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In this study, we proposed a family of m-point quaternary approximating subdivision schemes, characterized by an explicit formula involving three parameters. One of these parameters served as a shape control parameter, allowing for flexible curve design, while the other two parameters identify different members of the family and determined the smoothness of the resulting limit curves. We conducted a thorough analysis of the proposed schemes, covering their smoothness properties, polynomial generation, and reproduction capabilities. Additionally, we examined the behavior of the Gibbs phenomenon within the family both theoretically and graphically, highlighting the advantages of the proposed schemes in eliminating undesirable oscillations. A comparative study with existing subdivision schemes demonstrated the effectiveness and versatility of our approach. The results indicated that the proposed family offered enhanced smoothness and control, making it suitable for a wide range of applications in computer graphics and geometric modeling.

This research paper presents a comprehensive study on the intricate interplay of rotational, magnetic field and heat transfer effects in squeezing fluid flow, utilizing the foundational framework of Navier–Stokes equations. Fluid flow in confined geometries has significant applications in various industrial processes, including lubrication systems, microfluidics, and heat exchangers. The integration of rotational motion, magnetic fields, and heat transfer phenomena into this context holds immense potential for optimizing performance and efficiency. The study commences with the derivation and analysis of the coupled Navier–Stokes equations, incorporating the Coriolis force due to rotational motion, the Lorentz force arising from magnetic fields, and the energy equation for heat transfer. This comprehensive model accounts for the complex interactions between these physical phenomena, offering a holistic perspective on squeezing fluid flow dynamics. Numerical analysis are conducted to investigate the behavior of fluids under the influence of rotational, magnetic field, and heat transfer effects. The results reveal intricate flow patterns, including vortex formation and heat transfer enhancement. Furthermore, the impact of key parameters such as rotational speed, magnetic field strength, and temperature gradients on flow characteristics is systematically analyzed. The Dufour effect pertains to the transfer of energy resulting from a gradient in mass concentration, occurring as a correlated consequence of irreversible processes. It functions as the inverse of the Soret effect. The temperature changes due to the concentration gradient, amplifying as the Dufour number increases. The most significant temperature rise is observed at the midpoint of the fluid domain. Conversely, the Soret number behaves in a similar but opposite manner. Tables 4–10 has been created to offer a comprehensive analysis of the optimal alignment between Homotopy Analysis Method (HAM) and BVP4c. Specifically, it demonstrates a continuous decrease in skin friction for ξSo, ξRd, and ξR2, while showing an increase for ξSc, ξDo, and ξRem. Meanwhile, the skin friction remains constant for ξPr. The research centers on analyzing heat and mass transfer properties within an incompressible fluid confined between two rotating and stretching disks amidst an unpredictable flow pattern. The study employs a blend of analytical and numerical approaches to solve the governing equations, delving into the intricate fluid dynamics and the intricate processes of heat and mass transfer within this intricate system.

Abstract The nonlinear fractional Langevin equation is an important equation used in various areas of physics and mathematics. In this article, we focus on the singular-hybrid case of the generalized Langevin equations and prove the existence of a solution using fixed point theory. We also provide numerical and graphical simulations to demonstrate the convergence and existence of a solution for this singular problem. Finally, we present numerical examples in the form of tables and graphs to examine the accuracy of the method.

This research paper presents an in-depth investigation into the complex interactions governing stratified fluid flow in the presence of rotational and stretching phenomena, magnetic field influences, heat and mass transfer effects, and the incorporation of thermal and solutal-slips at fluid interfaces. Stratified fluid systems are commonly encountered in various natural and industrial processes, ranging from geophysical flows to chemical engineering applications, making a thorough understanding of these phenomena paramount.The study commences by formulating a comprehensive set of governing equations that account for the unique interplay of these multifaceted physical phenomena. Integrating thermal and solutal slip conditions at fluid interfaces introduces innovative prospects for precisely controlling and manipulating mass and heat transfer in these systems. This research paper extensively explores layered fluid dynamics, encompassing phenomena such as rotation and stretching, influences from magnetic fields, effects of heat and mass transfer, and the examination of thermal and solutal slip conditions. The insights obtained from this study not only enhance our comprehension of multi-physics fluid dynamics but also offer practical implications for streamlining processes across diverse fields. The principal findings and implications of this research can be summarized as follows: Various factors that affect fluid flow exhibit evident and discernible patterns. The mathematical model includes the Navier-Stokes equations with rotational and stretching terms, the Maxwell equations describing magnetic field effects, and the transport equations for heat and mass transfer. Additionally, thermal and solutal slip conditions at the fluid interfaces are incorporated to capture boundary layer dynamics.Numerical simulations are conducted to explore the intricate dynamics arising from this amalgamation of physical effects. The results reveal complex flow patterns, including stratification, boundary layer development, and vortical structures, offering insights into the behavior of fluids under the joint influence of rotational and stretching phenomena, magnetic fields, and thermal and solutal gradients. Furthermore, the impact of key parameters, such as rotation rate, stretching rate, magnetic field strength, and concentration gradients, is systematically analyzed to elucidate their effects on flow characteristics, heat and mass transfer rates, and concentration profiles within the stratified fluid layers. The Soret effect, or thermophoresis, causes nanoparticles to migrate away from hot regions due to fluid molecule flow and high energy levels. Elevating ξNt results in decreased heat and mass transfer. The Prandtl number shows minimal influence on heat and mass transfer when increased. The disk separation, and fluid viscosity, significantly affect the rotational spectrum. As the rotational parameter increases, heat and mass transfer decrease.

Abstract It observers that an object is submerged in a liquid, more pressure is applied to its bottom surface than its top surface, as a result pressure rises within depth of fluids. A buoyancy force is generated due to the pressure difference. In the current investigation, the mathematical formulation of bio-convective stagnation point flow of a radiative Maxwell fluid with multiple slip effect on an exponentially porous stretching surface is analyzed thoroughly. The buoyancy assisting and opposing conditions with magnetic field are discussed in the current investigation. Moreover, the energy and concertation equations are formulated by the utilization of Cattaneo–Christov theory and non-uniform heat source/sink. The flow model is developed in the form of partial derivatives, then use the similarity variables to convert the physical system into nonlinear ordinary derivative. The nonlinear system is tackled numerically with the help of Bvp4c approach on the MATLAB. The graphical upshots for various emerging parameter are observed with two aspects: buoyancy assisting $$\left( {\lambda > 0} \right)$$ λ > 0 and buoyancy opposing $$\left( {\lambda < 0} \right)$$ λ < 0 . The observation shows that the greater values of mixed convection parameter improves the fluid velocity for assisting flow $$\left( {\lambda > 0} \right)$$ λ > 0 , while declining trend is noticing for opposing flow situation $$\left( {\lambda > 0} \right)$$ λ > 0 . Further, it is worth noticing that stronger inputs of thermal and concentration relaxation parameter yield lesser thermal and concentration diffusivity, which reduces the temperature and nanoparticles concentration.

Abstract In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii’s fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results

Abstract The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ-Caputo fractional derivative with a modified argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ-Bielecki-type. Meanwhile, the fixed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ-Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the effectiveness of our main results for different cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems.

This study investigates the impact of the Stephan blowing and Cattaneo-Christov flux model on bioconvective flow of dusty hybrid nanofluid over a Riga plate in the presence of gyrotactic microorganisms and variable dust particles volume friction. Mass and heat phenomena are explored in the context of Stefan blowing impacts. The hybrid nanofluid consists of nanoparticles of MgO and Ag base fluid water. The Cattaneo-Christov mass and heat flux model has a significant impact on the bioconvective flow. The model predicts a slower temperature distribution and a higher concentration gradient than the traditional Fick's law and Fourier's law, respectively. The occurrence of gyrotactic microorganisms enhance the flow characteristics. This model is important for optimizing mass and heat transfer in a variety of engineering systems, including heating and cooling technologies, where effective thermal control is essential. It can be used by employing the regulated migration of microbes. By maximizing the removal of impurities, it helps with the design of sophisticated water purification systems in environmental engineering. The amalgamation of gyrotactic microorganisms and Stefan blowing effects augments the comprehension of intricate fluid dynamics, maybe resulting in advancements in microfluidic apparatuses and bio-inspired technologies. In order to transform the controlling PDEs into nonlinear ODEs, a new set of non-dimensional variables is used. The MATLAB (RKF-45th) technique is then used to resolve the ODEs numerically. As the Stephan blowing parameter (0.1≤Sb≤1.9) increases, the outcomes show that the flow distributions upsurge for both the dust and fluid phases, but the dust and fluid phase thermal distributions drop.

The Casson fluid flow over a stretching sheet is studied in this work in the presence of porous media with melting heat transfer and chemical reaction, combining nanoparticles of Ti4Al6V,AA7072 and AA7075 with a base fluid of sodium alginate (NaAlg). The porous medium Darcy Forchheimer effect is included in the momentum equation. Through the effects of melting, the process of heat transfer is described. In order to explore the features of heat transmission in the absenteeism of LTECs (local thermal equilibrium conditions), the current study employs a mathematical model that has been simplified. For both the solid and liquid phases, the LTNE model yields two different fundamental thermal gradients. This proposed model compares the YOM (Yamada-Ota model) and XM (Xue model), two well-known trihybrid nanofluid models, in terms of performance. After applying the proper transformations, modulated non-linear PDEs (partial differential equations) are simplified in ODEs (ordinary differential equations) and the bvp4c method is used to solve them numerically. A graphic discussion of the relevant factors' significance on the pertinent fields has been presented. It is observed that the Casson ternary hybrid nanofluid temperature and velocity distributions predominate at higher melting parameter. The solid phase's heat transport rate rises with an upsurge in the interphase heat transfer parameter.

Analytic solutions of nonlinear Volterra-Fredholm integral equations with generalized singular kernel arise in atomic scattering, electron emission, microscopy, radio astronomy, radar ranging, plasma diagnostics, and optical fiber evaluation and found a thorough effort among the scientific group. Due to the huge applications of such problems, its analytical solutions become a hot topic among researchers. In this framework, the current article addresses a novel, operational matrix-based iterative spectral scheme to examine the analytic solutions to such problems. First, the novel operational matrix is proposed to approximate the singular integral that appears in the problem, and then a fast scheme is proposed based on developed operational matrices. The well-known Picard iterative schemes are coupled to tackle the nonlinear term present in the problem. To show the reliability, efficiency, and appropriateness of the recommended method the generalized Abel’s, weekly singular integral equations and system are considered. The algorithm is consistent and mathematically validated through error-bound and convergence theorems. Finally, the efficiency of the method endorses that the proposed scheme is well-matched to deal with the solutions of other nonlinear physical mechanisms governed by integral equations. Moreover, it is noted that the convergence control parameters N and r are impacted significantly on the convergence of the proposed method.

The main objective of this article on Wireless sensor network of the Internet of Things (IoT). The wireless network, B bluetooth network, infrared network, and other networks are the key components of the Internet of Things (IoT). The major emphasis of this work was on the impact of Caputo-Fabrizio fractional derivative on worm propagation in heterogeneous susceptible-exposed-infected-recovered wireless IoT devices. We first determined the equilibrium points and fundamental reproduction number for the Caputo-Fabrizio HSEIR system, and then we discussed the stability of the system at the worm propagation equilibrium point. Using the Picard-Lindeof method, we determine the existence and unique solution for the fractional CF system of the heterogeneous SEIR model. Next, we use fixed point theory to judge the stability of the iterative method. We investigate the impact of the derivative order on the behaviour of the resultant functions and acquired numerical values by computing the model's findings for various fractional orders. In addition, we compute the integer-order model's results and contrast them with the results of the fractional-order model. We develop a periodically intermittent controller driven by white noise with the amazing benefits of reduced cost and more adaptable control technique to restrict the spread of worms in wireless IoT networks. To clearly define the conditions for stability in probability one, we employ the stochastic analysis technique. Our results show that the nonlinear worm propagation system may be stabilised by intermittent stochastic perturbation under the parameters of intermittent time linked to stochastic perturbation strength. Our theoretical conclusions may be used to analyse the observable processes of the worm, design countermeasures to prevent its spread, and evaluate the consequences of various system parameters.

Abstract Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler’s fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution’s existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example.

The physical phenomena of Darcy-Forchheimer flow of trihybrid nanofluid over a Riga plate with the influence of electrophoresis and thermophoresis on the particle deposition and non-uniform heat generation in a porous medium is discussed in this article. In a Marangoni convective flow, this study examined the impact of electrophoresis and thermophoresis on the rate at which aerosol particles deposited across a Riga plate. One of the most basic processes for moving tiny particles across a temperature gradient is known as thermophoresis particle deposition, and it is significant to both aero-solution and electrical engineering. Trihybrid nanofluid containing Cobalt iron oxide (COFe2O4), Manganese Zinc iron oxide (MnZnFeO4) and Molybdenum disulfides (MOS2) nanoparticles, and based fluid water is used. For the case of trihybrid nanoparticles, the Xue and Yamada-Ota nanofluid models have been expanded. The objective of this envisaged model is to compare the performance of two renowned ternary hybrid nanofluid models namely Xue and Yamada–Ota. The required similarity transformations are used to translate the set of governing equations into a collection of ODEs. The homotopy analysis method (HAM) is used to analytically solve these simplified equations. For the embedded non-dimensional parameters, the graphic exploration of the velocity, concentration and thermal fields is made. The study's primary outcomes are that as the Marangoni convection parameter is increased, the velocity profile is improved and the temperature and concentration distributions are lowered. The concentration profile decreases with increasing electrophoretic parameter, however the opposite behavior is seen with increasing electrophoretic parameter.

The current study examines the effect of heat generation on thermal Marangoni convective boundary layer flow of dusty trihybrid nanofluid across a flat surface with thermal radiation and non-linear mixed convection. We consider, two phase dusty liquid model with non-linear heat generation. The fluctuation of surface tension gradients leads to the discovery of Marangoni convection. It can be used for growing crystals, drying silicon wafers, stabilising soap films, and wielding. The main objective of this study is to ascertain the trihybrid nanofluid thermal mobility. A trihybrid nanofluid consisting of magnesium oxide (Mgo), titanium oxide (TiO2), silver (Ag) and water as the base fluid is used. This model can improve the efficiency and dependability of thermal systems in a variety of applications by helping to optimize heat transfer processes in materials processing, electronics cooling, and the creation of cutting-edge cooling technologies in energy systems. The existing PDEs are converted into nonlinear ODEs via similarity variables. The nonlinear ODEs are then solved numerically using shooting technique (RKF-45th approach). When the Marangoni convection parameter rises, higher surface tension gradients lead to stronger induced flows and more efficient heat transfer inside the liquid. As the temperature profiles of the dust and fluid phases drop, the distribution of these characteristics in the liquid becomes more homogeneous.

In this paper, we investigate the existence of a solution for the fractional q-integro-differential inclusion with new double sum and product boundary conditions. One of the most recent techniques of fixed point theory, i.e. endpoints property, and inequalities, plays a central role in proving the main results. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures. The paper ends with an example.

In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable.

In this paper, an analysis of a mathematical model of the coronavirus is carried out by using two fractal-fractional parameters. This dangerous virus infects a person through the mouth, eyes, nose or hands. This makes it so dangerous that no one can get rid of it. One of the main factors contributing to increasing infections of this deadly virus is crowding. We believe that it is necessary to model this effect mathematically to predict the possible outcomes. Hence, the study of neural network-based models related to the spread of this virus can yield new results. This paper also introduces the use of artificial neural networks (ANNs) to approximate the solutions, which is a significant contribution in this regard. We suggest employing this new method to solve a system of integral equations that explain the dynamics of infectious diseases instead of the classical numerical methods. Our study shows that, compared to the Adams-Bashforth algorithm, the ANN is a reliable candidate for solving the problems.

We aim to investigate an integro-differential inclusion using a novel computational approach in this research. The use of quantum calculus, and consequently the creation of discrete space, allows the computer and computational algorithms to solve our desired problem. Furthermore, to guarantee the existence of the solution, we use the endpoint property based on fixed point methods, which is one of the most recent techniques in fixed point theory. The above will show the novelty of our work, because most researchers use classical fixed point techniques in continuous space. Moreover, the sensitivity of the parameters involved in controlling the existence of the solution can be recognized from the heatmaps. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures in our examples that are presented at the end of the work.

This article exhibits a brief study of magneto ternary hybrid nanofluid flow with thermophoretic particle deposition and porous medium based on the extended version of model, known as the Yamada-Ota and Hamilton Crosser models. Thermophoretic particle deposition, which is important in both aero-solution and electrical engineering, is one of the most fundamental methods for transferring small particles across a heat gradient. This suggested model's goal is to evaluate how well Hamilton-Crosser and Yamada-Ota, two trihybrid nanofluid models, perform. Advanced heat transfer systems, in particular the creation of effective cooling and heating technologies, may benefit from this research. Researchers can develop more efficient heat exchangers, boost thermal management in electronic devices, and increase energy efficiency in industrial processes by better understanding how thermophoretic particles behave in complicated nanofluid systems under various conditions. By using the required similarity variables, the MATLAB solver bvp4c package handles the system of ODEs acquired from the leading PDEs to arrive at the numerical solution. The relevant parameters' effects on the relevant fields have been graphically explained. The results show that when inter-phase heat transfer parameter value grows, thermal profile and rate of heat transfer decrease of liquid phase. With regard to heat and mass transfer effectiveness, the Yamada-Ota model performs better than the Hamilton Crosser trihybrid nanofluid model.

A large number of physical phenomena can be described and modeled by differential equations. One of these famous models is related to the pantograph, which has been investigated in the history of mathematics and physics with different approaches. Optimizing the parameters involved in the pantograph is very important due to the task of converting the type of electric current in the relevant circuit. For this reason, it is very important to use fractional operators in its modeling. In this work, we will investigate the existence of the solution for the fractional pantograph equation by using a new $ \psi $-Caputo operator. The novelty of this work, in addition to the $ \psi $-Caputo fractional operator, is the use of topological degree theory and numerical results from simulations. Techniques in fixed point theory and the use of inequalities will also help to prove the main results. Finally, we provide two examples with some graphical and numerical simulations to make our results more objective. Our data indicate that the boundedness of the solution set for the desired problem depends on the choice of the $ \psi(\kappa) $ function.

In this paper, we investigate a multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains $ (a, \infty), a\geq 0 $. The existence and uniqueness of solution are established for a such problem by utilizing the Banach fixed point theorem in an applicable Banach space. In addition, stability of the types Ulam-Hyers ($ \mathcal UH $), Ulam-Hyers-Rassias ($ \mathcal UHR $) and semi-Ulam-Hyers-Rassias (s-$ \mathcal UHR $) are discussed by using nonlinear analysis topics. Finally, a concrete example includes some particular cases is enhanced to illustrate rightful of our results.

The World Health Organization declared COVID-19 a global pandemic in March 2020, which had a significant impact on global health and economies. There have been several Zika outbreaks in different regions such as Africa, Southeast Asia, and the Americas. Therefore, it is essential to study the dynamics of these two diseases, taking into account their memory and recurrence effects. A new fractal-fractional hybrid Mittag-Leffler model of COVID-19 and Zika co-dynamics is designed and studied to evaluate the effects of COVID-19 on Zika and vice-versa. The stability analysis of the local asymptotic type at disease-free equilibrium is conducted for the hybrid model. The existence of unique solutions to the model is established via some fixed point results. The fractal-fractional model is proved to be Hyers–Ulam stable. With the help of Newton polynomials, we obtain some numerical algorithms to approximate the solutions of the fractal-fractional hybrid Mittag-Leffler model graphically. The impact of fractional and fractal orders on the dynamics of each of the epidemiological classes is also assessed. In addition, empirical evidence from numerical simulations suggests that implementing measures to contain the transmission of the SARS-CoV-2 virus can significantly contribute to the reduction of co-infections involving the Zika virus. Therefore, it is imperative for healthcare systems to maintain a state of constant vigilance in order to detect any atypical patterns or probable occurrences of co-infections, particularly in areas where both diseases are widespread. Additionally, it is vital to consult the most recent directives provided by health authorities, as our comprehension of diseases may undergo advancements over the course of time.

The present study aims to consider a mathematical eco-epidemiological model involving two fractional operators. To this end, we provide approximate solutions to these fractional systems through the application of a numerical technique that is based on the rule of product integration. This feature contributes greatly to the efficiency and effectiveness of both methods. We have also presented some theoretical discussions related to the equilibrium points of the system. Further, several numerical simulations are presented in order to illustrate the impact of choosing different parameters on the dynamics of the model. It is demonstrated that the obtained numerical results are completely consistent with the expected theoretical results. Moreover, both techniques can be used to solve other problems in epidemiology and describe other problems in the future. The article's model has never been studied via the employed fractional operators, and this is a distinct point for our work and other existing research.

Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective.

Abstract In this manuscript, we study the existence and uniqueness of solutions for a new neutral hybrid nonlinear differential equation in the context of a fractional generalized operator in the sense of ψ-Caputo. To emphasize the novelty of the manuscript, a pure technique of the noncompactness measures is applied to a hybrid system based on the notion of the modulus of continuity in Darbo’s criterion that covers the existing results of other works published before. The Ulam–Hyers and generalized Ulam–Hyers stabilities are explored for the given neutral nonhybrid nonlinear problem. An application is prepared in the framework of an example to ensure the validity of theorems for different cases.

In computational mathematics, the comparison of convergence rate in different iterative methods is an important concept from theoretical point of view. The importance of this comparison is relevant for researchers who want to discover which one of these iterations converges to the fixed point more rapidly. In this article, we study the different numerical methods to calculate fixed point in digital metric spaces, introduce a new k-step iterative process and conduct an analysis on the strong convergence, stability and data dependence of the mentioned scheme. Some illustrative examples are given to show that this iteration process converges faster.

We prove existence and uniqueness of solutions to discrete fractional equations that involve Riemann-Liouville and Caputo fractional derivatives with three-point boundary conditions. The results are obtained by conducting an analysis via the Banach principle and the Brouwer fixed point criterion. Moreover, we prove stability, including Hyers-Ulam and Hyers-Ulam-Rassias type results. Finally, some numerical models are provided to illustrate and validate the theoretical results.

In this paper, for the first time, we study the existence and uniqueness of solutions of a Caputo variable-order initial value problem (IVP) in the impulsive settings. Our existence results are proved by using two fixed point theorems. The Ulam-Hyers stability of solutions is established for the variable order impulsive initial value problem. Finally, we provide an example to show the correctness of the results.

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.

Malaria disease, which is of parasitic origin, has always been one of the challenges for human societies in areas with poor sanitation. The lack of proper distribution of drugs and lack of awareness of people in such environments cause us to see many deaths every year, especially in children under the age of five. Due to the importance of this issue, in this paper, a new five-compartmental $ (c_1, c_2) $-fractal-fractional $ \mathcal{SIR} $-$ \mathcal{SI} $-model of malaria disease for humans and mosquitoes is presented. We use the generalized Mittag-Leffler fractal-fractional derivatives to design such a mathematical model. In different ways, we study all theoretical aspects of solutions such as the existence, uniqueness and stability. A Newton polynomial that works in fractal-fractional settings is shown, which allows us to get some numerical trajectories. From the trajectories, we saw that an increase in antimalarial treatment in consideration to memory effects reduces the peak of sick individuals, and mosquito insecticide spraying minimizes the disease burden in all compartments.

This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.

This paper reports a modified F-iterative process for finding the fixed points of three generalized α-nonexpansive mappings. We assume certain assumptions to establish the weak and strong convergence of the scheme in the context of a Banach space. We suggest a numerical example of generalized α-nonexpansive mappings which exceeds, properly, the category of functions furnished with a condition (C). After that, we show that our modified F-iterative scheme of this example converges to a common fixed point of three generalized α-nonexpansive mappings. As an application of our main findings, we suggest a new projection-type iterative scheme to solve variational inequality problems in the setting of generalized α-nonexpansive mappings. The main finding of the paper is new and extends many known results of the literature.

In the present paper, we analyze a coupled system of nonlinear three point boundary value problems (BVPs) consisting of a coupled system of higher order hybrid sequential differential equations formulated by fractional operators. We study several new conditions in the direction of existence theory for solutions of the given hybrid system under the weaker hypotheses based on the method of measure of non-compactness in degree theory. The required conditions confirming the uniqueness and stability are derived in relation to the coupled system of hybrid sequential BVPs. Lastly, we extend our hybrid sequential system to a real mathematical model of typhoid treatment with four compartments and by deriving numerical schemes for this generalized hybrid sequential system, we analyze the variations of different fractional orders and their impact on dynamical behaviors of the mentioned hybrid system of typhoid model.

Abstract In this paper, we conduct a research on a new version of the ( p , q ) $( p,q ) $ -Hermite–Hadamard inequality for convex functions in the framework of postquantum calculus. Moreover, we derive several estimates for ( p , q ) $(p,q)$ -midpoint and ( p , q ) $(p,q)$ -trapezoidal inequalities for special ( p , q ) $( p,q ) $ -differentiable functions by using the notions of left and right ( p , q ) $(p,q ) $ -derivatives. Our newly obtained inequalities are extensions of some existing inequalities in other studies. Lastly, we consider some mathematical examples for some ( p , q ) $(p,q)$ -functions to confirm the correctness of newly established results.

One of the most famous equations that are widely used in various branches of physics, mathematics, financial markets, etc. is the Langevin equation. In this work, we investigate the existence of the solution for two generalized fractional hybrid Langevin equations under different boundary conditions. For this purpose, the problem of the existence of a solution will become the problem of finding a fixed point for an operator defined in the Banach space. To achieve the result, one of the recent fixed point techniques, namely the α-ψ-contraction technique, will be used. We provide sufficient conditions to use this type of contraction in our main theorems. In the calculations of the auxiliary lemmas that we present, the Mittag-Leffler function plays a fundamental role. The fractional derivative operators used are of the Caputo type. Two examples are provided to demonstrate the validity of the obtained theorems. Also, some figures and a table are presented to illustrate the results.

This manuscript studies the existence and Ulam–Hyers stability results of solutions of q−fractional jerk type problem. 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. Also the stability in the sense of Ulam–Hyers stability is defined and studied. In conclusion, we display an case to appear the pertinence of the most comes about.

The goal of this research is to develop a novel analytic technique for obtaining the approximate and exact solutions of the Caputo time-fractional partial differential equations (PDEs) with variable coefficients. We call this technique as the Aboodh residual power series method (ARPSM), because it apply the Aboodh transform along with the residual power series method (RPSM). It is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates the computation of the fractional derivatives each time. As ARPSM just requires the idea of an infinite limit, we simply need a few computations to get the coefficients. This technique solves nonlinear problems without the He's polynomials and Adomian polynomials, so the small size of computation of this technique is the strength of the scheme, which is an advantage over the homotopy perturbation method and the Adomian decomposition method. The absolute and relative errors of five linear and non-linear problems are numerically examined to determine the efficacy and accuracy of ARPSM for time-fractional PDEs with variable coefficients. In addition, numerical results are also compared with other methods such as the RPSM and the natural transform decomposition method (NTDM). Some graphs are also plotted for various values of fractional orders. The results show that our technique is easy to use, accurate, and effective. Mathematica software is used to calculate the numerical and symbolic quantities in the paper.

Abstract In this paper, we study a new class of non-hybrid single-valued fractional boundary value problems equipped with integro-non-hybrid-multiterm-multipoint-multistrip conditions and a fully hybrid integro-multi-valued fractional boundary value problem by some new methods including the Kuratowski measures based on Sadovskii’s theorem, Krasnoselskii–Zabreiko criterion, and Dhage’s technique. We generalize the Gronwall inequality in relation to a non-hybrid single-valued fractional boundary value problem, and then we investigate the stability notions in two versions. To examine the correctness of the results, we provide some examples.

This paper discusses the existence, uniqueness and stability of solutions for a nonlinear fractional differential system consisting of a nonlinear Caputo-Hadamard fractional initial value problem (FIVP). By using some properties of the modified Laplace transform, we derive an equivalent Hadamard integral equation with respect to one-parametric and two-parametric Mittag-Leffer functions. The Banach contraction principle is used to give the existence of the corresponding solution and its uniqueness. Then, based on a Lyapunov-like function and a K-class function, the generalized Mittag-Leffler stability is discussed to solve a nonlinear Caputo-Hadamard FIVP. The findings are validated by giving an example.

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.

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.

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.

In this paper, we study a general system of fractional hybrid differential equations with a nonlinear $ \phi_p $-operator, and prove the existence of solution, uniqueness of solution and Hyers-Ulam stability. We use the Caputo fractional derivative in this system so that our system is more general and complex than other nonlinear systems studied before. To establish the results, Green functions are used to transform the considered hybrid boundary problem into a system of fractional integral equations. Then, with the help of the topological degree theorem, we derive some sufficient conditions that ensure the existence and uniqueness of solutions for the proposed system. Finally, an example is presented to show the validity and correctness of the obtained results.

Abstract The approximate controllability of second-order integro-differential evolution control systems using resolvent operators is the focus of this work. We analyze approximate controllability outcomes by referring to fractional theories, resolvent operators, semigroup theory, Gronwall’s inequality, and Lipschitz condition. The article avoids the use of well-known fixed point theorem approaches. We have also included one example of theoretical consequences that has been validated.

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.