Abstract: Abstract We prove the convergence of certain second-order numerical methods to weak solutions of the Navier–Stokes equations satisfying, in addition, the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli–Kohn–Nirenberg. More precisely, we treat the space-periodic case in three space dimensions and consider a full discretization in which the classical Crank–Nicolson method (θ-method with \(\theta =1/2\) ) is used to discretize the time variable. In contrast, in the space variables, we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions toward weak solutions (satisfying a precise local energy balance) without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes, providing alternate proofs of the existence of “physically relevant” solutions in three space dimensions. PubDate: 2022-11-28

Abstract: Abstract This paper investigates the quasiconsensus problem of fractional-order heterogeneous multiagent systems, the distributed impulsive control protocol is designed for the multiagent system. In contrast to some existing results, the impulsive moments are determined by preset events, i.e., the event-triggered mechanism is used. Based on the fractional-order Lyapunov stability theory and fractional-order differential inequality, the quasiconsensus criteria are derived; furthermore, the prescribed error bound is given. Then, Zeno behavior for the considered event-triggered control method is excluded. Finally, numerical examples are given to shown the effectiveness of the proposed method. PubDate: 2022-11-24

Abstract: Abstract Nonorthogonal polynomials have many useful properties like being used as a basis for spectral methods, being generated in an easy way, having exponential rates of convergence, having fewer terms and reducing computational errors in comparison with some others, and producing most important basic polynomials. In this regard, this paper deals with a new indirect numerical method to solve fractional optimal control problems based on the generalized Lucas polynomials. Through the way, the left and right Caputo fractional derivatives operational matrices for these polynomials are derived. Based on the Pontryagin maximum principle, the necessary optimality conditions for this problem reduce into a two-point boundary value problem. The main and efficient characteristic behind the proposed method is to convert the problem under consideration into a system of algebraic equations which reduces many computational costs and CPU time. To demonstrate the efficiency, applicability, and simplicity of the proposed method, several examples are solved, and the obtained results are compared with those obtained with other methods. PubDate: 2022-11-24

Abstract: Abstract This paper investigates a globally coupled map lattice. Rigorous proofs to the existence of chaos in the sense of both Li–Yorke and Devaney in two controlled globally coupled map lattices are presented. In addition, the existence of Li–Yorke chaos and Devaney chaos for a general discrete dynamical system in \({\mathbf {R}}^{N}\) and \(l^{\infty}\) is considered. For illustration, two examples are provided. PubDate: 2022-11-16

Abstract: Abstract In this paper, we replace the standard numerical approach of estimating parameters in a mathematical model using numerical solvers for differential equations with a physics-informed neural network (PINN). This neural network requires a sequence of time instances as direct input of the network and the numbers of susceptibles, vaccinated, infected, hospitalized, and recovered individuals per time instance to learn certain parameters of the underlying model, which are used for the loss calculations. The established model is an extended susceptible-infected-recovered (SIR) model in which the transitions between disease-related population groups, called compartments, and the physical laws of epidemic transmission dynamics are expressed by a system of ordinary differential equations (ODEs). The system of ODEs and its time derivative are included in the residual loss function of the PINN in addition to the data error between the current network output and the time series data of the compartment sizes. Further, we illustrate how this PINN approach can also be used for differential equation-based models such as the proposed extended SIR model, called SVIHR model. In a validation process, we compare the performance of the PINN with results obtained with the numerical technique of non-standard finite differences (NSFD) in generating future COVID-19 scenarios based on the parameters identified by the PINN. The used training data set covers the time between the outbreak of the pandemic in Germany and the last week of the year 2021. We obtain a two-step or hybrid approach, as the PINN is then used to generate a future COVID-19 outbreak scenario describing a possibly next pandemic wave. The week at which the prediction starts is chosen in mid-April 2022. PubDate: 2022-10-27

Abstract: Abstract Peridynamic (PD) theories have become widespread in various research areas due to the ability of modeling discontinuity formation and evolution in materials. Bond-based peridynamics (BB-PD), notwithstanding some modeling limitations, is widely employed in numerical simulations due to its easy implementation combined with physical intuitiveness and stability. In this paper, we review and investigate several aspects of bond-based peridynamic models. We present a detailed description of peridynamics theory, applications, and numerical models. We display the employed BB-PD integral kernels together with their differences and commonalities; then we discuss some consequences of their mathematical structure. We critically analyze and comment on the kinematic role of nonlocality, the relation between kernel structure and material impenetrability, and the role of PD kernel nonlinearity in crack formation prediction. Finally, we propose and present the idea of extending BB-PD to fluids in the framework of fading memory material, drawing some perspectives for a deeper and more comprehensive understanding of the peridynamics in fluids. PubDate: 2022-10-23

Abstract: Abstract We devise a numerical scheme for computing arc-length parameterized curves of low bending energy that are confined to convex domains. We address the convergence of the discrete formulations to a continuous model and the unconditional stability of an iterative scheme. Numerical simulations confirm the theoretical results and lead to a classification of observed optimal curves within spheres. PubDate: 2022-10-21

Abstract: Abstract Parametric interpolatory curves play a vital part in geometric modeling. Cubic Catmull–Rom spline is a well-known tool for constructing parametric interpolatory curves, but it cannot be modified once its control points are fixed. We propose a novel quartic Catmull–Rom spline with free parameters to tackle this issue. The quartic Catmull–Rom spline owns shape adjustability based on inheriting the features of the cubic Catmull–Rom spline. Some modeling examples show that the shape of the quartic Catmull–Rom spline can realize both global adjustment and local adjustment by changing the free parameters. In addition, we give three schemes for optimizing the shape of the quartic Catmull–Rom spline, which can generate the spline with minimal internal energy, the shape-preserving spline, and the monotonicity-preserving spline. Numerical examples indicate that the proposed schemes are effective and the quartic Catmull–Rom spline is more practical than the cubic Catmull–Rom spline in data interpolation. PubDate: 2022-10-21

Abstract: Abstract In this work, the Chebyshev collocation scheme is extended for the Volterra integro-differential equations of pantograph type. First, we construct the operational matrices of pantograph and derivative based on Chebyshev polynomials. Also, the obtained operational matrices are utilized to approximate the derivatives of unknown functions. Furthermore, a detailed analysis of convergence is discussed in the weighted square norm. We conduct some numerical experiments to verify the high performance of the suggested numerical approach. The results show that the computational scheme is accurate. PubDate: 2022-10-01

Abstract: Abstract In this study, the time-dependent potential coefficient in a higher-order PDE with initial and boundary conditions is numerically constructed for the first time from a nonlocal integral condition. Even though the inverse identification problem investigated in this study is ill-posed, it has a unique solution. For discretizing the direct problem and finding stable and accurate solutions, we employ the Quintic B-spline (QBS) collocation and Tikhonov regularization methods, respectively. The following nonlinear minimization problem is solved using MATLAB. The collected findings demonstrate that accurate and stable solutions can be found. PubDate: 2022-09-01

Abstract: Abstract Image denoising approaches based on partial differential modeling have attracted a lot of attention in image processing due to their high performance. The nonlinear anisotropic diffusion equations, specially Perona–Malik model, are powerful tools that improve the quality of the image by removing noise while preserving details and edges. In this paper, we propose a powerful and accurate local meshless algorithm to solve the time-fractional Perona–Malik model which has an adjustable fractional derivative making the control of the diffusion process more convenient than the classical one. In order to overcome the complexities of the problem, a suitable combination of the compactly supported radial basis function method and operator splitting technique is proposed to convert a complex time-fractional partial differential equation into sparse linear algebraic systems that standard solvers can solve. The numerical results of classical and fractional models are explored in different metrics to demonstrate the proposed scheme’s effectiveness. The numerical experiments confirm that the method is suitable to denoise digital images and show that the fractional derivative increases the model’s ability to remove noise in images. PubDate: 2022-09-01

Abstract: Abstract In this paper, we focus on the development and study of the finite difference/pseudo-spectral method to obtain an approximate solution for the time-fractional diffusion-wave equation in a reproducing kernel Hilbert space. Moreover, we make use of the theory of reproducing kernels to establish certain reproducing kernel functions in the aforementioned reproducing kernel Hilbert space. Furthermore, we give an approximation to the time-fractional derivative term by applying the finite difference scheme by our proposed method. Over and above, we present an appropriate technique to derive the numerical solution of the given equation by utilizing a pseudo-spectral method based on the reproducing kernel. Then, we provide two numerical examples to support the accuracy and efficiency of our proposed method. Finally, we apply numerical experiments to calculate the quality of our approximation by employing discrete error norms. PubDate: 2022-08-23

Abstract: Abstract In this paper, we construct a new linear second-order finite difference scheme with two parameters for space-fractional Allen–Cahn equations. We first prove that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Secondly, we analyze the maximum-norm error. Thirdly, we can see that the proposed scheme is unconditionally energy-stable by defining the modified energy and selecting the appropriate parameters. Finally, two numerical examples are presented to verify the theoretical results. PubDate: 2022-08-20

Abstract: Abstract The goal of this paper is to present a new class of operators satisfying the Prešić-type rational η-contraction condition in the setting of usual metric spaces. New fixed point results are also obtained for these operators. Our results generalize, extend, and unify many papers in this direction. Moreover, two examples are derived to support and document our theoretical results. Finally, to strengthen our paper and its contribution to applications, some convergence results for a class of matrix difference equations are investigated. PubDate: 2022-08-17

Abstract: Abstract In this paper, we establish an SIVR model with diffusion, spatially heterogeneous, latent infection, and incomplete immunity in the Neumann boundary condition. Firstly, the threshold dynamic behavior of the model is proved by using the operator semigroup method, the well-posedness of the solution and the basic reproduction number \(\Re _{0}\) are given. When \(\Re _{0}<1\) , the disease-free equilibrium is globally asymptotically stable, the disease will be extinct; when \(\Re _{0}>1\) , the epidemic equilibrium is globally asymptotically stable, the disease will persist with probability one. Then, we introduce the patient’s treatment into the system as the control parameter, and the optimal control of the system is discussed by applying the Hamiltonian function and the adjoint equation. Finally, the theoretical results are verified by numerical simulation. PubDate: 2022-07-18

Abstract: We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the p-Laplace equation or the Modica–Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of right-hand sides. PubDate: 2022-07-15

Abstract: Abstract In this paper, we investigate the generalized fractional system (GFS) with order lying in \((1, 2)\) . We present stability analysis of GFS by two methods. First, the stability analysis of that system using the Gronwall–Bellman (G–B) Lemma, the Mittag–Leffler (M–L) function, and the Laplace transform is introduced. Secondly, by the Lyapunov direct method, we study the M–L stability of our system with order lying in \((1, 2)\) . Using the modified predictor–corrector method, the solutions of GFSs are calculated and they are more complicated than the classical fractional one. Based on linear feedback control, we investigate a theorem to control the chaotic GFSs with order lying in \((1, 2)\) . We present an example to verify the validity of control theorem. We state and prove a theorem to calculate the analytical formula of controllers that are used to achieve synchronization between two different chaotic GFSs. An example to study the synchronization for systems with orders lying in \((1, 2)\) is given. We found an agreement between analytical results and numerical simulations. PubDate: 2022-07-15

Abstract: Abstract This paper proposes a local meshless radial basis function (RBF) method to obtain the solution of the two-dimensional time-fractional Sobolev equation. The model is formulated with the Caputo fractional derivative. The method uses the RBF to approximate the spatial operator, and a finite-difference algorithm as the time-stepping approach for the solution in time. The stability of the technique is examined by using the matrix method. Finally, two numerical examples are given to verify the numerical performance and efficiency of the method. PubDate: 2022-06-28

Abstract: Abstract This paper is concerned with the Cauchy problem for semilinear wave equation with space-dependent scattering damping and combined nonlinearities. The blowup results of solution are established by introducing proper test functions. Moreover, upper bound lifespan estimates of a solution to the Cauchy problem with small initial values are derived. To the best of our knowledge, the results in Theorems 1.1–1.2 are new. PubDate: 2022-06-28

Abstract: Abstract The aim of the manuscript is to present the concept of a graphical double controlled metric-like space (for short, GDCML-space). The structure of an open ball of the proposed space is also discussed, and the newly presented ideas are explained with a new technique by depicting appropriately directed graphs. Moreover, we present some examples in a graph structure to prove that our results are sharp compared to those in the previous papers. Further, the existence of a solution to the boundary value problem originating from the transverse oscillations of a homogeneous bar (TOHB) is obtained theoretically. PubDate: 2022-06-17