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In this paper, we numerically investigate the fractional quadratic and cubic logistic models involving the Caputo–Fabrizio operator. We construct the successive iterations using the theory of fractional calculus and Lagrange polynomials. Then, we handled the exact solutions of these models. The validity of the accuracy and efficiency will be satisfied through some numerical results. [ABSTRACT FROM AUTHOR]

In this paper, a quadratic B-spline collocation method is developed to solve a singularly perturbed semilinear reaction–diffusion problem with a discontinuous source term. The discontinuous source term leads to a jump in the second-order derivative of the exact solution at the discontinuous point. A quadratic B-spline collocation method on a Shishkin-type mesh is used to discretized the singularly perturbed problem on the left and right sides of the discontinuous point, respectively. The collocation equations at the discontinuous point are obtained using the conditions satisfied at the discontinuous point. It is shown that the scheme is stable and almost second-order uniformly convergent. Numerical experiments support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a collocation method for the approximate solution of two-dimensional Hammerstein integral equations over polygonal regions in ℝ 2 . For a reformulation of the equation, we triangulate the polygonal region and use a special type of quadratic interpolation on each triangle. This produces a quadrature formula with a higher degree of precision than expected, leading to a numerical method that is superconvergent at the collocation nodes. We prove the convergence of the method and give error estimates. The implementation is also discussed and the applicability of the proposed scheme is presented on numerical examples. The paper concludes with a discussion on the efficiency of the proposed method, also highlighting some ideas for future research in this area. [ABSTRACT FROM AUTHOR]

In one of his meetings with members of the Vienna Circle, Wittgenstein discusses Hermann Weyl’s brief conversion to intuitionism and criticizes his arguments against applying the law of the excluded middle to generalizations over the natural numbers. Like Weyl, however, Wittgenstein rejects the classical model theoretic conception of generality when it comes to infinite domains. Nonetheless, he disagrees with him about the reasons for doing so. This paper provides an account of Wittgenstein’s criticism of Weyl that is based on his differing understanding of what a general statement over infinite domains consists in. This difference in their conception of generality is argued to be central to the middle Wittgenstein’s overall stance on intuitionism as well. While Weyl (and other intuitionists) reject the law of the excluded middle on grounds of constructivity, Wittgenstein argues that general statements over infinite domains do not express propositions in the first place. The origin of this position as well as its consequences for contemporary debates on generality are further assessed. [ABSTRACT FROM AUTHOR]

As users have higher and higher requirements for the quality of experience, traditional cloud computing is gradually unable to meet the needs of user equipments. Hence mobile edge computing networks mounted by unmanned aerial vehicles are introduced to improve user experience and reduce energy consumption. However, most current work is based on neural networks, which require large amounts of labeled data or long training times. Given these challenges, this paper proposes an energy-efficient multi-stage alternating optimization scheme to reduce the weighted energy consumption of the entire network. We analyze the energy consumption of each device and formulate a non-convex optimization problem. Considering the impact of task offloading, resource allocation, and path planning on network energy consumption, we transform the energy consumption problem into three subproblems. And use the coordinate descent algorithm, interior point method, and successive convex approximation method to optimize them alternately. The simulation results show that the proposed optimization scheme can significantly reduce the network's total energy consumption. [ABSTRACT FROM AUTHOR]

In this paper, we first derive a theoretical basis for spherical conformal parameterizations between a simply connected closed surface S and a unit sphere S 2 by minimizing the Dirichlet energy on C ¯ with stereographic projection. The Dirichlet energy can be rewritten as the sum of the energies associated with the southern and northern hemispheres and can be decreased under an equivalence relation by alternatingly solving the corresponding Laplacian equations. Based on this theoretical foundation, we develop a modified Dirichlet energy minimization with nonequivalence deflation for the computation of the spherical conformal parameterization between S and S 2 . In addition, under some mild conditions, we verify the asymptotically R-linear convergence of the proposed algorithm. Numerical experiments on various benchmarks confirm that the assumptions for convergence always hold and demonstrate the efficiency, reliability and robustness of the developed modified Dirichlet energy minimization. [ABSTRACT FROM AUTHOR]

Internet of Things (IoT) applications have invaded several domains (supply chain, healthcare, etc.). To enhance the quality of the provided service in terms of latency, response time, etc., service providers such as Amazon, Google, and Microsoft turned to running IoT tasks near the end user by invoking the fog computing concept. Fog computing extends cloud services to the edge of the network. It provides a variety of computing resources in the form of fog nodes, which offer multiple services known as fog services. These latter are used to store and process the data generated by IoT devices. Fog services are characterized by their high reusability. It enables the construction of a composite service to provide complicated IoT tasks. In this paper, we introduce an adaptive requirements-aware approach for configuring IoT applications in fog computing. The configuration is based on a Composite Fog Service (CFS) model and is restricted by a set of constraints. The proposed approach is implemented in a Smart Car Parking (SCP) scenario. Simulation results reveal the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-point boundary-value problem with a Riemann–Liouville–Caputo fractional derivative of order α ∈ (1 , 2) . We solve this boundary-value problem in a sequence of processes, first using the shooting technique based on the secant iterative method, we convert the boundary-value problem into an initial-value problem, then the initial-value problem is transformed into an equivalent Volterra integral equation with weakly singular kernel. Finally, we find the approximate solution of the resultant equation by using a discretization scheme on a uniform mesh. The method's convergence analysis has been thoroughly established, and it demonstrates that the scheme is first-order convergent. To show the effectiveness of the suggested approach, numerical results are provided. [ABSTRACT FROM AUTHOR]

This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a second order parameter-uniform numerical method for a singularly perturbed Volterra delay-integro-differential equation on a Bakhvalov-type mesh. The equation is discretized by using the variable two-step backward differentiation formula of the first derivative term and the trapezoidal formula of the integral term. The stability and convergence of the numerical method in the discrete maximum norm are proved. Finally, the theoretical results are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

Many of today's problems require techniques that involve the solution of arbitrarily large systems A x = b . A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor decomposition. The numerical experiments support the fact that this algorithm converges especially fast when the matrix of the linear system is Laplacian-Like. These matrices that follow the tensor structure of the Laplacian operator are formed by sums of Kronecker product of matrices following a particular pattern. Moreover, this set of matrices is not only a linear subspace it is a Lie sub-algebra of a matrix Lie Algebra. In this paper, we characterize and give the main properties of this particular class of matrices. Moreover, the above results allow us to propose an algorithm to explicitly compute the orthogonal projection onto this subspace of a given square matrix A ∈ R N × N. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Mickens' methodology to construct a second-order nonstandard finite difference (NSFD) method, which preserves dynamical properties including positivity, local asymptotic stability and especially, global asymptotic stability of a general single-species model. This NSFD method is based on a novel weighted non-local approximation of the right-hand side function in combination with the renormalization of the denominator function. The weight guarantees the dynamic consistency and the nonstandard denominator function ensures the convergence of order 2 of the NSFD method. The result is that we obtain a second-order and dynamically consistent NSFD method. It is proved that the NSFD method is simple and efficient and can be extended for solving a broad range of mathematical models arising in real-world applications. Also, we combine the constructed second-order NSFD method with Richardson's extrapolation technique to generate high-order numerical approximations. Finally, the theoretical findings are illustrated and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider the exponential stabilization for a series system of N heat equations, where one boundary control is only applied to the leading (first) heat equation, and the boundary temperature of the previous heat is fed into the boundary heat flux of the next one. With N invertible transformations, we present a boundary feedback control so that the closed-loop system is equivalent to an exponentially stable target system. Some numerical simulation results are presented to validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we study a uniform finite difference method for the first-order singularly perturbed Volterra integro-differential problem depending on a parameter. We prove that the method is uniform second-order convergent except for a logarithmic factor in the discrete maximum norm. Numerical results are presented which are in agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper focuses on new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space in cooperation with Navier-Stokes equations and the Nédélec’s edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order Nédélec’s edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order Nédélec’s edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented, which confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper discusses the Richardson extrapolation technique of an upwind finite difference scheme on a modified Bakhvalov-type mesh (named Vulanović–Bakhvalov mesh) for a model singularly perturbed convection–diffusion problem in two dimensions. We give a rigorous convergence analysis for the Richardson extrapolation method, which is shown that the use of Richardson extrapolation technique improves the ε -uniform accuracy of the presented scheme in the discrete maximum norm from O N - 1 to O N - 2 , where N is the number of mesh-intervals. Then we also propose a new numerical method to obtain the optimization mesh parameters of the Vulanović–Bakhvalov mesh. Finally some numerical experiments are provided to validate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on a new error analysis and a recovering technique of frequently-used mixed FEMs for a dynamical incompressible magnetohydrodynamics (MHD) system. The methods use the standard inf-sup stable Taylor–Hood/MINI velocity-pressure space pairs to solve the Navier–Stokes equations and the Nédélec's edge element for solving the magnetic field. We establish new and optimal error estimates. In particular, we prove that the method provides the optimal accuracy for the MINI element in L 2 -norm and for the Taylor-Hood element in H 1 -norm. The analysis is based on a modified Maxwell projection and the corresponding estimates in negative norms, while all the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order Nédélec's edge approximation in analysis. In addition, at any given time step, we develop a simple recovery technique for numerical approximation to the magnetic field of one order higher accuracy in the spatial direction. [ABSTRACT FROM AUTHOR]

In this paper, we consider exponentially fitted peer methods for the numerical solution of first order differential equations and we investigate how the frequencies can be tuned in order to obtain the maximal benefit. We will show that the key is analyzing the error's behavior. Formulae for optimal frequencies are computed. Numerical experiments show the properties of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider singularly perturbed differential equations having delay on the convection and reaction terms. The considered problem exhibits boundary layer on the left or right side of the domain depending on the sign of the coefficient of the convection term. We approximated the terms containing the delay using Taylor series approximation. The resulting singularly perturbed boundary value problem is treated using exponentially fitted operator mid-point upwind finite difference method. The stability of the scheme is analysed and investigated using maximum principle and by constructing barrier functions for solution bound. We formulated the uniform converges of the scheme. The scheme converges uniformly with linear order of convergence. To validate the theoretical analysis and finding, we consider three examples exhibiting boundary layer on the left and right side of the domain. The obtained numerical result in this paper is accurate and parameter uniformly convergent. [ABSTRACT FROM AUTHOR]

The popularity of Internet of Things (IoT) devices in smart homes has raised significant concerns regarding data security and privacy. Traditional machine learning (ML) methods for anomaly detection often require sharing sensitive IoT data with a central server, posing security and efficiency challenges. In response, this paper introduces FedGroup, a novel Federated Learning (FL) method inspired by FedAvg. FedGroup revolutionizes the central model’s learning process by updating it based on the learning patterns of distinct groups of IoT devices. Our experimental results demonstrate that FedGroup consistently achieves comparable or superior accuracy in anomaly detection when compared to both federated and non-federated learning methods. Additionally, Ensemble Learning (EL) collects intelligence from numerous contributing models, leading to enhanced prediction performance. Furthermore, FedGroup significantly improves the detection of attack types and their details, contributing to a more robust security framework for smart homes. Our approach demonstrates exceptional performance, achieving an accuracy rate of 99.64% with a minimal false positive rate (FPR) of 0.02% in attack type detection, and an impressive 99.89% accuracy in attack type detail detection. [ABSTRACT FROM AUTHOR]

This paper presents a sequence of variable time step deferred correction (DC) methods constructed recursively from the second-order backward differentiation formula (BDF2) applied to the numerical solution of initial value problems for first-order ordinary differential equations (ODE). The sequence of corrections starts with the BDF2 then considered as DC2. We prove that this improvement from a p-order solution (DCp) results in a p + 1 -order accurate solution (DC p + 1 ). This one-order increment in accuracy holds for the least stringent BDF2 0-stability conditions. If we introduce additional requirements for the ratio of consecutive variable time step sizes, then the order increment is 2, allowing a direct transition from DCp to DC p + 2 . These requirements include the constant time step DCp methods. We also prove that all these DCp methods are A-stable. We briefly discuss two other DC variants to illustrate how a proper transition from DCp to DC p + 1 is critical to maintaining A-stability at all orders. Numerical experiments based on two manufactured (closed-form) solutions confirmed the accuracy orders of the DCp – for DCp, p = 2 , 3 , 4 , 5 – both with constant or alternating time step sizes. We showed that the theoretical conditions required to obtain an increment of orders 1 and 2 are satisfied in practice. Finally, a test case shows that we can estimate the error on the DCp solution with the DC p + 1 solution, and a last test case that our new methods maintain their order of accuracy for a stiff system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to design appropriate nonstandard finite difference (NSFD) schemes for two population mathematical models based on coupled nonlinear ordinary differential equations. Our work clarifies existing constructions of NSFD schemes for these two population models, which are not in full compliance with Mickens' methodology. We select the denominator functions for the discrete first-order derivatives depending on the existence of conservation laws, by following empirical rules suggested by Mickens. We fix nonlocal discretizations that preserve positivity of the schemes, irrespective of the value of the step size. Thus, our NSFD schemes are dynamically consistent with the two population models. We conduct a numerical study to assess the performance of the NSFD method. [ABSTRACT FROM AUTHOR]

In this paper we study the deviation of the error estimation for the numerical solution of two-point boundary value problems based on defect correction principle. We show that the order of the deviation of the error is O(h4). The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part. [ABSTRACT FROM AUTHOR]

In the present paper, fluid flow and heat transfer for micropolar fluids over a stretching sheet through Darcy porous medium are studied. The heat transfer phenomenon is considered with the isothermal wall as well as the isoflux boundary conditions. The fluid flow and heat transfer phenomena are modeled in the form of coupled nonlinear partial differential equations. The numerical solution is obtained using a nonstandard finite difference approximation on a quasi-uniform mesh. The numerical results obtained by the present approach are compared to those obtained by the Runge–Kutta fourth order method to demonstrate the accuracy of the present method. The numerical results obtained by both methods show excellent agreement. The effect of various physical parameters, namely the Reynolds number, Prandtl number, micropolar material parameters, injection/suction parameter, heat index parameter on bulk fluid speed, temperature distribution, and spin behavior of microstructures are demonstrated and discussed graphically. The simplicity of the finite difference approximation makes the selected technique more significant in the numerical study of micropolar fluid. The boundary layer thickness reduces with the increasing value of injection/suction parameter, Reynolds number, and micropolar parameter. The thermal boundary layer also reduces with the increment in the value of the micropolar parameter, Prandtl number, heat index parameter while microrotation increases with the increasing value of the injection/suction parameter. [ABSTRACT FROM AUTHOR]

In this paper, we consider high order finite difference schemes and fast algorithms for one- and two-dimensional space fractional diffusion equations (SFDEs). We first derive a class of 4-point weighted and shifted Grünwald difference (4WSGD) operators to approximate Riemann–Liouville fractional derivatives. A class of spatially jth ( j ≥ 3 ) order CN-4WSGD schemes for SFDEs with variable coefficients are proposed, where CN stands for Crank-Nicolson. As the coefficient matrix of the resulted linear system is Toeplitz-like but non-diagonally dominant, we theoretically analyze the stability and the accuracy of CN-4WSGD schemes by using the generating function of Toeplitz matrix. Moreover, we establish a diagonal-times-Toeplitz-splitting iteration method to solve the linear system, which is proved to be asymptotic uniformly convergent without imposing any extra conditions. Numerical examples are presented to verify the accuracy of the numerical schemes for both smooth and lower smooth solutions and test the efficiency of the proposed iteration methods. [ABSTRACT FROM AUTHOR]

This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme is A-stable, satisfies a B-convergence property, and that the correction on a scheme DC2j of order 2j of accuracy leads to a scheme DC2j + 2 of order 2j + 2. The order of accuracy is guaranteed by a deferred correction condition. Numerical experiments with standard stiff and non-stiff ODEs are performed with the DC2,..., DC10 schemes. The results show a high accuracy of the method. The theoretical orders of accuracy are achieved together with a satisfactory stability. [ABSTRACT FROM AUTHOR]

In this paper, a singularly perturbed Volterra integro- differential equation is being surveyed. On a piecewise-uniform Shishkin mesh, a fitted mesh finite difference approach is applied using a composite trapezoidal rule in the case of integral component and a finite difference operator for the derivative component. The proposed technique acquires a uniform convergence in accordance with the perturbation parameter. To improve the accuracy of the computed solution, an extrapolation, specifically Richardson extrapolation, is used measured in the discrete maximum norm and almost second-order convergence is attained. Further numerical results are provided to assist the theoretical estimates. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the numerical solution for the discretized system from time-dependent space fractional diffusion equations (TS-FDEs) with initial-boundary value. The coefficient matrix of the discretized system is full and posses Toeplitz-like structure. Then an incomplete circulant and skew-circulant splitting (ICSCS) iteration method is present. The main advantage of the ICSCS iteration method is that one only needs to carry out the iteration by two fast Fourier transforms and computing a product of a diagonal matrix and a vector. The convergence conditions are analyzed in detail. Numerical examples are used to show the feasibility and effectiveness of the ICSCS iteration method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a monotone numerical method on uniform mesh for solving singularly perturbed three-point reaction–diffusion boundary value problems. Firstly, properties of the exact solution are analyzed. Difference schemes are established by the method of the integral identities with the appropriate quadrature rules with remainder terms in integral form. It is then proved that the method is second-order uniformly convergent with respect to singular perturbation parameter, in discrete maximum norm. Finally, one numerical example is presented to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]