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In this study, we investigate the effect of Clarke's subdifferential type on the optimal control results for fractional differential systems of order 1

The existence and uniqueness of solutions to Hilfer fractional neutral delay integro‐differential equations subject to nonlocal conditions are discussed in this study. The main results of this study are based in part on the fixed point techniques of Banach contraction and Krasnoselskii's fixed point theorem from calculus theory. First, we determine whether or not the fractional system has a mild solution. The uniqueness of the mild solution is further illustrated by expanding our results. The optimal control problems are governed by a new class of neutral delay integro‐differential equations in Banach spaces, and we also develop sufficient conditions for the approximate controllability of a nonlinear fractional system. An example is given at the end to strengthen the compatibility of the results. [ABSTRACT FROM AUTHOR]

In this paper, the possibility of existing a novel class of compact charged spheres based on a charged perfect fluid within the realm of f(Q)$f(Q)$ gravity theory is explored. The authors started by proposing physically meaningful explicit formulas for the potential, denoted λ(r)$\lambda (r)$, and the electric field to find a close‐form solution. More precisely, the change of the dependent variable approach by exploiting the transformation Y(r)=eζ(r)/2$Y(r)=e^{\zeta (r)/2}$ is applied. Successively, the field equations analytically are solved and generate the most general solution, which leads us to examine various significant aspects of the stellar system. These aspects comprise the regularity of gravitational potentials, energy density and pressure, electric charge, the mass‐radius relationship, subluminal sound velocities in the radial direction, and the adiabatic index for charged compact stars. For a more in‐depth system study, mass measurements using contour diagrams are carried out. This mainly involves varying the variable parameters β1$\beta _1$ and E0$E_0$ to distinguish their effect on the mass distribution within the stellar structure. What is more, the electric charge controls the stability of the stellar system is shown, which yields that a stable system can possess a maximum charge of order 1020C$10^{20}\nobreakspace \text{C}$. The results strongly argue that charged stars could conceivably exist in nature and that such a deviation from traditional theories may be seen in future astrophysical observations. In this paper, the possibility of existing a novel class of compact charged spheres based on a charged perfect fluid within the realm of f(Q)$f\!(\mathcal{Q})$ gravity theory is explored. The authors started by proposing physically meaningful explicit formulas for the potential, denoted λ$\lambda$(r), and the electric field to find a close‐form solution. More precisely, the change of the dependent variable approach by exploiting the transformation Y(r) = eζ(r)∕2 is applied. Successively, the field equations analytically are solved and generate the most general solution, which leads us to examine various significant aspects of the stellar system. These aspects comprise the regularity of gravitational potentials, energy density and pressure, electric charge, the mass‐radius relationship, subluminal sound velocities in the radial direction, and the adiabatic index for charged compact stars. For a more in‐depth system study, mass measurements using contour diagrams are carried out. This mainly involves varying the variable parameters β1 and E0 to distinguish their effect on the mass distribution within the stellar structure. What is more, the electric charge controls the stability of the stellar system is shown, which yields that a stable system can possess a maximum charge of order 1020 C. The results strongly argue that charged stars could conceivably exist in nature and that such a deviation from traditional theories may be seen in future astrophysical observations. [ABSTRACT FROM AUTHOR]

This study concentrates on time fractional convection–diffusion equations (TFCDEs) with variable coefficients and their numerical solutions. Caputo derivative is used to calculate the time fractional order derivatives. In order to give an approximate solution to the TFCDE, an effective approach is proposed utilizing Fibonacci wavelet and block pulse functions. The Fibonacci wavelets operational matrices of fractional order integration are constructed. By combining the collocation technique, they are used to simplify the fractional model to a collection of algebraic equations. The suggested approach is quite practical for resolving issues of this nature. The comparison and analysis with other approaches demonstrate the effectiveness and precision of the suggested approach. [ABSTRACT FROM AUTHOR]

In the present article, the intelligence based numerical investigation by Levenberg–Marquardt algorithm with Back‐propagated Artificial Neural Networks (LMA–BANN) is exploited to analyze the nonlinear radiative stagnation point flow of cross nanofluid system (NRS‐CNFS) past a stretching surface. The designed NRS‐CNFS, initially represented by system of partial differential equations (PDEs), is converted into system of non‐linear ODEs through the applicability of mathematical conversion analysis. The desired reference solution in the form of a dataset for LMA–BANN is achieved from the Adam method for NRS‐CNFS's different scenarios by varying the magnetic parameter (Ha), Prandlt number (Pr), thermal radiation parameter(R), Eckert number (Ec), thermophoresis diffusion coefficients (Nt), and non‐dimensional activation energy (E). The obtained results are construed for the NRS‐CNFS through the performances of the testing, validation, and training. In addition, the comparison is provided using the LMA ‐BANN analysis, which is validated in terms of regression, fitness estimations, and histograms along with MSE performances. Hence, the article provides an impeccable innovative approach to problem‐solving by implementing a soft computation paradigm. It promotes the use of an effective and dependable alternative framework that is based on soft computing environments and emphasizes descriptive analysis to successfully handle the problems brought on by varied physical features. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on approximate controllability results for fractional Sobolev type Volterra‐Fredholm integro‐differential inclusions of order 1 < r < 2. By applying the results and ideas belongs to the cosine function of operators, fractional calculus and fixed point approach, the main results are established. Initially, we establish the approximate controllability of the considered fractional system, then continue to examine the system with the concept of nonlocal conditions. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

This paper explains the importance and benefits of using AI in fluid mechanics, emphasizing its capability of finding the solution of fluid flow systems through iterative optimization techniques. It explores how AI can enhance the understanding of fluid dynamics to improve engineering processes. In the presented studies, the heat and mass transport of the Casson nanofluidic flow model on a nonlinear slanted extending sheet (HMT‐CNFM) via AI‐based Levenberg Marquard methodology with backpropagated trained neural networks (TNN‐BLMM) is studied. By applying an appropriate transformation, the governing PDEs representing HMT‐CNFM are converted into a system of nonlinear ODEs. The dataset for Levenberg Marquard methodology with backpropagated trained neural network (TNN‐BLMM) for all six scenarios of this proposed model via computational power of the Lobatto IIIA scheme using the "bvp4c" package in MATLAB and then graphically illustrate all these six scenarios through nftool to attain mean square error, regression, error histogram, performance, and fit curve. Training, testing, and validation processes of NN‐BLMM are organized for the investigation of the HMT‐CNFM model. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order 1 < r < 2. By applying the results and facts belongs to the cosine function of operators, fractional calculus, the measure of noncompactness and fixed point approach, the main results are established. Initially, we focus the existence of mild solution, and later we establish the controllability of the considered fractional system. Finally, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In our paper, we primarily concentrate on the existence and controllability results for fractional evolution inclusions of order 1 < r < 2 with Clarke's subdifferential type. By applying the facts related to the measure of noncompactness, fractional calculus, and fixed‐point technique, the principal results are proved. Initially, we are concentrating the existence and continue to prove the controllability of the fractional evolution system. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

In our article, we primarily concentrate on the approximate controllability results for Sobolev type fractional stochastic evolution hemivariational inequalities. By applying the facts related to fractional calculus, and fixed‐point technique, the principal results are proved. Initially, we are concentrating the existence and continue to prove the controllability of the fractional evolution system. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

The fluid flow between the rotating disk has wide range of applications and are used in medical equipment's, car brake system, gas turbines, manufacturing of glass and plastic films. Inspired from these applications, we scrutinized the behavior of an axisymmetric expansion–contraction rotating disk on three‐dimensional flow of hybrid nanoparticles in the presence of microorganisms. Here, we considered copper and silica nanoparticles suspended in base fluid water. It is known that, the inequality of concentration in chemically reactive species affecting mass transfer mechanism and activation energy plays an important role in increasing the production speed of chemical reactions. Hence, the Arrhenius activation energy and chemical reaction effects are added in the mass equation. Autorelated transformations were adopted to reduce PDE's to ODE's and then, we utilized the RKF‐45 by adopting shooting skill to solve reduced ODE's. The analysis of rate of heat and mass transfer is done through graphs. Furthermore, change in velocity, thermal and concentration profiles for various nondimensional parameters are deliberated briefly by using suitable plots. Result reveals that rise in wall expansion/contraction parameter supports up the velocity gradients, higher the values of bioconvection Peclet number and concentration difference parameter declines the microorganism profile and Nusselt number decreases for rise in values of thermophoresis parameter. [ABSTRACT FROM AUTHOR]

In the Nidovirales order of the Coronaviridae family, where the coronavirus (crown‐like spikes on the surface of the virus) causing severe infections like acute lung injury and acute respiratory distress syndrome. The contagion of this virus categorized as severed, which even causes severe damages to human life to harmless such as a common cold. In this manuscript, we discussed the SARS‐CoV‐2 virus into a system of equations to examine the existence and uniqueness results with the Atangana–Baleanu derivative by using a fixed‐point method. Later, we designed a system where we generate numerical results to predict the outcome of virus spreadings all over India. [ABSTRACT FROM AUTHOR]

This study aims to develop a supervised learning artificial recurrent neural network algorithm supported by Bayesian regularization called (ARNN‐BR) to analyze the impact of physical parameters, including radius of curvature (κ$\kappa $), Casson parameter (β$\beta $), heat generation parameter (λ$\lambda $) and radiation parameter (Rd${{R}_d}$) on velocity fʹ(η), and temperature profiles θ(η) in Casson nanofluid consisting of carbon nanotubes (CNTs‐CNF) model for single and multiwalled CNTs across a curved stretched surface. The numerical dataset of the proposed model has been constructed by varying various parameters for five scenarios that are used in a Bayesian regularization‐based intelligent computing method to build networks for approximating the numerical solutions of CNTs‐CNF model. It is observed that increment in the dimensionless radius of curvature (κ$\kappa $) causes to rise an increase in the velocity profile fʹ(η) for both SWCNTs and MWCNTs. However, a contrasting trend is observed when the Casson parameter (β$\beta $) is increased to higher values. The temperature θ(η) of fluid increases as the heat generation parameter (λ$\lambda $) and radiation parameter (Rd${{R}_d}$) increase. However, an opposite behavior is noticed when the dimensionless radius of curvature (κ$\kappa $) varies. The effectiveness and significance of designed Bayesian regularization based artificial recurrent neural networks (ARNN‐BR) is demonstrated through regression index measurements, error histogram studies, auto‐correlation analysis and convergence curves showing a minimal level of mean square error (E‐11 to E‐04) for the comprehensive simulations of CNTs‐CNF model. The designed ARNN‐BR algorithm is employed in many domains such as voice recognition, machine translation, identification of neurological brain illnesses as well as for automated translation of texts across different languages. [ABSTRACT FROM AUTHOR]