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In this study, a novel reaction–diffusion model for the spread of the new coronavirus (COVID-19) is investigated. The model is a spatial extension of the recent COVID-19 SEIR model with nonlinear incidence rates by taking into account the effects of random movements of individuals from different compartments in their environments. The equilibrium points of the new system are found for both diffusive and non-diffusive models, where a detailed stability analysis is conducted for them. Moreover, the stability regions in the space of parameters are attained for each equilibrium point for both cases of the model and the effects of parameters are explored. A numerical verification for the proposed model using a finite difference-based method is illustrated along with their consistency, stability and proving the positivity of the acquired solutions. The obtained results reveal that the random motion of individuals has significant impact on the observed dynamics and steady-state stability of the spread of the virus which helps in presenting some strategies for the better control of it.

We establish uniform error estimates of finite difference methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter ε (ε Є (0,1]). When ε -> 0⁺, NLSW collapses to the standard NLS. In the small perturbation parameter regime, i. e., 0 < ε < 1, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with O(ε²)-wavelength at O(ε⁴) and O(ε²) amplitudes for well-prepared and ill-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in ε of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, and the semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in ε, at the order of O(h² + r) and O(h² + r 2/3 ) with time step r and mesh size h for well-prepared and ill-prepared initial data, respectively, for both CNFD and SIFD in the l² -norm and discrete semi-H¹ norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two, and three dimensions. To derive these uniform error bounds, we combine ε-dependent error estimates of NLSW, ε-dependent error bounds between the numerical approximate solutions of NLSW and the solution of NLS, together with error bounds between the solutions of NLSW and NLS. Other key techniques in the analysis include the energy method, cut-off of the nonlinearity, and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi-H¹ norm estimate. Finally, numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes.

The aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

Artificial electromagnetic surfaces, metasurfaces, control light in the desired manner through the introduction of abrupt changes of electromagnetic fields at interfaces. Current modelling of metasurfaces successfully exploits generalised sheet transition conditions (GSTCs), a set of boundary conditions that account for electric and magnetic metasurface-induced optical responses. GSTCs are powerful theoretical tools but they are not readily applicable for arbitrarily shaped metasurfaces. Accurate and computationally efficient algorithms capable of implementing artificial boundary conditions are highly desired for designing free-form photonic devices. To address this challenge, we propose a numerical method based on conformal boundary optics with a modified finite difference time-domain (FDTD) approach which accurately calculates the electromagnetic fields across conformal metasurfaces. Illustrative examples of curved meta-optics are presented, showing results in good agreement with theoretical predictions. This method can become a powerful tool for designing and predicting optical functionalities of conformal metasurfaces for new lightweight, flexible and wearable photonic devices. Predicting and modelling the responses of free-from photonics devices remain challenging with conventional computational tools. Here, the authors propose an efficient algorithm based on conformal boundary optics and modified finite difference time-domain to calculate the electromagnetic fields across conformal metasurfaces.

In this paper, we present a method based on radial basis function (RBF)-generated finite differences (FD) for numerically solving diffusion and reaction–diffusion equations (PDEs) on closed surfaces embedded in R d . Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.

In this paper, we study the Caputo–Hadamard fractional partial differential equation where the time derivative is the Caputo–Hadamard fractional derivative and the space derivative is the integer-order one. We first introduce a modified Laplace transform. Then using the newly defined Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution to this kind of linear equation. Furthermore, we study the regularity and logarithmic decay of its solution. Since the equation has a time fractional derivative, its solution behaves a certain weak regularity at the initial time. We use the finite difference scheme on non-uniform meshes to approximate the time fractional derivative in order to guarantee the accuracy and use the local discontinuous Galerkin method (LDG) to approximate the spacial derivative. The fully discrete scheme is established and analyzed. A numerical example is displayed which support the theoretical analysis.

We consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models. Two classes of discretization methods are presented, including standard finite element methods and quadrature-based finite difference methods. We discuss the applicability of these approaches to nonlocal problems having various singular kernels and study basic numerical analysis issues. We illustrate the similarities and differences of the resulting nonlocal stiffness matrices and discuss whether discrete maximum principles can be established. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. While it is known that the nonlocal models converge to corresponding differential equations in the local limit, we elucidate how such limiting behaviors may or may not be preserved in various discrete approximations. Our findings thus offer important insight into applications and simulations of nonlocal models.

In this paper, a novel meshless method that can handle porous flow problems with singular source terms is developed by virtually constructing node control domains. By defining a connectable node cloud, this method uses the integral of the diffusion term and generalized finite-difference operators to derive overdetermined equations of the node control volumes. An empirical method of calculating reliable node control volumes and a triangulation-based method to determine the connectable point cloud are developed. The method focuses only on the volume of the node control domain rather than the specific shape, so the construction of node control domains is virtual and does not increase the computational cost. To our knowledge, this is the first study to construct node control volumes in the meshless framework, termed the node control domain-based meshless method (NCDMM), which can also be regarded as an extended finite-volume method (EFVM). Taking two-phase porous flow problems as an example, discrete NCDMM schemes satisfying local mass conservation are derived by integrating the generalized finite-difference schemes of governing equations on each node control domain. Finally, commonly used low-order finite-volume method (FVM)-based nonlinear solvers for various porous flow models can be directly employed in the proposed NCDMM, significantly facilitating general-purpose application. Theoretically, the proposed NCDMM has the advantages of previous meshless methods for discretizing computational domains with complex geometries, as well as the advantages of traditional low-order FVMs for stably handling a variety of porous flow problems with local mass conservation. Four numerical cases are implemented to test the computational accuracy, efficiency, convergence, and good adaptability to the calculation domain with complex geometry and various boundary conditions.

Producing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant |J| everywhere, the number of pixels (2D) or voxels (3D) with |J|<0 has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, |J| is commonly approximated using a central difference, but this strategy can yield positive |J|’s for transformations that are clearly not diffeomorphic—even at the pixel or voxel resolution level. To show this, we first investigate the geometric meaning of different finite difference approximations of |J|. We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of |J| is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of |J|’s must be positive to ensure that the entire domain is invertible and free of folding at the pixel level. For a 3D transformation, ten unique finite differences approximations of |J|’s are required to be positive. Our proposed digital diffeomorphism criteria solves several errors inherent in the central difference approximation of |J| and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism.

A numerical scheme based on polynomials and finite difference method is developed for numerical solutions of two-dimensional linear and nonlinear Sobolev equations. In this approach, finite difference method is applied for the discretization of time derivative whereas space derivatives are approximated by two-dimensional Lucas polynomials. Applying the procedure and utilizing finite Fibonacci sequence, differentiation matrices are derived. With the help of this technique, the differential equations have been transformed to system of algebraic equations, the solution of which compute unknown coefficients in Lucas polynomials. Substituting the unknowns constants in Lucas series, required solution of targeted equation has been obtained. Performance of the method is verified by studying some test problems and computing E2, E∞ and Erms (root mean square) error norms. The obtained accuracy confirms feasibility of the proposed technique.