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We focus on the control of unknown partial differential equations (PDEs). The system dynamics is unknown, but we assume we are able to observe its evolution for a given control input, as typical in a reinforcement learning framework. We propose an algorithm based on the idea to control and identify on the fly the unknown system configuration. In this work, the control is based on the state-dependent Riccati approach, whereas the identification of the model on Bayesian linear regression. At each iteration, based on the observed data, we obtain an estimate of the a-priori unknown parameter configuration of the PDE and then we compute the control of the correspondent model. We show by numerical evidence the convergence of the method for infinite horizon control problems. [ABSTRACT FROM AUTHOR]

In the context of this study, we focus on the Volterra integro-differential equation incorporating a delay term. Employing a Picard iteration scheme, we establish the existence and uniqueness of the solution, presenting the quadrature method as a tool for numerically approximating the analytical solution. Our investigation contributes to the comprehensive understanding of nonlinear Volterra integro-differential equations with delay terms. We verify the convergence of our numerical approximation method, and we provide two numerical examples, ensuring the precision of our findings. Through, varying the delay term, and underscore its substantial influence on the equation's solution. [ABSTRACT FROM AUTHOR]

Due to the high approximation power and simplicity of computation of smooth radial basis functions (RBFs), in recent decades they have received much attention for function approximation. These RBFs contain a shape parameter that regulates their approximation power and stability but its optimal selection is challenging. To avoid this difficulty, this paper follows a novel and computationally efficient strategy to propose a space of radial polynomials with even degree that well approximates flat RBFs. The proposed space, Hn, is the shifted radial polynomials of degree 2n. By obtaining the dimension of Hn and introducing a basis set, it is shown that Hn is considerably smaller than P2n while the distances from RBFs to both Hn and P2n are nearly equal. For computation, by introducing new basis functions, two computationally efficient approaches are proposed. Finally, the presented theoretical studies are verified by the numerical results. [ABSTRACT FROM AUTHOR]

AbstractIn this work, we propose fractional-order deterministic mathematical models for human brain information dynamics, including the major effect on sensory memory. The two models are top-down and bottom-up processing. Mathematical models are analysed for different cases for long and short-term memory effects with the effect of experience and prior knowledge. Studies are conducted both qualitatively and quantitatively to analyse systems using fixed-point theory results. The impact of the global derivative, linear growth, and Lipschitz criteria are used to check the existence and uniqueness of the model for the biological feasibility of the system. Positivity and boundedness of the unique solution were also treated. The global stability of the proposed model is constructed by constructing the first and second derivatives for the Lyapunov function using wave analysis according to the equilibrium point. Numerical simulations of the proposed method in the range of fractional orders are conducted to demonstrate the implications of fractional and fractal orders. To further explore the impact of various parameters on the information processing process in the human brain, information in long-term memory, repetition, and rehearsal Chaotic modelling of the brain information dynamics model is also derived for different cases that are stable and bound to feasible regions. Results demonstrate the strong memory effect by using nonlocal and non-singular kernels at different fractional order values and fractal dimensions, which support experimental and theoretical observations.

In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ3[a,b]{\varpi }_{3}\left[a,b]. From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

AbstractThis study investigates the behavior of micropolar Eyring-Powell ferrofluids over a stretchable surface, considering heterogeneous-homogeneous chemical reactions, thermal radiation, and magnetic dipole effects. The governing partial differential equations are transformed into ordinary differential equations (ODEs) using similarity transformations, which are then solved using the shooting method with Runge-Kutta updates. The effects of various parameters on the flow and temperature fields are computed and visualized using MATLAB 2023. The results show that the velocity field decreases with increasing ferro-magnetic parameter β and fluid material parameter ß, while the temperature field increases with thermal radiation R, but decreases with higher dissipation parameter λ1 and Prandtl number Pr. The results are in excellent agreement with previously published data, validating the accuracy of our approach. [ABSTRACT FROM AUTHOR]

Rotating-disk generated flows find potential applications in industry such as cooling of electronics, car-breaking system, wind turbine engineering and others. Current article formulates heat transfer in the steady revolving flow of viscoplastic fluid obeying Bingham model over a rotating disk. Simulations are conducted by applying partial slip wall conditions, which are Robbin-type and non-linear in velocity components. Contribution of viscous dissipation, representing the conversion of mechanical energy into thermal energy caused by internal friction, is added in the analysis. Furthermore, the phenomenon of Joule heating due to the consideration of magnetic field affected flow is accounted for. The self-similar problem that arises after implementing Von-Kármán variables is computationally addressed using the MATLAB tool bvp4c. Significance of yield stress on the Von-Kármán flow under slip boundary assumption is elucidated through graphical results. Additionally, the implications of frictional heating and Joule heating on the development of a thermal boundary layer in a Bingham fluid are clarified. As the yield stress τ y increases, the velocity profiles meet their respective far field conditions at smaller vertical distances, and therefore a higher torque is required by the disk. Noticeably, considering the slip effect not only lowers the torque that the disk requires, but it also boosts disk's rate of cooling—a desirable outcome in practical applications. The reliability of the mathematical model is assessed by performing uncertainty analysis of the computational data. [ABSTRACT FROM AUTHOR]

One out of eight people in the United Arab Emirates (UAE) are estimated to have diabetes. Modeling the relationship between developing this health condition and complications is an important step towards understanding this disease. In this paper, we investigate a system of fractional differential equations to estimate the size of diabetic population with and without complication in the UAE. We provide the well‐posedness analysis of the model in the sense of Hadamard. We also provide the numerical approximations of the solution as a means of prediction of the future size of UAE diabetic population and the economic burden of the country that may be used by the government to take necessary preventative measure. [ABSTRACT FROM AUTHOR]

Background: Infectious diseases are a leading cause of death and disability worldwide, so it is crucial to plan for their potential impact in order to implement an efficient response. Examining the seasonality and distribution of influenza viruses in Ghana, as well as susceptible demographic groups and circulating strains of the virus, were the objectives of this study. Methods: We worked with a modified version of the Susceptible-Exposed-Infectious-Recovered-Vaccinated (SEIR-V) transmission model to forecast the possible outcomes of the influenza pandemic in Ghana. Using the fourth-order Runge-Kutta method, we were able to get numerical simulations for changing the model parameters. We analyzed forecasts for the illness transmission rate β, vaccination rate ρ, and recovery rate γ on a daily and cumulative basis. The average fundamental reproduction number for the parameters β and γ was also rendered graphically. Results: We effectively forecasted the trajectory of influenza-related morbidity using our model, which paves the way for future approaches of controlling and monitoring the flu in our study area. In order to restrict the seasonal influenza, we have provided visual evidence that vaccinated patients and a quarantine in Ghana for at least the next 10 days are needed. It has been noted that the recovery rates of non-vaccinated patients and the vaccination rate work together to reduce the contagious disease. Conclusion: Using precise parameter approximations, theoretical epidemic analysis has proven to be an effective method for predicting and managing the spread of pandemics such as seasonal influenza virus. This model has been transformed into an epidemic model by adding the hospitalized-vaccination compartment for patients with confirmed infections to the SEIR-V model. [ABSTRACT FROM AUTHOR]

Generating numerical solutions for boundary value problems (BVPs) using numerical methods can be a difficult task since the solutions may involve complex mathematical formulations. Hence, various types of B-splines collocation methods were developed to produce better numerical approximations with much simpler approaches. The present study introduced two new types of extended Cubic Hybrid B-spline Collocation Method (CHBSM) namely the Extended Cubic Hybrid B-spline Collocation Method (ECHBSM) and Extended Trigonometric Cubic Hybrid B-spline Collocation Method (ETCHBSM) for solving second order linear two-point BVPs. These methods were tested on three examples of linear two-point BVPs of order two. For comparison purposes, three established collocation methods which are the Cubic B-spline Collocation Method (CBSM), Cubic Trigonometric B-spline Collocation Method (CTBSM) and Cubic Hybrid B-spline Collocation Method (CHBSM) were also applied on these examples. The numerical results were tabulated and analysed to make comparisons with the analytical solutions and the conventional methods from past literatures. For CHBSM, ECHBSM and ETCHBSM, optimization was applied to the free parameter, γ using a simple proposed approach. The result demonstrated that CHBSM yields the best approximation with the analytical solutions. [ABSTRACT FROM AUTHOR]

Summary: Fluid mechanics is a critical field in both engineering and science. Understanding the behavior of fluids requires solving the Navier–Stokes equation (NSE). However, the NSE is a complex partial differential equation that can be challenging to solve, and classical numerical methods can be computationally expensive. In this paper, we propose enhancing physics‐informed neural networks (PINNs) by modifying the residual loss functions and incorporating new computational deep learning techniques. We present two enhanced models for solving the NSE. The first model involves developing the classical PINN for solving the NSE, based on a stream function approach to the velocity components. We have added the pressure training loss function to this model and integrated the new computational training techniques. Furthermore, we propose a second, more flexible model that directly approximates the solution of the NSE without making any assumptions. This model significantly reduces the training duration while maintaining high accuracy. Moreover, we have successfully applied this model to solve the three‐dimensional NSE. The results demonstrate the effectiveness of our approaches, offering several advantages, including high trainability, flexibility, and efficiency. [ABSTRACT FROM AUTHOR]

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

The paper introduced an image-denoising algorithm based on the fractal–fractional integral operator for removing Gaussian noise in images. Using this algorithm fractional masks have been constructed. The capacity of the fractal–fractional integral mask to smooth the Gaussian noisy images for varied noise levels has been demonstrated through experiments. Peak signal-to-noise ratio (PSNR) is used for denoising images to analyse performance. The acquired experimental results demonstrate that fractal–fractional masks have comparable capabilities to some recently developed masks and are computationally efficient.

Different from the construction process of the traditional finite difference method, we derive a large class of high-accuracy methods for approximating derivatives. Since Taylor expansion is avoided, the requirement for function smoothness in the new methods is greatly reduced. We analyze the approximation errors of the proposed methods and compare their approximation effects. As a typical application, we use the proposed methods to solve Elliptic, Hyperbolic and Parabolic problems numerically. For time-dependent problems, a class of new fully implicit difference schemes and the fourth-order Runge-Kutta method are both used for discretization. When using fully implicit difference schemes, unconditional stability and convergence are proved for hyperbolic and parabolic problems with the periodic boundary conditions. A large number of numerical experiments are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a collocation method based on the Haar wavelet is presented for the solution of both linear and nonlinear first-order neutral delay differential equations. The Haar functions are used to approximate the first-order derivative, and the approximate solution is obtained by using initial condition and integration. Some examples from the literature are used to test the suggested method efficiency and applicability. A comparison of exact and approximate solutions is given in figures for different numbers of collocation points. The root mean square and maximum absolute errors are calculated for different numbers of collocation points. The rate of convergence is calculated which is approximately equal to 2. The comparison of the present method with the other numerical methods is also given. The results demonstrate that the Haar wavelet collocation method is simple and effective for solving first-order linear and nonlinear neutral delay differential equations. [ABSTRACT FROM AUTHOR]

We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of polynomials of degree N ∈ N in space. We prove that we can choose a sequence of controls f N associated with the approximate control problem in such a way that they converge, as N → ∞ , to a control of the continuous wave equation. Unlike other numerical approximations tried in the literature, this one does not require regularization techniques and can be easily adapted to other equations and systems where the controllability of the continuous model is known. The method is illustrated with several examples in 1-d and 2-d in a square domain. We also give numerical evidence of the highly accurate approximation inherent to spectral methods. [ABSTRACT FROM AUTHOR]

In this work, we propose fractional-order deterministic mathematical models for human brain information dynamics, including the major effect on sensory memory. The two models are top-down and bottom-up processing. Mathematical models are analysed for different cases for long and short-term memory effects with the effect of experience and prior knowledge. Studies are conducted both qualitatively and quantitatively to analyse systems using fixed-point theory results. The impact of the global derivative, linear growth, and Lipschitz criteria are used to check the existence and uniqueness of the model for the biological feasibility of the system. Positivity and boundedness of the unique solution were also treated. The global stability of the proposed model is constructed by constructing the first and second derivatives for the Lyapunov function using wave analysis according to the equilibrium point. Numerical simulations of the proposed method in the range of fractional orders are conducted to demonstrate the implications of fractional and fractal orders. To further explore the impact of various parameters on the information processing process in the human brain, information in long-term memory, repetition, and rehearsal Chaotic modelling of the brain information dynamics model is also derived for different cases that are stable and bound to feasible regions. Results demonstrate the strong memory effect by using non-local and non-singular kernels at different fractional order values and fractal dimensions, which support experimental and theoretical observations. [ABSTRACT FROM AUTHOR]

In this article, we will investigate the Galerkin sinc approximation to solve the single pseudo-Poisson problem and in this method we will reach a linear system. We will solve this system by carefully choosing the length of the steps and the number of nodal points and with two methods; the first method, which is the usual method, is theoretically flawed and not practical and computationally efficient. Therefore, to solve the mentioned system, we introduce the orthogonalization technique, which solves both theoretical and computational problems. Numerical approximation is obtained whose accuracy is exponential and of order 0(exp (-c √N) where N is a transaction parameter and c is a constant independent of N. In the final part, we give some numerical examples of individual pseudo-Poisson problems to demonstrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

Kuiper's statistic is a good measure for the difference of ideal distribution and empirical distribution in the goodness-of-fit test. However, it is a challenging problem to solve the critical value and upper tail quantile, or simply Kuiper pair, of Kuiper's statistics due to the difficulties of solving the nonlinear equation and reasonable approximation of infinite series. In this work, the contributions lie in three perspectives: firstly, the second order approximation for the infinite series of the cumulative distribution of the critical value is used to achieve higher precision; secondly, the principles and fixed-point algorithms for solving the Kuiper pair are presented with details; finally, finally, a mistake about the critical value cnα for (α,n)=(0.01,30) in Kuiper's distribution table has been labeled and corrected where n is the sample capacity and α is the upper tail quantile. The algorithms are verified and validated by comparing with the table provided by Kuiper. The methods and algorithms proposed are enlightening and worth of introducing to the college students, computer programmers, engineers, experimental psychologists and so on.

In the present research, a prey-predator system that classifies predator species into premature and veteran species was proposed. Mathematical models of the system of species that include the impact of panic and a variety of functional reactions to the species have been recently reviewed. The consequence of fear is a reduction factor in the fundamental reproduction of the prey species. Relying on the prior research, we propose a mathematical predator-prey model employing the concept of fractional order system that contains two crucial components: intra-specific competition under mature predator density and fear, which affects the progeny of the prey community. Incorporating self-similarity characteristics is the significance of adopting a fractional order system. Following that, using memory kernel, we have interpret this model into the perspective of Caputo, that is, Atangana_Baleanu_Caputo (A_B_C) kind of fractional order derivatives. Singularity and non-locality are two of the main properties of fractional operators that make it simpler to figure out the natural behaviour of complicated systems. Details regarding the unique solution to the established model's derivation are provided. The dichotomy operator investigates the stability of the system under Hyper-Ulam`s stability method. The numerical interpolation to the suggested model generates the best precise approximate result. Through computer simulations, a thorough investigation on the predator species exhibiting intra-specific factor and prey species exhibiting fear factor is investigated. Using Adam's semi-analytical technique, we were able to derive the numerical estimates and determine the solution. The purpose to employ the Adams-Moulton approach is that it will provide more implicitly accurate results. Extensive numerical experiments are carried out using MATLAB software in order to portray the dynamical behaviour of the system. These observations lead us to the conclusion that total population expansion in the environment would be managed by the successful adoption of the fear factor in prey and the intra-species rivalry in predators.

We present a high‐accuracy numerical method based on a decoupled dimensionality reduction scheme for Maxwell eigenvalue problem in spherical domains. Using the orthogonality of vector spherical harmonics and the variable separation approach, we decompose the original problem into two classes of decoupled one‐dimensional TE mode and TM mode. For the TE mode, we establish a variational formulation and its discrete scheme and give the error estimations of the approximate eigenvalues and eigenfunctions. For the TM mode, it is different from TE mode which naturally meets the divergence‐free condition and will not generate some spurious eigenvalues. We design a numerical algorithm based on a parameterized method to filter out the spurious eigenvalues. Finally, some numerical results are presented to confirm the theoretical results and validate the algorithms. [ABSTRACT FROM AUTHOR]

Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the infsup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of random periodic solutions of the SDEs with coefficients satisfying the one-sided Lipschitz condition. By using the tamed Euler-Maruyama (EM) method we give a numerical approximation for the random periodic solution and reveal that the numerical approximation converges to the exact solution with an order a ∈ (0,1/2) in the mean square sense. Examples are given to verify the theoretical result. [ABSTRACT FROM AUTHOR]

The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory when input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB's intimate relations with RD, we provide substantial insights into its solution structure, highlighting caveats in its finite-dimensional treatments. Sub-optimal solutions are seen to collide or exchange optimality at its bifurcations. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB's first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, we not only detect IB bifurcations but also identify their type in order to handle them accordingly. Rather than approaching the IB's optimal tradeoff curve from sub-optimal directions, the latter allows us to follow a solution's trajectory along the optimal curve under mild assumptions. We thereby translate an understanding of IB bifurcations into a surprisingly accurate numerical algorithm. [ABSTRACT FROM AUTHOR]

In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps $ \delta t $ the recommended method converges at a rate of $ O(\delta t^2 + h^2) $. Based on graphic results and numerical examples, the application of the model is also examined.

This research aims to propose a numerical methodology that employs a non-increasing sequence to approximate the solution for a large class of ordinary differential equations like the ones described in complex non-linear heat transfer through porous fins. It also aims to compare this proposed methodology with an earlier one using a non-decreasing sequence of elements and to provide an upper-bound estimation for the error. The comparison between the non-increasing and non-decreasing sequences showed excellent agreement when applied to an example of convection and radiation in porous fins.

We propose an approach for the precise numerical integration of a stochastic SVIR model defined by a stochastic differential equation (SDE) with non‐globally Lipschitz continuous coefficients and multiplicative noise. This equation, based on a compartmental epidemic model, describes a continuous vaccination strategy with environmental noise effects. By means of an appropriate invertible continuous transformation, we link the solution to the stochastic SVIR model to the solution of an auxiliary random differential equation (RDE) that has an Ornstein–Uhlenbeck process as the only input parameter of the system. In this way, based on this explicit conjugacy between both equations, new pathwise numerical schemes are constructed for the SVIR model. In particular, we propose an exponential method that outperforms other integrators in the literature and is able to approximate, with high stability, meaningful probabilistic features of the continuous system, including its stationary distribution and ergodicity. A simulation study is presented to illustrate the practical performance of the introduced methods, and a comparative analysis with other integrators commonly used for the simulation of epidemiological models is performed. [ABSTRACT FROM AUTHOR]

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability. [ABSTRACT FROM AUTHOR]

The paper deals with the problem of representation of Horn's hypergeometric functions via continued fractions and branched continued fractions. We construct the formal continued fraction expansions for three ratios of Horn's hypergeometric functions H 7. The method employed is a two-dimensional generalization of the classical method of constructing a Gaussian continued fraction. It is proved that the continued fraction, which is an expansion of each ratio, uniformly converges to a holomorphic function of two variables on every compact subset of some domain of C 2 , and that this function is an analytic continuation of such a ratio in this domain. To illustrate this, we provide some numerical experiments at the end. [ABSTRACT FROM AUTHOR]

In order to reflect the dispersal of pollutants in non-adjacent areas and the large-scale movement of individuals, this paper proposes an epidemic model of nonlocal dispersal with air pollution, where the transmission rate is related to the concentration of pollutants. This paper checks the uniqueness and existence of the global positive solution and defines the basic reproduction number, R 0 . We simultaneously explore the global dynamics: when R 0 < 1 , the disease-free stable point is global asymptotic stability; when R 0 > 1 , the disease is uniformly persistent. Additionally, in order to approximate R 0 , a numerical method has been introduced. Illustrative examples are used to verify the theoretical outcomes and show the effect of the dispersal rate on the basic reproduction number R 0 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the energy minimization model arising in the ensemble Kohn--Sham density functional theory for metallic systems, in which a pseudo-eigenvalue matrix and a general smearing approach are involved. We study the invariance of the energy functional and the existence of the minimizer of the ensemble Kohn--Sham model. We propose an adaptive twoparameter step size strategy and the corresponding preconditioned conjugate gradient methods to solve the energy minimization model. Under some mild but reasonable assumptions, we prove the global convergence for the gradients of the energy functional produced by our algorithms. Numerical experiments show that our algorithms are efficient, especially for large scale metallic systems. In particular, our algorithms produce convergent numerical approximations for some metallic systems, for which the traditional self-consistent field iterations fail to converge. [ABSTRACT FROM AUTHOR]

In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps δ t the recommended method converges at a rate of O (δ t 2 + h 2) . Based on graphic results and numerical examples, the application of the model is also examined. [ABSTRACT FROM AUTHOR]

In this paper, a unique and novel numerical approach—the fractional-order Caputo-Fabrizio derivative in the Caputo sense—is developed for the solution of fractional differential equations with a non-singular kernel. After converting the differential equation into its corresponding fractional integral equation, we used Simpson's 1/3 rule to estimate the fractional integral equation. A thorough study is then conducted to determine the convergence and stability of the suggested method. We undertake numerical experiments to corroborate our theoretical findings.

The struggle to find the analytic solution of several differential equations leads to several issues like difficulty in finding solutions, singularities, convergent issues, and stability. Because of these problems, most of the researchers come up with explicit approaches such as the Runge Kutta method, Euler’s method, and Taylor’s polynomial method for finding numerical solutions to the ordinary differential equation. In this work, we combine both the Aitken methods and Newton’s interpolation method (NIM) to solve first-order differential equations. The numerical results obtained provide minimal error. The result is supported by solving an example.

In this article, the time-space fractional diffusion equation is solved by using the fractional operator in Atangana–Baleanu–Caputo (ABC) sense based on the Mittag-Leffler function involving non-singular and non-local kernels. This study mainly develop the numerical method for time-space linear and non-linear ABC fractional diffusion equation by utilising centred difference approximation with second-order accuracy of Riesz fractional derivative in space and trapezoidal formula for fractional integral approximation for a system of ABC integral equations. Also, we examine the stability and show that the proposed scheme converges at the rate of [Formula: see text] having space mesh size h and time step size t. Numerical simulations are presented based on the presented method. We finished the article with a conclusion.

This paper is devoted to the robust approximation with a variational phase field approach of multiphase mean curvature flows with possibly highly contrasted mobilities. The case of harmonically additive mobilities has been addressed recently using a suitable metric to define the gradient flow of the phase field approximate energy. We generalize this approach to arbitrary nonnegative mobilities using a decomposition as sums of harmonically additive mobilities. We establish the consistency of the resulting method by analyzing the sharp interface limit of the flow: A formal expansion of the phase field shows that the method is of second order. We propose a simple numerical scheme to approximate the solutions to our new model. Finally, we present some numerical experiments in dimensions 2 and 3 that illustrate the interest and effectiveness of our approach, in particular for approximating flows in which the mobility of some phases is zero. [ABSTRACT FROM AUTHOR]

The infinite time reachable set of a strictly stable linear control system is the Hausdorff limit of the finite time reachable set of the origin as time tends to infinity. By definition, it encodes useful information on the long-term behavior of the control system. Its characterization as a limit set gives rise to numerical methods for its computation that are based on forward iteration of approximate finite time reachable sets. These methods tend to be computationally expensive, because they essentially perform a Minkowski sum in every single forward step. We develop a new approach to computing the infinite time reachable set that is based on the invariance properties of the control system and the desired set. These allow us to characterize a polyhedral outer approximation as the unique solution to a linear program with constraints that incorporate the system dynamics. In particular, this approach does not rely on forward iteration of finite time reachable sets. [ABSTRACT FROM AUTHOR]

This article presents a variational and numerical analysis for a thermal fluid–elastic structure interaction problem related to the blood flow through a vessel. We prove the existence and the uniqueness of the weak solution to the mathematical model associated with this physical problem and we establish some estimates that give the regularity for the unknown functions. Since the variational problem introduced in order to obtain these results does not provide enough regularity in the elastic domain, we approximate it with a family of viscoelastic problems, depending on a small parameter ε . The viscoelastic problems contain an additional term that corresponds to the regularity “uncovered” by the initial variational problem. This approximation is justified by an error estimate theorem, followed by a convergence result. We associate to any viscoelastic problem a numerical scheme. The additional viscoelastic term allows us to establish suitable estimates for the solution to the numerical scheme, used for obtaining stability properties. Relying on these properties, we prove the convergence of the numerical scheme to the viscoelastic problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for approximating the solution of the linear Fredholm integral equation of the second kind which is defined on a closed contour τ in the complex plane. The right-hand side of the equation is a piecewise continuous function that is numerically defined on a finite set of points on τ. To approximate the solution, we use a linear combination of B-spline functions and Heaviside step functions defined on τ. We discuss both theoretical and practical aspects of the pointwise convergence of the method, including its performance in the vicinity of the points where discontinuities occur. [ABSTRACT FROM AUTHOR]

We define the concept of energy-variational solutions for the Ericksen–Leslie equations in three spatial dimensions. This solution concept is finer than dissipative solutions and satisfies the weak-strong uniqueness property. For a certain choice of the regularity weight, the existence of energy-variational solutions implies the existence of measure-valued solutions and for a different choice, we construct an energy-variational solution with the help of an implementable, structure-inheriting space-time discretization. Computational studies are performed in order to provide some evidence of the applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a unique and novel numerical approach--the fractional-order Caputo-Fabrizio derivative in the Caputo sense--is developed for the solution of fractional differential equations with a non-singular kernel. After converting the differential equation into its corresponding fractional integral equation, we used Simpson's 1/3 rule to estimate the fractional integral equation. A thorough study is then conducted to determine the convergence and stability of the suggested method. We undertake numerical experiments to corroborate our theoretical findings. [ABSTRACT FROM AUTHOR]

The Era of data is transubstantiating into a Big Data model in this technological world in the early 21st century. In 2005, Roger Mougalas coined a combination of data for this future world of the human race. The information helps to find specific solutions for any physical problem under Catastrophic circumstances in high populations such as Covid-19. To store massive data and historical events in a computer, the possibility of damage occurred to the complete data. Hence, viruses are a crucial threat to such data worth millions and billions. For this purpose, we spend enormous costs and efforts to build defensive strategies to save that information. Analyzing the expansion and extension of viruses helps to protect data and prevent viruses. In this manuscript, we study optimal control analysis for the suggested model in the sense of the Atangana–Baleanu derivative (AB-derivative). We employed a fixed point theorem to analyze the solutions for the fractional order computer virus model. We verified the results numerically and expressed them graphically.

The present attempt is to design a novel approach for the numerical solution of fractional telegraph equation. The novelty of the paper exist in solving the time fractional telegraph equation with differential quadrature method based on cubic B-spline (MHB-DQM) for the fractional parameter 1

The paper deals with the problem of representation of Horn's hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn's hypergeometric function H 4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H , H ⊂ C 2 , and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn's double hypergeometric series H 4 is considered, and the expression of solutions of some systems of partial differential equations is indicated. [ABSTRACT FROM AUTHOR]