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We explored a class of quantum calculus boundary value problems that include fractional $ q $-difference integrals. Sufficient and necessary conditions for demonstrating the existence and uniqueness of positive solutions were stated using fixed point theorems in partially ordered spaces. Moreover, the existence of a positive solution for a boundary value problem with a Riemann-Liouville fractional derivative and an integral boundary condition was examined by utilizing a novel fixed point theorem that included a $ \mathfrak{a} $-$ \eta $-Geraghty contraction. Several examples were provided to demonstrate the efficacy of the outcomes.

In this manuscript, our work was about a qualitative study for a class of multi-complex orders nonlinear fractional differential equations (FDEs). Our methodology utilized the topological degree theory and studied a novel operator tailored for non-singular FDEs with $ \mathrm{T} $-Riemann-Liouville (T-RL) fractional order derivatives. The primary objective was to prove the existence and uniqueness of solutions for both initial and boundary value problems within the intricated framework. With the help of topological degree theory, we contributed to a wider understanding of the structural aspects governing the behavior of the considered FDE. The novel operator proposing for non-singular FDEs added a unique dimension to our analytical problem, offering a versatile and effective means of addressing the challenges posed by these complex systems for their theoretical analysis. For the practical implications of our theoretical framework, we presented two concrete examples that reinforced and elucidated the key concepts introduced. These examples underscored our approach's viability and highlighted its potential applications in diverse scientific and engineering domains. Through this comprehensive exploration, we aimed to contribute significantly to advancing the theoretical foundation related to the study of multi-complex nonlinear FDEs. Moreover, a fixed-time terminal sliding mode control (TSMC) has been developed. This proposed control strategy for eliminating leukemic cells while maintaining normal cells was based on a chemotherapeutic treatment that was not only effective but also widely acknowledged to be safe. This strategy was evaluated using the fixed-time Lyapunov stability theory, and simulations were included to illustrate its performance in terms of tracking and convergence.

The fundamental problem in electronic cooling systems is the implementation of a cavity such that it can be used to provide localized cooling to specific components, such as CPUs or GPUs, enhancing their performance and longevity. It can also be used in microfluidic devices for controlled drug delivery, where precise control of fluid flow is crucial. The present article numerically explores the free convection non-Newtonian Casson hybrid nanofluid phenomena that occur within an H-shaped cavity while heated from the middle. The heating efficiency and heat flow in a cavity are influenced by perpendicular hot walls that connect two vertical closed channels. A numerical solution is obtained by implementing the Galerkin finite element method to solve the partial differential equation. The numerical outcomes are depicted on the contour of streamlines and isotherms for different parameters in the following ranges: 0.1 ≤ η ≤ 0.4, 0.005≤ϕhp≤0.020, 0.1 ≤ γ ≤ 2, and 103 ≤ Ra ≤ 106 at fixed Pr = 6.2. In addition, line graphs show rate of heat transfer within the enclosure using the average Nusselt number for these parameters. Increased aspect ratios (η = 0.4) result in a minimal rate of heat transfer enhancement, whereas decreasing η leads to a significantly higher average Nusselt number and maximum heat transfer within the cavity. The convective rate of heat transfer increases with the presence of hybrid nanoparticles inside an H-shaped cavity for all Rayleigh numbers. The rotation of the Casson hybrid nanofluid also rises as the volume ratio of nanoparticles increases. For a fixed aspect ratio (A.R) of 0.1, the heat dissipation is 6.91% at a lower ϕhp value of 0.005 at a fixed Ra value of 105. However, it increases to 7.072% for a higher ϕhp value of 0.02 at Ra = 105. With increasing Ra number, ϕhp, and γ, the number NuAve increases.

The present study communicates with the thermal analysis of an incompressible micropolar fluid by considering two different prescribed conditions corresponding to two different surfaces. On the porous surface of two different stretchable sheets, the unsteady flow problem of a micropolar fluid is explored by taking the variable impacts of the magnetic field. For both considered sheets, the prescribed thermal analysis is carried out with the contribution of joule heating, viscous dissipation, and radiation. Two prescribed cases of surface temperature and heat flux are discussed for linear stretched surface. Similarly, prescribed cases of exponential surface temperature and exponential heat flux are investigated for the case of an exponential stretchable surface. An effective technique of bvp4c in MATLAB is utilized to numerically tackle the governing equations of the problem. Both the distributions of temperature and velocity and the field of microrotation are graphically scrutinized relative to different geometries and several pertinent parameters. This study of thermal analysis with prescribed conditions elucidates that for an exponential surface, three distributions of flow phenomenon (temperature, microrotation, velocity) manifest more prominent results in comparison to the stretchable surface with linear velocity.

This study provides a comprehensive exploration of the qualitative analysis of a hybrid system of pantograph equations with fractional order and a p-Laplacian operator. The existence of the solution of the system is explicitly established within the context of Riemann–Liouville’s fractional order operator, employing the Arzelà–Ascoli theorem for validation. The establishment of uniqueness criteria is accomplished by the utilization of the Banach contractive technique. In addition, the examination of solution stability is conducted using the Hyers–Ulam (HU) stability technique. In order to enhance the credibility of our main conclusions, we have included a representative and illustrative example in the concluding section of the study. This work serve to offer a thorough and applicable comprehension of the mathematical framework that has been proposed.

Abstract In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes.

In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem.

Abstract The fractional operator of Caputo-Fabrizio has significant advantages in various physical flow problems due to the implementations in manufacturing and engineering fields such as viscoelastic damping in polymer, image processing, wave propagation, and dielectric polymerization. The current study has the main objective of implementation of Caputo-Fabrizio fractional derivative on the flow phenomenon and heat transfer mechanism of trigonometric non-Newtonian fluid. The time-dependent flow mechanism is assumed to be developed through a vertical infinite plate. The thermal radiation’s effects are incorporated into the analysis of heat transfer. With the help of mathematical formulations, the physical flow system is expressed. The governing equations of the flow system acquire the dimensionless form through the involvement of the dimensionless variables. The application of Caputo-Fabrizio derivative is implemented to achieve the fractional model of the dimensionless system. An exact solution of the fractional-based dimensionless system of the equations is acquired through the technique of the Laplace transform. Physical interpretation of temperature and velocity distributions relative to the pertinent parameters is visualized via graphs. The current study concludes that the velocity distributions exhibit an accelerating nature corresponding to the increasing order of the fractional operator. Moreover, the graphical results are more significant corresponding to the greater time period.

This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form \align\Delta^{\gamma}u(\kappa)&+\Theta[\kappa+\gamma,w(\kappa+\gamma)] =&\Phi(\kappa+\gamma)+\Upsilon(\kappa+\gamma)w^{\nu}(\kappa+\gamma) +\Psi[\kappa+\gamma,w(\kappa+\gamma)],\quad\kappa\in\mathbb{N}_{1-\gamma}, u_0 =&c_0, where $\mathbb{N}_{1-\gamma}=\{1-\gamma,2-\gamma,3-\gamma,\cdots\}$, $0

Drug usage has always been a top concern for parents and government officials, harming younger people's lives in circumstances that cannot be undone. This paper considers a modified ABC-fractional-order Ice-smoking dynamical system for the theoretical and numerical results. In the theoretical aspect, we study the solution existence, uniqueness work, and Hyers-Ulam (HU) stability of the presumed extended fractional order dynamical system of Ice smoking. Recursive sequences that follow one another are designed to verify the existence of a solution to the provided Ice-smoking model. Functional analysis principles and findings are applied to showcase the uniqueness of the solution and the stability under the Hyers-Ulam (HU) framework. We offer simulations and a comparison by graphs by using Lagrange polynomials. The simulations demonstrate the scheme's applicability, future forecasting, and validations of the results of modified ABC-fractional orders.

In this article, a coupled nonlinear problem of hybrid fractional differential equations (HFDEs) is presented for the qualitative work and numerical results. Two types of operators are involved in the research problem. One of them is DβrR which represent Riemann–Liouville's (RL) fractional derivatives while the operator L(DϱrR) is a series operator and DϱrR's are RL operators such that βr,ϱr∈(0,1]. These operators are joined by Φp operator. As a result, we have a nonlinear coupled system of FDEs. The newly established nonlinear system is studied for the existence, uniqueness criteria, stability of the solutions, and numerical computations. For the theoretical results, we take help from the available literature about the fixed point (FP) techniques. Then a computational scheme is developed with the help of Lagrange's interpolation technique. An application of the problem as a particular case is presented in the sense of the Leukemia mathematical model. The model presents the infection propagation. Leukemia can be managed by providing a chemotherapeutic treatment generally accepted to be safe, and a fractional-order fixed-time terminal sliding mode control has been developed to achieve this goal of removing Leukemic cells while keeping a sufficient number of normal cells. In order to evaluate the proposed controller stability, the fixed-time Lyapunov stability theory is employed. To better illustrate the study, comparison simulations are shown, demonstrating that the suggested control approach has higher tracking and convergence performance.

The study focuses on the instability of local linear convective flow in an incompressible boundary layer caused by a rough rotating disk in a steady MHD flow of viscous nanofluid. Miklavčič and Wang's (Miklavčič and Wang, 2004) [9] MW roughness model are utilized in the presence of MHD of Cu-water nanofluid with enforcement of axial flows. This study will investigate the instability characteristics with the MHD boundary layer flow of nanofluid over a rotating disk and incorporate the effects of axial flow with anisotropic and isotropic surface roughness. The resulting ordinary differential equations (ODEs) are obtained by using von Kàrmàn (Kármán, 1921) [3] similarity transformation on partial differential equations (PDEs). Subsequently, numerical solutions are obtained using the shooting method, specifically the Runge-Kutta technique. Steady-flow profiles for MHD and volume fractions of nanoparticles are analyzed by the partial-slip conditions with surface roughness. Convective instability for stationary modes and neutral stability curves are also obtained and investigated by the formulation of linear stability equations with the MHD of nanofluid. Linear convective growth rates are utilized to analyze the stability of magnetic fields and nanoparticles and to confirm the outcomes of this analysis. Stationary disturbances are also considered in the energy analysis. The investigation indicates the correlation between instability modes Type I and Type II, in the presence of MHD, nanoparticles, and the growth rates of the critical Reynolds number. An integral energy equation enhances comprehension of the fundamental physical mechanisms. The factors contributing to convective instability in the system are clarified using this approach.

In this article, we study the qualitative analysis of a fractional coupled hybrid pantograph system with a p-Laplacian operator. We developed the criteria for the existence of solutions for the proposed system with boundary conditions in the context of Caputo’s operator. In addition, existence and uniqueness criteria are carried out with the help of Banach’s and Arzelá Ascoli theorems by contractive approach. Furthermore, the Hyers–Ulam stability method was applied for stability analysis. For the sake of the reliability of the main results, we provided some examples at the end of the article for illustrative purposes.

In this study, we give the notion of a piecewise modified Atangana-Baleanu-Caputo (mABC) fractional derivative and apply it to a tuberculosis model. This novel operator is a combination of classical derivative and the recently developed modified Atangana-Baleanu operator in the Caputo's sense. For this combination, we have considered the splitting of an interval $ [0, t_2] $ for $ t_2\in\mathbb{R}^+ $, such that, the classical derivative is applied in the first portion $ [0, t_1] $ while the second differential operator is applied in the interval $ [t_1, t_2] $. As a result, we obtained the piecewise mABC operator. Its corresponding integral is also given accordingly. This new operator is then applied to a tuberculosis model for the study of crossover behavior. The existence and stability of solutions are investigated for the nonlinear piecewise modified ABC tuberculosis model. A numerical scheme for the simulations is presented with the help of Lagrange's interpolation polynomial is then applied to the available data.

People’s ability to easily travel across cities is thought to play a major role in the spread of infectious diseases. This research develops and analyzes a new Susceptible-Infected-Recovered-Susceptible (SIRS) model that accounts for transport-related infection, media coverage, and three types of noise (white, telegraph, and Lévy) to examine the role of transport in disease transmission. Thus, several theoretical and numerical uses of the revised model are investigated. Using Lyapunov functions, we check whether or not the model has a positive global solution. The infection either dies out or stays put beyond that point. The researched analytical results are validated by developing and using a numerical scheme to analyze and simulate the effects of various factors on the model’s dynamics.

Abstract In a porous medium, we have examined sinusoidal two-dimensional transport enclosed porous peristaltic boundaries having an Eyring Powell fluid with a water containing $$\text{Al}_{2}{\text{O}}_{3}$$ Al 2 O 3 . The determining momentum and temperature equations are solved semi-analytically by using regular perturbation method and Mathematica. In present research only free pumping case and small amplitude ratio is studied. Mathematical and pictorial consequences are investigated for distinct physical parameters of interest like porosity, viscosity, volume fraction and permeability to check the effects of flow velocity and temperature.

The scrutinization of entropy optimization in the various flow mechanisms of non-Newtonian fluids with heat transfer has been incredibly enhanced. Through the investigation of irreversibility sources in the steady flow of a non-Newtonian Willaimson fluid, an analysis of entropy generation is carried out in this current work. The current study has an essential aspect of investigating the heat transfer mechanism with flow phenomenon by considering convective-radiative boundary conditions. A horizontal MHD channel is assumed with two parallel plates to develop a mathematical model for the flow phenomenon by considering the variable viscosity of the fluid. The contribution of physical impacts of thermal radiation, Joule heating, and viscous dissipation is interpolated in the constitutive energy equation. The complete flow of the current analysis is established in the form of ordinary differential equations which further take the form of the dimensionless system through the contribution of the similarity variables. A graphical scrutinization of the physical features of the flow phenomenon in relation to the pertinent parameters is proposed. This study reveals that the higher magnitude of radiation parameter and Brinkman number dominates the system's entropy. Moreover, the temperature distribution experiences an increasing mechanism with improved conduction-radiation parameter at the lower plate.

This paper seeks to present an optimization method to estimate the solutions of nonlinear oscillation equations of fractional order. The mentioned method is based on Bernstein polynomials (Bps). In the presented numerical approach, the operational matrices of the ordinary and fractional derivatives of Bernstein polynomials are utilized to estimate the solution of the model under the study. In this technique, the unknown function is expanded in terms of Bps. By using the residual function and its 2-norm, the problem under consideration is converted into a constrained nonlinear optimization one. So that, the constraint equations are obtained from the given initial conditions and the object function is obtained from the residual function. Finally, we obtain the unknown coefficients optimally by a set of unknown Lagrange multipliers. The main advantage of this approach is that it reduces such problems to those optimization problems, which greatly simplifies them and also leads to obtain a good approximate solution for them. The accuracy and efficiency of the presented method are supported by some examples. At the end, we compare the numerical results with other results.

This analysis deals the steady and the incompressible MHD fluid flow by sinusoidal walls of a hexagonal cavity with a cylindrical obstacle at the center. Finite element method (FEM), a numerical modeling method is used to examine the heat transfer and fluid flow controlled by the energy equation, the continuity equation and Navier-Stokes equations, which are converted to dimensionless form through suitable parameterization, which are tackled by finite element method. The temperature distribution and velocity fields are displayed for various parameters which are Richardson number, Hartmann number, Reynolds number, velocities’ amplitudes ratio, temperatures’ amplitudes ratio and phase deviation. This graphical study shows good convergent results of temperature and velocity for variation of involved parameters. At the end, the significant effects of the heat transfer rate are discussed in terms of the Nusselt number.

The engineers are fascinated in fluid flows and heat transfer in engineering optics and devices. For such purposes, fluid flows are computed in different geometries. The engineers normally look for reduced heat transfer, enhanced cooling rate and reduced friction drag. Thus, the focus of the present article is to undertake the flow of viscous or Newtonian fluid through joined cavities which is very important in many engineering fields. The Navier-Stokes equations are operated for studying behavior of flow, while Fourier's law of heat conduction is employed for the analysis of heat flow in connected cavities. The consequences show that high pressure, lowest velocity and high temperature are observed at the thin connection of cavities.

Abstract In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results.

In this article, we investigate some necessary and sufficient conditions required for the existence of solutions for modified ABC-fractional differential equations (mAB-FDEs) with p-Laplacian operator. We also study the uniqueness and Hyers-Ulam stability (HU-stability) for the solutions of the presumed mABC-FDEs system. For the recently developed mABC-operator, such a problem has not yet studied also our problem is more general than those in the available literature.

In this study, we discuss the passivity analysis for Markovian jumping Neural Networks of neural-type. The results are demonstrated using phases of linear matrix inequalities as well as an improved Lyapunov-Krasovskii functional (LKF) of the triple integral terms and quadruple integrals. The information of the mode-dependent of all delays have been taken into account in the constructed Lyapunov–Krasovskii functional and novel stability criterion is derived. The value of selecting as many Lyapunov matrices that are mode-dependent as possible is demonstrated. The effectiveness and decreased conservatism of the aforementioned theoretical results are eventually demonstrated by a numerical example.

Nonlinear differential equations are widely used in everyday scientific and engineering dynamics. Problems involving differential equations of fractional order with initial and phase changes are often employed. Using a novel norm that is comfortable for fractional and non-singular differential equations containing Atangana-Baleanu-Caputo fractional derivatives, we examined a new class of initial values issues in this study. The Perov fixed point theorems that are utilized in generalized Banach spaces form the foundation for the new findings. Examples of the numerical analysis are provided in order to safeguard and effectively present the key findings.

A mathematical model of discrete fractional equations with initial condition is constructed to study the tumor-immune interactions in this research. The model is a system of two nonlinear difference equations in the sense of Caputo fractional operator. The applications of Banach’s and Leray–Schauder’s fixed point theorems are used to analyze the existence results for the proposed model. Additionally, we developed several kinds of Ulam’s stability results for the suggested model. The tumor-immune fractional map’s dynamic behavior is numerical analyzed for some special cases. Further, adaptive control law is proposed to stabilize the fractional map and a control scheme is introduced to enhance the synchronization of the fractional model.

Due to realistic implementations of heat transfer flows, this study communicates with the two cases of the prescribed thermal process. A comparative analysis of the time-dependent flow of a micropolar fluid between two problems of linear stretching sheet and exponential stretching sheet is demonstrated. In the context of the linear stretching sheet, the investigation covers situations involving prescribed surface temperature (PST) and prescribed heat flux (PHF). Likewise, for the exponential stretching sheet, the study examines cases of prescribed exponential order temperature (PEST) and prescribed exponential order heat flux (PEHF). For both problems, the two cases of the thermal process are examined for heat transfer analysis considering joule heating and nonlinear thermal radiation. The two-dimensional flow in the presence of a variable magnetic field is explored for both the linear and exponential stretching sheet problems. Graphical representations are employed to provide a comparative illustration of flow properties concerning relevant parameters for both cases. From the graphical and tabular analysis, we deduce that an exponential stretching sheet provides more consequential results as compared to a linear stretching sheet. Furthermore, the material parameter demonstrates an increase in the velocity field for both of the considered sheets.

In this paper, a general system of quadratically perturbed system of modified fractional differential equations (FDEs) is considered for the solution existence, solution uniqueness, stability results, numerical scheme and computational applications. The presumed perturbed system is more general and several preexisting problems become its special cases. Fixed point results are applied for the theoretical results. The Lagrange’s polynomial is used to approximate the nonlinear system and a useful numerical scheme is established. For an application, a complex system of chaotic attractor is given and is computational studied via some graphical presentation.

Abstract In this paper, we propose a generalized Gronwall inequality in the context of the ψ-Hilfer proportional fractional derivative. Using Picard’s successive approximation and the definition of Mittag–Leffler functions, we construct the representation formula of the solution for the ψ-Hilfer proportional fractional differential equation with constant coefficient in the form of the Mittag–Leffler kernel. The uniqueness result is proved by using Banach’s fixed-point theorem with some properties of the Mittag–Leffler kernel. Additionally, Ulam–Hyers–Mittag–Leffler stability results are analyzed. Finally, numerical examples are provided to demonstrate the theory’s application.

In this article, we investigate some necessary and sufficient conditions required for the existence of solutions for mABC-fractional differential equations (mABC-FDEs) with initial conditions; additionally, a numerical scheme based on the the Lagrange's interpolation polynomial is established and applied to a dynamical system for the applications. We also study the uniqueness and Hyers-Ulam stability for the solutions of the presumed mABC-FDEs system. Such a system has not been studied for the mentioned mABC-operator and this work generalizes most of the results studied for the ABC operator. This study will provide a base to a large number of dynamical problems for the existence, uniqueness and numerical simulations. The results are compared with the classical results graphically to check the accuracy and applicability of the scheme.

This work is concerned with the study of the existing solution for the fractional (p,q)-difference equation under first order (p,q)-difference boundary conditions in generalized metric space. To achieve the solution, we combine some contraction techniques in fixed point theory with the numerical techniques of the Lipschitz matrix and vector norms. To do this, we first associate a matrix to a desired boundary value problem. Then we present sufficient conditions for the convergence of this matrix to zero. Also, we design some algorithms to use the computer for calculate the eigenvalues of such matrices and different values of (p,q)-Gamma function. Finally, by presenting two numerical examples, we examine the performance and correctness of the proposed method. Some tables and figures are provided to better understand the issues.

This article investigates the modeling and numerical simulation of Magnetohydrodynamic (MHD) buoyancy-driven convection flow in a differentially heated, square enclosure. Left vertical side is given a high temperature and the right vertical side is sustained at a low temperature. Horizontal sides of the enclosure are insulated. A constant magnetic field is presumed horizontally. Findings of the governing differential equations are explored numerically considering the impact of Magneto-hydrodynamic (MHD). Problem is deciphered by Galerkin finite element approach in COMSOL Multiphysics. Numerical solutions are computed for different values of Rayleigh number ranging 103≤Ra≤107 and Hartmann number ranging 0≤Ha≤40. Rate of heat that passes from the heated side is affected by increasing Rayleigh and Hartmann numbers. Comportment of MHD free convection heat flow from transient to steady state is numerically examined for a period of 0 to 1 s. The numerical solutions are discussed in respect of streamlines, iso-contours, and isotherms. In addition, physical quantities such as velocity and Nusselt number are studied. It is seen that with increasing values of Rayleigh number there is increase in local Nusselt number distribution on heated side of the cavity. Velocity distribution in the flow domain decreases in variations with increasing Hartmann number.

In this paper, we discussed the existence, uniqueness and Ulam-type stability of solutions for sequential coupled hybrid fractional differential equations with two derivatives. The uniqueness of solutions is established by means of Banach's contraction mapping principle, while the existence of solutions is derived from Leray-Schauder's alternative fixed point theorem. Further, the Ulam-type stability of the addressed problem is studied. Finally, an example is provided to check the validity of our obtained results.

This paper studies the exponential stability for random impulsive non-linear singular differential systems. We established some new sufficient conditions for the proposed singular differential system by using the Lyapunov function method with random impulsive time points. Further, to validate the theoretical results' effectiveness, we finally gave two numerical examples that study with graphical illustration and an additional example involving matrices with complex entries, proving the results to be true in that case as well.

Based on both the characteristics of shear thinning and shear thickening fluids, the Sutterby fluid has various applications in engineering and industrial fields. Due to the dual nature of the Sutterby fluid, the motive of the current study is to scrutinize the variable physical effects on the Sutterby nanofluid flow subject to shear thickening and shear thinning behavior over biaxially stretchable exponential and nonlinear sheets. The steady flow mechanism with the variable magnetic field, partial slip effects, and variable heat source/sink is examined over both stretchable sheets. The analysis of mass and heat transfer is carried out with the mutual impacts of thermophoresis and Brownian motion through the Buongiorno model. Suitable transformations for both exponential and nonlinear sheets are implemented on the problem's constitutive equations. As a result, the nonlinear setup of ordinary differential equations is acquired which is further numerically analyzed through the bvp4c technique in MATLAB. The graphical explanation of temperature, velocity, and concentration distributions exhibits that the exponential sheet provides more significant results as compared to the nonlinear sheet. Further, this study revealed that for the shear thickening behavior of Sutterby nanofluid, the increasing values of Deborah number increase the axial velocity.

Fractional calculus yields numerous implementations in different fields such as biological materials, physical memory, oscillation, wave propagation, and viscoelastic dynamics. Due to the significant applications of fractional calculus, the current study deals with the fractional derivative base study of a Brinkman hybrid nanofluid with an inclined magnetic field. A three-parametric Prabhakar fractional derivative with the involvement of the Mittag-Leffler function is implemented. A vertical plate moving with exponential velocity is considered to be the source of the flow mechanism. Moreover, the effects of exponential heating are incorporated into the thermal analysis. An appropriate group of dimensionless ansatz is adopted to get the dimensionless setup of equations. The Prabhakar fractional operator is implemented in the dimensionless equations which are further tackled by an effectual Laplace transform technique. An inverse Stehfest method and Tzou's method are implemented to tackle the inversion of the Laplace transform. This study exhibits that the fractional constraints minimize both the fields of temperature and velocity. Moreover, the velocity distribution deteriorates corresponding to the improved Brinkman parameter. The Brinkman parameter and the fluid's viscosity are directly related to each other. With the improved Brinkman parameter, the viscosity of the fluid increases. As a result, the fluid motion decreases.

In this paper, a novel influenza $ \mathcal{S}\mathcal{I}_N\mathcal{I}_R\mathcal{R} $ model with white noise is investigated. According to the research, white noise has a significant impact on the disease. First, we explain that there is global existence and positivity to the solution. Then we show that the stochastic basic reproduction $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}} {_r} $ is a threshold that determines whether the disease is cured or persists. When the noise intensity is high, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} < 1 $ and the disease goes away; when the white noise intensity is low, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} > 1 $, and a sufficient condition for the existence of a stationary distribution is obtained, which suggests that the disease is still there. However, the main objective of the study is to produce a stochastic analogue of the deterministic model that we analyze using numerical simulations to get views on the infection dynamics in a stochastic environment that we can relate to the deterministic context.

In this paper, we apply the fractal-fractional derivative in the Atangana-Baleanu sense to a model of the human immunodeficiency virus infection of CD$ 4^{+} $ T-cells in the presence of a reverse transcriptase inhibitor, which occurs before the infected cell begins producing the virus. The existence and uniqueness results obtained by applying Banach-type and Leray-Schauder-type fixed-point theorems for the solution of the suggested model are established. Stability analysis in the context of Ulam's stability and its various types are investigated in order to ensure that a close exact solution exists. Additionally, the equilibrium points and their stability are analyzed by using the basic reproduction number. Three numerical algorithms are provided to illustrate the approximate solutions by using the Newton polynomial approach, the Adam-Bashforth method and the predictor-corrector technique, and a comparison between them is presented. Furthermore, we present the results of numerical simulations in the form of graphical figures corresponding to different fractal dimensions and fractional orders between zero and one. We analyze the behavior of the considered model for the provided values of input factors. As a result, the behavior of the system was predicted for various fractal dimensions and fractional orders, which revealed that slight changes in the fractal dimensions and fractional orders had no impact on the function's behavior in general but only occur in the numerical simulations.

Abstract Results reported in this paper establish the existence of solutions for a class of generalized fractional inclusions based on the Caputo–Hadamard jerk system. Under some inequalities between multi-functions and with the help of special contractions and admissible maps, we investigate the existence criteria. Fixed points and end points are key roles in this manuscript, and the approximate property for end points helps us to derive the desired result for existence theory. An example is prepared to demonstrate the consistency and correctness of analytical findings.

The influence of hybrid nano-fluid on heat transport in a semi-annular channel is investigated numerically. The hybrid nano-fluid is a liquid water with suspension of SWCNT and MWCNT. Constant heat fluxes are applied to the walls of channel. The finite volume approach is used to solve the governing mathematical equations. The volume percent of SWCNT is fixed at 0.01% and the volume fraction of MWCNT is varied from 0.01% to 0.1%. Heat transport is shown to diminish as the volume fraction of MWCNT increases. The heat transferred from the wall to the near fluid is influenced by the curvature of the walls. The results are displayed as velocity contours and isotherms. For certain volume fractions of nanoparticles, local Nusselt number distributions are given. It has been discovered that walls with a smaller curvature have a stronger convection heat transfer. Pressure is increased when the volume percentage of nanoparticles increases.

In this article, we consider a Covid-19 model for a population involving diabetics as a subclass in the fractal–fractional (FF) sense of derivative. The study includes: existence results, uniqueness, stability and numerical simulations. Existence results are studied with the help of fixed-point theory and applications. The numerical scheme of this paper is based upon the Lagrange’s interpolation polynomial and is tested for a particular case with numerical values from available open sources. The results are getting closer to the classical case for the orders reaching to 1 while all other solutions are different with the same behavior. As a result, the fractional order model gives more significant information about the case study.

In this article, we consider a fractal-fractional tobacco mathematical model with generalized kernels of Mittag-Leffler functions for qualitative and numerical studies. From qualitative point of view, our study includes; existence criteria, uniqueness of solution and Hyers-Ulam stability. For the numerical aspect, we utilize Lagrange's interpolation polynomial and obtain a numerical scheme which is further illustrated simulations. Lastly, a comparative analysis is presented for different fractal and fractional orders. The numerical results are divided into four figures based on different fractal and fractional orders. We have found that the fractional and fractal orders have a significant impact on the dynamical behaviour of the model.

Abstract A thermostat model described by a second-order fractional difference equation is proposed in this paper with one sensor and two sensors fractional boundary conditions depending on positive parameters by using the Lipschitz-type inequality. By means of well-known contraction mapping and the Brouwer fixed-point theorem, we provide new results on the existence and uniqueness of solutions. In this work by use of the Caputo fractional difference operator and Hyer–Ulam stability definitions we check the sufficient conditions and solution of the equations to be stable, while most researchers have examined the necessary conditions in different ways. Further, we also establish some results regarding Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability for our discrete fractional-order thermostat models. To support the theoretical results, we present suitable examples describing the thermostat models that are illustrated by graphical representation.

Abstract In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form.

In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.

In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu > 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.

Abstract In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q-differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution by means of the Schauder fixed point theorem. Then, the stability of solution will be discussed under the perturbations of boundary condition, a function existing in the problem, and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Abstract In this paper, we consider initial value problems for two different classes of implicit ϕ-Hilfer fractional pantograph differential equations. We use different approach that is based on α − ψ $\alpha -\psi $ -contraction mappings to demonstrate the existence and uniqueness of solutions for the proposed problems. The mappings are defined in appropriate cones of positive functions. The presented examples demonstrate the efficiency of the used method and the consistency of the proposed results.

Abstract In this paper, we introduce a new structure of the generalized multi-point thermostat control model motivated by its standard model. By presenting integral solution of this boundary problem, the existence property along with the uniqueness property are investigated by means of a special version of contractions named μ-φ-contractions and the Banach contraction principle. Then, on the given nonlinear generalized BVP of thermostat, the Bernstein polynomials are introduced and numerical solutions obtained by them are presented. At the end, three different structures of nonlinear thermostat models are designed and the results are examined.

Abstract The fundamental goal of the study under consideration is to establish some of the existence criteria needed for a particular fractional inclusion model of cantilever beam in the setting of quantum calculus using new arguments of existence theory. In this way, we investigate a fractional integral equation that corresponds to the aforementioned boundary value problem. In a more concrete sense, we design new multi-valued operators based on this integral equation, which belong to the certain subclasses of functions, called α-admissible and α-ψ-contractive multi-functions, in combination with the AEP-property. Also, we use some inequalities such as Ω-inequality and set-valued version inequalities. Moreover, we add a simulative example for a numerical analysis of our results obtained in this study.