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In this manuscript, we describe fractional differential equations with neutral, integral boundary conditions and mixed delay using Atangana–Baleanu derivatives, which include the generalized Mittag-Leffler kernel. We determine the existence and uniqueness of results and analysis by fixed point method. Moreover, we explained the stability of the fractional differential equation in the frame of Ulam–Hyers. Then, we investigate an example with various values and illustrate the outcomes graphically. • We consider the nonlinear fractional mixed integro-differential equations with fractional boundary conditions. • The existence result is obtained by using standard fixed point theory. • An example is given to illustrate our analytical and numerical results. By using MATLAB coding graph is drawn. [ABSTRACT FROM AUTHOR]

In this manuscript, we describe fractional differential equations with neutral, integral boundary conditions and mixed delay using Atangana–Baleanu derivatives, which include the generalized Mittag-Leffler kernel. We determine the existence and uniqueness of results and analysis by fixed point method. Moreover, we explained the stability of the fractional differential equation in the frame of Ulam–Hyers. Then, we investigate an example with various values and illustrate the outcomes graphically. • We consider the nonlinear fractional mixed integro-differential equations with fractional boundary conditions. • The existence result is obtained by using standard fixed point theory. • An example is given to illustrate our analytical and numerical results. By using MATLAB coding graph is drawn. [ABSTRACT FROM AUTHOR]

In this research communication the Boussinesq like equation namely 2 + 1 − dimensional Wazwaz Kaur Boussinesq equation is studied and gotten their Jacobi elliptic function (doubly periodic) solutions employing the simple algebraic Jacobi elliptic function method. The said method makes the two transformations of nonlinear partial differential equations to ordinary differential equations to solvable system of algebraic equations. The gotten doubly periodic solutions are modular transformed to show the degeneration into non-topological solitons, compound non-topological singular solitons and complex singular solitons. The three dimensional surfaces are drawn for few elliptic solutions and solitons. • The (2+1) dimensional Wazwaz Kaur Boussinesq equation is studied. • By employing Jacobi elliptic function method doubly periodic solutions are obtained. • The degeneration of Jacobi elliptic solutions to non-topological soliton is shown. [ABSTRACT FROM AUTHOR]

Smoking is now regarded as an epidemic because of the excessive mortality rate and large costs related to it. In line with the arena of the health enterprize, smoking is the third most common cause of mortality among human beings and the most preventable cause of sickness. Intending to study this problem, this manuscript frequently stresses the usage of the hybrid fractional differential operator to evaluate the dynamics of the fractional order quitting smoking version. We study a fractional-order version with a Constant Proportional-Caputo Fabrizio (CPCF) operator to investigate and examine the dynamical transmission of smoking. The qualitative analysis, and well-posedness of the fractional order system are examined. The analysis of the Volterra-type Lyapunov function for global stability is verified with the first and derivative tests. Also, Banach contraction characteristics are used to fulfil the criteria for the uniqueness of the exact solution. A complete discussion of additional analysis of the CPCF operator is furnished. In order to demonstrate the impacts of modifying the fractional order and to validate the theoretical findings, numerical simulations are carried out using an iterative Laplace transform method for a variety of fractional orders. The stability of the iterative scheme is also verified by using Picard's stable condition from the fixed point theorem. The CPCF operator has a broader definition than the Caputo–Fabrizio (CF) operator, and it elaborates on new insights into the mechanisms of smoking transmission throughout society. • A fractional order proportional-Caputo Fabrizio operator is used. • The qualitative analysis and well-posedness of the fractional system are studied. • Analysis of equilibrium gives the uniqueness and existence of the proposed model. • The stability of the iterative scheme is verified by using Picard's stable condition. [ABSTRACT FROM AUTHOR]

• A non-linear Hilfer system has been studied for mild solution with the aid of Banach contraction principle. • Mönch fixed point theorem ensures the exact controllability of the system concerned. • Existence of optimal pair verified for the Hilfer nonlinear system. • Some numerical approaches have been done to exemplify the criteria of obtained results using the software 'mathematica'. • Comparison analysis was done with the major classifications of fractional systems. In this article, the controllability results of the non-dense Hilfer neutral fractional derivative (HNFD) are presented. The results are acknowledged using semigroup theory, fractional calculus, Banach contraction principle, and Mönch technique. Moreover, a numerical analysis is given to enhance our model. [ABSTRACT FROM AUTHOR]

The present study is influenced by the wide applications of the Schrödinger equations. Its occurrence can be easily seen in electromagnetic wave propagation, quantum mechanics, plasma physics, nonlinear optics, underwater acoustics, etc. Solving equations of this type is always difficult. In the current paper, we have discussed a very easy numerical technique which is also known as the Kansa method along with polyharmonic radial basis function for the numerical study of generalized 2-D nonlinear Schrödinger equations. The stability analysis of the present method is discussed. The efficiency and accuracy of the present method are demonstrated by considering three numerical cases along with different types of boundary conditions. [ABSTRACT FROM AUTHOR]

Diabetes is a silent illness that is endangering public health in society. Diabetes is a chronic disease affecting millions of people worldwide, and understanding the underlying mechanisms of glucose homeostasis is crucial for managing this condition. Diabetes is a significant public health issue due to the early morbidity, mortality, shortened life expectancy, and financial and other expenses to the patient, their careers, and the health care system. In this study, we propose a mathematical model consisting of β − cells, insulin, glucose, and growth hormone that incorporates the fractional operator. Using the Lyapunov function, we treated a global stability analysis and investigated the impact of a new wave of dynamical transmission on the equilibrium points of the second derivative. With the Lipschitz criteria and linear growth, the exact singular solution for the proposed model has been determined. Furthermore, we present a detailed analysis of infections, and numerical simulations are conducted using the Mittag-Leffler Kernel mathematical framework to illustrate the theoretical conclusions for various orders of the fractional derivative. Controllability and observability of the linear system are treated for close loop design to check the relation between the glucose and insulin systems. Overall, our results provide a comprehensive understanding of glucose homeostasis and its underlying mechanisms, contributing to the development of effective diabetes management strategies. The proposed model and mathematical framework offer a valuable tool for investigating complex systems and phenomena, with applications beyond the field of diabetes research and helpful to designing the closed loop for the glucose–insulin system. [ABSTRACT FROM AUTHOR]

The new idea of Atangana and Baleanu introduced recently for fractional derivatives has received tremendous attention from researchers of various fields. However, this idea has not been applied for heat dissipation in transmission line of electrical circuit. Although the phenomenon of heat dissipation potentially damages the electrical devices which leads to catastrophic failure and shortens the life expectancy of costly electrical components. This manuscript signifies an electro-analog model of fractional diffusion process which enables to simulate heat dissipation in circuit board. The modeling of the problem has been established through modern operators of fractional calculus via the governing differential equation. By invoking Laplace transform on fractional governing equation of heat dissipation in transmission line of circuit, the analytic solutions are investigated through the exponential and Mittage Leffler kernel of non-integer order differentiations. The duality of analytic solutions has been compared with Caputo–Fabrizio and Atangana–Baleanu fractional differentiations graphically using embedded pertinent parameters. The comparison of both approaches of fractional differentiations has disclosed the heat transfer of circuit board and diffusion process with dissipation vividly. [ABSTRACT FROM AUTHOR]

In this study, we formulated and analyzed a deterministic mathematical model for the co-infection of COVID-19 and tuberculosis, to study the co-dynamics and impact of each disease in a given population. Using each disease's corresponding reproduction number, the existence and stability of the disease-free equilibrium were established. When the respective threshold quantities R C , and R T are below unity, the COVID-19 and TB-free equilibrium are said to be locally asymptotically stable. The impact of vaccine (i.e., efficacy and vaccinated proportion) and the condition required for COVID-19 eradication was examined. Furthermore, the presence of the endemic equilibria of the sub-models is analyzed and the criteria for the phenomenon of backward bifurcation of the COVID-19 sub-model are presented. To better understand how each disease condition impacts the dynamics behavior of the other, we investigate the invasion criterion of each disease by computing the threshold quantity known as the invasion reproduction number. We perform a numerical simulation to investigate the impact of threshold quantities (R C , R T) with respect to their invasion reproduction number, co-infection transmission rate (β c t) , and each disease transmission rate (β c , β t) on disease dynamics. The outcomes established the necessity for the coexistence or elimination of both diseases from the communities. Overall, our findings imply that while COVID-19 incidence decreases with co-infection prevalence, the burden of tuberculosis on the human population increases. [ABSTRACT FROM AUTHOR]

The yellow virus in the red chili fractional order model is investigated in this scientific study. The Caputo–Fabrizio and Fractal fractional derivative operator, which incorporates an antiretroviral treatment compartment, are used to investigates this pandemic occurrence. It is essential to figure out how to develop methods to halt the spread of the yellow virus in red chili. While measures are being taken to curb the pandemic of the yellow virus, the more contagious yellow virus found in red chilies is emerging in several areas. It is essential to develop methods for preventing the spread of the yellow virus. To maintain a certain level of protection while simulating the yellow virus's spread in red chili plants. We investigated the potential for an epidemic in red chili plants as a case study. In this study, we use fixed point theory to create existence, uniqueness, and stability conditions for the fractional order yellow virus in the red chilli model using the Caputo Fabrizio operator. For these kinds of biological models, the fractional derivative technique is quite new. By using the iterative Laplace transform approach, we have also discovered the initial approximate solutions for a suggested model that are readily available. This method combines the Laplace transform method with one of the most dependable methods, the new iterative method. Finally, we analysed parameters that represent the progression of sickness and used charts to display the numerical simulations. [ABSTRACT FROM AUTHOR]

This study is devoted to developing mathematical models associated with the Helmholtz equation as a second-order oscillator involved with nonlinear Caputo fractional difference equations. This study also focuses on determining the approximate solution of this model via the Ulam stability conception. Some properties of the mathematical model dealt with in this study are also presented. Numerical simulations are presented to justify the existence of stability results. • A new formulation of the mathematical model and analysis in Helmholtz Duffing oscillator with the damping term is presented. on the natural frequencies. • Using this Helmholtz–Duffing oscillator with the damping term, we proved the Hyers–Ulam stability to ensure existence and uniqueness. • We analyzed practical examples describing the application of quadcopter. [ABSTRACT FROM AUTHOR]

In this paper, we examine the multi-derivative nonlinear fractional integro-differential equations involving Atangana-Baleanu fractional derivative of Riemann-Liouville sense. We study the elementary results about the existence and difference solution on different data, based on T. R. Prabhakar fractional integral operator ε σ , η , ϑ ; c + γ involving generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and Gronwall-Bellman inequality. 2010 Subject Classification: 34A60,33E12,34G20. • The existence outcomes for fractional multi-derivative nonlinear integrodifferential equations involving A-B fractional derivative of R-L sense. • We study the elementary results about the existence and difference solution on different data, based on T. R. Prabhakar fractional integral operator. • The results are obtained by using Krasnoselskii's fixed point theorem and Gronwall-Bellman inequality. [ABSTRACT FROM AUTHOR]

In this article, we mainly focus on the existence and approximate controllability results for the fractional semilinear impulsive control system of order r ∈ (1 , 2). We consider two different sets of sufficient conditions. In the first set, we derive the results by using the theories on the fractional calculus, compactness of the cosine family, and Schauder's fixed point theorem. In the second set, we prove the main results by using Gronwall's inequality, avoids the usage of the compactness of cosine family and fixed point theorems. By introducing the suitable assumptions, we discuss the existence and uniqueness of mild solutions for the fractional semilinear impulsive system. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion. [ABSTRACT FROM AUTHOR]

In this manuscript, we mainly focus on the approximate controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order 1 < r < 2. The primary outcomes of our manuscript are proved by using the results and facts associated with fractional calculus, multivalued maps, and Bohnenblust and Karlin's fixed point theorem. Firstly, we focus the approximate controllability and then we extend the discussion to the system with nonlocal conditions. Lastly, we provide an application for the illustration of the obtained theoretical results. • Approximate controllability of fractional evolution inclusions of order 1 < r < 2 are proved. • The results are established by using the results on fractional calculus and multivalued maps. • We use Bohnenblust and Karlin's fixed point theorem to prove the main theorems. • Initially, we prove the approximate controllability of fractional evolution system. • Then, we extend the discussion to the system with nonlocal conditions. • Finally, we provide an application for the illustration of the obtained theory. [ABSTRACT FROM AUTHOR]

A rotating MHD flow of electrically conducting Oldroyd-B fluid through non-Darcy porous medium across a stretching/shrinking surface is investigated in purview of bioconvection effects. The fluid velocity field, temperature field, concentration of nano materials and that of bio microorganisms have been simultaneously formulated in the form of coupled nonlinear partial differential equations. The quest for improved thermal performance in practicable situations has motivated to utilize some real fluids like Oldroyd-B nanofluids with support from bioconvection effect to stabilize nanoparticle fraction. The zero normal flux of the nanoparticles and convective boundary condition are imposed. The substitution of appropriate similarity functions has resulted a set of ordinary differential equations (ODE). The sparse system of transformed ODEs was simulated numerically with bvp4c solver facilitated by MATLAB commercial software. Computations were executed for specific parameters to perceive the response of velocity, temperature, concentration of nanoparticles and concentration of microorganisms, skin friction coefficient and local heat transfer rate with variation of influential parameters. It is detected that fluid temperature profile rises when Deborah number, Biot number, parameters for buoyancy ratio, porosity, rotation, magnetic field strength, thermophoresis diffusion are enhanced. [ABSTRACT FROM AUTHOR]

• The neutral fractional delay system governed by nonlocal conditions is presented and proved. • Noncompact measure and Mönch's technique, fractional calculus techniques are used. • Established the sufficient conditions to confirm the existence of neutral delay differential system. • An illustration of derived results is given. The study of neutral fractional delay system governed by nonlocal conditions is presented and proved. With the aid of fractional theory, noncompact measure and Mönch's technique, we established some sufficient conditions to confirm the existence of neutral delay differential system. An illustration of derived results is also offered. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability of Sobolev-type fractional stochastic evolution hemivariational inequalities of order r ∈ (1 , 2). Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, and fixed point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study. • Approximate controllability of fractional hemivariational inequalities are studied. • Stochastic theory, multimaps, and fixed point approach are used in the main results. • We begin by proving the existence of mild solutions of the considered system. • Next, approximate controllability of fractional stochastic control system are proved. • At last, some examples are provided to define the primary results. [ABSTRACT FROM AUTHOR]

Studying the dynamical system that represents the transmission of an outbreak from prey to predatory is crucial and significant because the results may be used to highlight a variety of real-world issues. Infections and predator-prey interplay have been linked to provide a complicated cumulative influence as regulators of prey and predator size range in predator-prey ecology. This paper presents a novel implementation of intelligent computation to analyze the dynamics of a fractional epidemiological model with disease infection in both the population (FEM-DIBP) using supervised machine learning (SMLs) optimized with the Bayesian Regularization methodology (BRM). The FOLotkaVoltera solver based on Grunwald-Letnikov is used to build the dataset for the FEM-DIBP for observation approach of the SML model of the system. Also state variables trajectories and Lorenz curves are constructed using FOLorenz to analyze the dynamics of the epidemical model. The BRM learned SMLs models are used in the train, test, and validating procedures to find the FEM-DIBP solutions to various situations depending on changing epidemical factors. The effectiveness of SML in solving FEMDIBP is demonstrated by estimated regression measurements, error histogram visualizations, and mean-squared error analyses. The suitability of the created BRMNNs for such a simulated situation is demonstrated by the most appealing mathematical findings for A.E within the range of 10 − 2 to 10 − 6. • A novel implementation of intelligent computation to analyze the dynamics of a fractional epidemiological model • Used supervised machine learning (SMLs) optimized with the Bayesian Regularization methodology • The FOLotkaVoltera solver based on Grunwald-Letnikov is used to build the dataset. • State variables trajectories and Lorenz curves are constructed to analyze the dynamics of the epidemical model. [ABSTRACT FROM AUTHOR]

In the study, the sudden act of the cancer model was studied utilizing the fractional operator and its applications to discretize the conformable cancer model. A collection of nonlinear fractional differential equations make up the fractional-order model. We also look at the fractional-order model, which examines how chemotherapeutic attention medications, when combined, interact with the immune system and cancer cells. Both qualitative and quantitative research has been conducted on the cancer model. The proposed fractional-order cancer model's first and second derivatives of Lyapunov stability, positivity, and boundedness are all checked using nonlinear analysis. A generalized Mittag Leffler kernel is used through the ABC derivative, which is also used to check the dynamical behavior of the model. Also, utilizing alternative dimensions of fractional order and validating the effect of fractional parameters, a comparison was made to check the efficiency of the scheme for all compartments. Finally, the physiological results are analyzed and showing off is scooped up to demonstrate cancer disease and corroborate logical findings, which will be useful for future study and reducing cancer's social effect. [ABSTRACT FROM AUTHOR]

In this work, we analyse the study of neutral fractional differential equation in an arbitrary Hilbert space. An associated integral equation is studied and approximate integral equation is obtained. We demonstrate the existence and uniqueness of an approximate solution by using analytic semigroup theory and the Fixed point method. In the application part, we discuss the approximation and the convergence results for such an approximation. • We analyse the study of neutral fractional differential equation in an arbitrary Hilbert space. An associated integral equation is studied and approximate integral equation is obtained. • We demonstrate the existence and uniqueness of an approximate solution by using analytic semigroup theory and the fixed-point method. • In the application part, we discuss the approximation and the convergence results for such an approximation. [ABSTRACT FROM AUTHOR]

• Approximate controllability results of Atangana-Baleanu fractional neutral stochastic systems with infinite delay are studied. • By using multivalued maps, fractional calculus, and Bohnenblust-Karlin fixed point techniques, the main results are proved. • We begin by proving the existence of mild solutions of the considered system • Next, the approximate controllability of the fractional stochastic control system are proved. • Then the results extended to the theory of nonlocal conditions. • At last, an example is provided to define the primary results. In this paper, we investigate the approximate controllability results of Atangana-Baleanu fractional neutral stochastic systems with infinite delay. Using principles and ideas from stochastic analysis, the theory of multivalued maps, fractional calculus, and Bohnenblust-Karlin fixed point theorem, a new set of sufficient conditions are formulated and proved for the approximate controllability of the fractional stochastic control system. We then apply our findings to the theory of nonlocal conditions. Finally, an example is given to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a new set of sufficient conditions for the approximate controllability of fractional evolution stochastic integrodifferential delay inclusions of order r ∈ (1 , 2) with nonlocal conditions in Hilbert space. Martelli's fixed point theorem, multivalued functions, cosine and sine families, fractional calculus and operator semigroups are used to establish the results under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the applicability of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

• The basic motivation of our article is the approximate controllability issues of delay evolution equations for Sobolev type with order 1

• This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. • The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. • Initially, we discuss the approximate controllability of the fractional evolution system. • Then, we extend our results to the concept of nonlocal conditions. • Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion. This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. Initially, we discuss the approximate controllability of the fractional evolution system. Then, we extend our results to the concept of nonlocal conditions. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion. [ABSTRACT FROM AUTHOR]

In this paper, a new HIV CD4+T cells reaction-diffusion model in two dimensions has been introduced. Two novel and efficient positivity preserving finite difference schemes for the numerical solution has been used. The positivity property is of great importance in epidemic models because negative values have no meaning. The stability and consistency of the proposed positivity preserving schemes have been discussed briefly. A comparison of the proposed schemes with an extensively used Euler scheme has been provided. The numerical simulations of all the schemes have been presented with the help of a numerical test and found that the Euler method shows false behavior which is not a part of the continuous system. Moreover, both proposed positivity preserving schemes illustrate the behavior which is consistent with the continuous system. [ABSTRACT FROM AUTHOR]

In this article, we study an impulsive neutral fractional integro-differential equation (FIDE) via Atangana-Baleanu fractional derivative. The fixed point approach is employed to prove the existence results of the equation with initial and non-local conditions. The application of the results obtained are verified using suitable examples of different fractional orders. [ABSTRACT FROM AUTHOR]

• Fractional modified Kawahara model with Mittag–Leffler memory is discussed. • The solution of the fractional modified Kawahara model is derived by using HATM. • Existence and uniqueness analysis of the solution of the model is examined. • The numerical results are demonstrated in graphical forms. In this work, we study a fractional extension of modified Kawahara equation by using Atangana–Baleanu fractional operator in the sense of Caputo (ABC). The fractional modified Kawahara equation is very useful to describe plasma waves and capillary-gravity water waves. We show existence and uniqueness of the solution of fractional modified Kawahara equation by making use of the fixed-point theorem. We obtain the solution of the fractional modified Kawahara equation with aid of the homotopy analysis transform technique. The outcomes of the investigation are demonstrated in graphical and tabular forms to show the influence of order of ABC fractional operator and variables on the displacement profile. [ABSTRACT FROM AUTHOR]

• The Caputo–Fabrizio derivative is applied to solve a Brinkman nanoliquid (suspension of nanomaterials in a base fluid) problem. • Studied the effect of copper nanoparticles on the MHD free convection transient motion of C 6 H 9 NaO 7 Brinkman nanoliquid lies over a vertical surface with time-dependent velocity, temperature, and concentration. • The mathematical problem is modeled for nanofluid and generalized by using the Caputo–Fabrizio fractional derivatives and solved with the Laplace transform. • The effect of different parameters has been analyzed through graphs and variations in Nusselt number, skin friction and Sherwood number are shown in tabular form. Amongst the several definitions of fractional operator (memory operator), these days the Caputo–Fabrizio operator (CF) without singular kernel is a useful extension of the classical Caputo derivative operator as the latter includes a singular mathematical expression also known as the kernel in its definition, leading to some difficulties in finding solutions to the corresponding differential equation, whereas the kernel of the former (CF) is non-singular. In this paper, the idea of CF derivative has been applied to solve a Brinkman nanoliquid (suspension of nanomaterials in a base fluid) problem. More exactly, the present article studies the effect of copper nanoparticles on the MHD free convection transient motion of C 6 H 9 NaO 7 Brinkman nanoliquid lies over a vertical surface with time-dependent velocity, temperature, and concentration. Formulation of the problem in the fractional form (CF) is done and then closed-form solution is obtained. Graphs are drawn for different embedded parameters and variations in Nusselt number, skin friction and Sherwood number are shown in tabular form. C 6 H 9 NaO 7 is a food product taken from seaweed or brown algae. C 6 H 9 NaO 7 is used in the food industry for making gel-like foods such as pimento stuffing in prepared cocktail olives. In a liquid, C 6 H 9 NaO 7 also acts as a thickener. C 6 H 9 NaO 7 Brinkman nanoliquid in this direction (food technology) using CF derivative has not been investigated yet. This study will further provide new directions to carry this research using Atangana–Baleanu fractional derivatives for other non-Newtonian C 6 H 9 NaO 7 nanoliquids used in different foods. [ABSTRACT FROM AUTHOR]