transform and the Adomian decomposition method to derive a fascinating explicit pattern for the outcomes of the KdV, mKdV, K(2,2) and K(3,3) models that involve the Caputo fractional derivative operator and the Atangana-Baleanu fractional derivative operator in the Caputo sense. The novel exact-approximate solutions are derived from the formulation of trigonometric, hyperbolic, and exponential function forms. Laser and plasma sciences may benefit from these solutions. It is demonstrated that this approach produces a simple and effective mathematical framework for tackling nonlinear problems. To provide additional context for these ideas, simulations are performed, employing a computationally packaged program to assist in comprehending the implications of solutions.

In this paper, new intuitionistic fuzzy fixed point results for sequence of intuitionistic fuzzy set-valued maps in the structure of b-metric spaces are examined. A few nontrivial comparative examples are constructed to keep up the hypotheses and generality of our obtained results. Following the fact that most existing concepts of Ulam-Hyers type stabilities are concerned with crisp mappings, we introduce the notion of stability and well-posedness of functional inclusions involving intuitionistic fuzzy set-valued maps. It is a familiar fact that solution of every functional inclusion is a subset of an appropriate space. In this direction, intuitionistic fuzzy fixed point problem involving (α,β)-level set of an intuitionistic fuzzy set-valued map is initiated. Moreover, novel sufficient criteria for existence of solutions to an integral inclusion are investigated to indicate a possible application of the ideas presented herein.

In this research, the Shehu transform is coupled with the Adomian decomposition method for obtaining the exact-approximate solution of the plasma fluid physical model, known as the Zakharov-Kuznetsov equation (briefly, ZKE) having a fractional order in the Caputo sense. The Laplace and Sumudu transforms have been refined into the Shehu transform. The action of weakly nonlinear ion acoustic waves in a plasma carrying cold ions and hot isothermal electrons is investigated in this study. Important fractional derivative notions are discussed in the context of Caputo. The Shehu decomposition method (SDM), a robust research methodology, is effectively implemented to generate the solution for the ZKEs. A series of Adomian components converge to the exact solution of the assigned task, demonstrating the solution of the suggested technique. Furthermore, the outcomes of this technique have generated important associations with the precise solutions to the problems being researched. Illustrative examples highlight the validity of the current process. The usefulness of the technique is reinforced via graphical and tabular illustrations as well as statistics theory.

The Shehu homotopy perturbation transform method (SHPTM) via fuzziness, which combines the homotopy perturbation method and the Shehu transform, is the subject of this article. With the assistance of fuzzy fractional Caputo and Atangana-Baleanu derivatives operators, the proposed methodology is designed to illustrate the reliability by finding fuzzy fractional equal width (EW), modified equal width (MEW) and variants of modified equal width (VMEW) models with fuzzy initial conditions (ICs). In cold plasma, the proposed model is vital for generating hydro-magnetic waves. We investigated SHPTM's potential to investigate fractional nonlinear systems and demonstrated its superiority over other numerical approaches that are accessible. Another significant aspect of this research is to look at two significant fuzzy fractional models with differing nonlinearities considering fuzzy set theory. Evaluating various implementations verifies the method's impact, capabilities, and practicality. The level impacts of the parameter ℏ and fractional order are graphically and quantitatively presented, demonstrating good agreement between the fuzzy approximate upper and lower bound solutions. The findings are numerically examined to crisp solutions and those produced by other approaches, demonstrating that the proposed method is a handy and astonishingly efficient instrument for solving a wide range of physics and engineering problems.

The present research investigates the Susceptible-Infected-Recovered (SIR) epidemic model of childhood diseases and its complications with the Atangana-Baleanu fractional derivative operator in the Caputo sense (ABC). With the aid of the Elzaki Adomian decomposition method (EADM), the approximate solutions of the aforesaid model are discussed by exerting the Adomian decomposition method. By employing the fixed point postulates and the Picard–Lindelöf approach, the stability, existence, and uniqueness consequences of the model are demonstrated. Furthermore, we illustrate the essential hypothesis for disease control in order to find the role of unaware infectives in the spread of childhood diseases. Besides that, simulation results and graphical illustrations are presented for various fractional-orders. A comparison analysis is shown with the previous findings. It is hoped that ABC fractional derivative and the projected algorithm will provide new venues in futuristic studies to manipulate and analyze several epidemiological models.

A susceptible diabetes comorbidity model was used in the mathematical treatment to explain the predominance of mellitus. In the susceptible diabetes comorbidity model, diabetic patients were divided into three groups: susceptible diabetes, uncomplicated diabetics, and complicated diabetics. In this research, we investigate the susceptible diabetes comorbidity model and its intricacy via the Atangana-Baleanu fractional derivative operator in the Caputo sense (ABC). The analysis backs up the idea that the aforesaid fractional order technique plays an important role in predicting whether or not a person will develop diabetes after a substantial immunological assault. Using the fixed point postulates, several theoretic outcomes of existence and Ulam's stability are proposed for the susceptible diabetes comorbidity model. Meanwhile, a mathematical approach is provided for determining the numerical solution of the developed framework employing the Adams type predictor–corrector algorithm for the ABC-fractional integral operator. Numerous mathematical representations correlating to multiple fractional orders are shown. It brings up the prospect of employing this structure to generate framework regulators for glucose metabolism in type 2 diabetes mellitus patients.

For a multi-term time-fractional diffusion equation comprising Hilfer fractional derivatives in time variables of different orders between 0 and 1, we have studied two problems (direct problem and inverse source problem). The spectral problem under consideration is self-adjoint. The solution to the given direct and inverse source problems is formulated utilizing the spectral problem. For the solution of the given direct problem, we proposed existence, uniqueness, and stability results. The existence, uniqueness, and consistency effects for the solution of the given inverse problem were addressed, as well as an inverse source for recovering space-dependent source term at certain T. For the solution of the challenges, we proposed certain relevant cases.

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.

Fractional calculus has been the target of the work of many mathematicians for more than a century. Some of these investigations are of inequalities and fractional integral operators. In this article, a novel fractional operator which is known as weighted generalized proportional Hadamard fractional operator with unknown attribute weight is proposed. First, a fractional formulation is constructed, which covers a subjective list of operators. With the aid of the above mentioned operators, numerous notable versions of Pólya-Szegö, Chebyshev and certain related variants are established. Meanwhile, new outcomes are introduced and new theorems are exhibited. Taking into account the novel generalizations, our consequences have a potential association with the previous results. Furthermore, we demonstrate the applications of new operator with numerous integral inequalities by inducing assumptions on weight function ϖ and proportionality index φ. It is hoped that this research demonstrates that the suggested technique is efficient, computationally, very user-friendly and accurate.

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.

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.

In this work, a new strategy to derive inequalities by employing newly proposed fractional operators, known as a Hilfer generalized proportional fractional integral operator ($\widehat{\mathcal{GPFIO}}$). The presented work establishes a relationship between weighted extended \v{C}eby\v{s}ev version and P\'{o}lya-Szeg\"{o} type inequalities, which can be directly used in fractional differential equations and statistical theory. In addition, the proposed technique is also compared with the existing results. This work is important and timely for evaluating fractional operators and predicting the production of numerous real-world problems in varying nature.

In statistical analysis, oftentimes a probability density function is used to describe the relationship between certain unknown parameters and measurements taken to learn about them. As soon as there is more than enough data collected to determine a unique solution for the parameters, an estimation technique needs to be applied such as “fractional calculus”, for instance, which turns out to be optimal under a wide range of criteria. In this context, we aim to present some novel estimates based on the expectation and variance of a continuous random variable by employing generalized Riemann-Liouville fractional integral operators. Besides, we obtain a two-parameter extension of generalized Riemann-Liouville fractional integral inequalities, and provide several modifications in the Riemann-Liouville and classical sense. Our ideas and obtained results my stimulate further research in statistical analysis.

The investigation of the proposed techniques is effective and convenient for solving the integrodifferential and difference equations. The present investigation depends on two highlights; the novel Hermite-Hadamard type inequalities for $\mathcal{K}$-conformable fractional integral operator in terms of a new parameter $\mathcal{K}>0$ and weighted version of Hermite-Hadamard type inequalities for exponentially convex functions in the classical sense. By using an integral identity together with the Hölder-İşcan and improved power-mean inequality we establish several new inequalities for differentiable exponentially convex functions. This generalizes the Hadamard fractional integrals and Riemann-Liouville into a single form. Our contribution expands some innovative studies in this line. Moreover, two suitable examples are presented to demonstrate the novelty of the results established, the first one about the contributions of the modified Bessel functions and the other is about $\sigma$-digamma function. Finally, various applications for some special means as arithmetic, geometric and logarithmic are given.

The key purpose of this study is to suggest a delta Riemann-Liouville (RL) fractional integral operators for deriving certain novel refinements of Pólya-Szegö and Čebyšev type inequalities on time scales. Some new Pólya-Szegö, Čebyšev and extended Čebyšev inequalities via delta-RL fractional integral operator on a time scale that captures some continuous and discrete analogues in the relative literature. New explicit bounds for unknown functions concerned are obtained due to the presented inequalities.

In the article, we introduce the concept of the exponentially $tgs$-convex function and discover two new conformable fractional integral identities concerning the first-order differentiable convex mappings. By using these identities, we establish several new right-sided Hermite-Hadamard type inequalities for the exponentially $tgs$-convex functions via conformable fractional integrals. Our outcomes for conformable fractional integral operators are also applied to some special means.

The main aim of the present paper is to present a novel approach base on the exponentially convex function to broaden the utilization of celebrated Hermite-Hadamard type inequality. The proposed technique presents an auxiliary result of constructing the set of base functions and gives deformation equations in a simple form. The auxiliary result in the convexity has provided a convenient way of establishing the convergence region of several novel results. The strategy is not limited to the small parameter, such as in the classical method. The numerical examples obtained by the proposed approach indicate that the approach is easy to implement and computationally very attractive. The implementation of this numerical scheme clearly exhibits its effectiveness, reliability, and easiness regarding the applications in error estimates for weighted mean, the integral formula, rth moments of a continuous random variable, application to weighted special means and in developing the variants by extraordinary choices of n and θ as well as its better approximation.

The investigation of the proposed methods is effective and convenient for solving the integrodifferential and difference equations. In this note, we introduce the generalized $\mathcal{K}$-fractional integral in terms of a new parameter $\mathcal{K}>0$ for exponentially convex functions. This paper offers some novel inequalities of Ostrowski-type using the generalized $\mathcal{K}$-fractional integral. In the application viewpoint, we proved several corollaries that investigate for proving Hermite-Hadamard inequalities for generalized $\mathcal{K}$-fractional integral operator. Some numerical examples are offered to explain the obtained results. Moreover, some applications of proposed results are presented to the demonstration of the efficiency of the proposed technique. The numerical results show that our approach is superior to some related methodologies.

In this paper, some new inequalities of Simpson's type are set up for the classes of functions whose derivatives of absolute are preinvex by means of ĸ-fractional integrals. Additionally, by extraordinary choices of n and ĸ, we give some diminished outcomes. Meanwhile, we also provide the inequalities for $\mathcal{F}$-divergence measures and in probabilistic versions.