We present and analyse numerical methods with operator splitting procedures, applied to an epidemic model which takes into account the space-dependence of the infection. We derive conditions on the time step, under which the numerical methods preserve the non-negativity and monotonicity properties of the exact solution. Our results are illustrated by numerical experiments.

A classical approach to solving two-block separable convex optimization could be the symmetric alternating direction method of multipliers (S-ADMM). However, its convergence may not be guaranteed for a general multi-block case without additional assumptions. Bai et al. proposed a variant of S-ADMM entitled the generalized symmetric ADMM (GS-ADMM), in which the variables are regrouped into two groups

We propose novel second-order accurate methods for nonlinear space-fractional PDEs with time-dependent boundary conditions. The matrix transfer technique is used for spatial discretization of the parabolic PDEs to obtain systems of ordinary differential equations. The treatment of the boundary conditions gave rise to a source term which depends on the temporal variable only. Methods based on rational

In this work, a Sinc-Galerkin method is considered and analyzed for solving the fourth-order partial integro-differential equation with a weakly singular kernel. The time derivative and Riemann-Liouville fractional integral term are approximated via the Crank-Nicolson method and the trapezoidal convolution quadrature rule, respectively. Then a fully discrete scheme is formulated via using the Sinc-Galerkin

Selecting between explicit and implicit time marching schemes, as well as non-iterative explicit-implicit ones, is well understood on a linear stability basis. Implicit schemes are favored when either searching for steady-states or performing time accurate simulations of stiff equations due to their larger time steps. When nonlinear stability properties must be preserved, however, solution monotonicity

In this paper, we investigate the Sine-Gordon Equations with the Riesz space fractional derivative. Firstly, a fourth-order conservative difference scheme is proposed for the one-dimensional problem. The unique solvability, conservation and boundedness of the difference scheme are rigorously demonstrated. It is proved that the scheme is convergent at the order of O(τ2+h4) in the l∞ norm, where τ,h

Canonical correlation analysis (CCA) is a famous data analysis method that has been successfully used in many areas. CCA extracts meaningful information from a pair of data sets, by seeking pairs of linear combinations from two sets of variables with maximum correlation. Mathematically, CCA resorts to solving a large-scale generalized eigenvalue problem. However, as the dimension of the data sets is

The Richards' equation models the water flow through porous media in groundwater aquifers and petroleum reservoir simulations. This is a degenerate parabolic differential equation, in which no analytical solution is known. Most numerical methods use implicit time schemes supporting large time steps, but involving near degenerate nonlinear systems. To solve them, we need robust and efficient linearization

In this work we construct a multiderivative implicit-explicit (IMEX) scheme for a class of stiff ordinary differential equations (ODEs). Our solver is high-order accurate and has an asymptotic preserving (AP) property for a large class of singularly perturbed ODEs. In this context, the AP property means that the singular limit is discretely preserved when a stiff parameter ε goes to zero. The proposed

The linear complementarity problem arising from a free boundary problem can be equivalently reformulated as a fixed-point equation. We present a modified modulus-based multigrid method to solve this fixed-point equation. This modified method is a full approximation scheme using the modulus-based splitting iteration method as the smoother and avoids the transformation between the auxiliary and the original

A high-order compact alternating direction implicit scheme is considered to solve the two-dimensional time-fractional integro-differential equation with weak singularity near the initial time in this paper. The L1 formula and trapezoidal PI rule on nonuniform meshes, which greatly improve the temporal accuracy compared to the method on uniform grids, are adopted to approximate the Caputo derivative

In Cai, He, and Zhang (2017), we studied an improved Zienkiewicz-Zhu (ZZ) a posteriori error estimator for conforming linear finite element approximation to diffusion problems. The estimator is more efficient than the original ZZ estimator for non-smooth problems, but with comparable computational costs. This paper extends the improved ZZ estimator for discontinuous linear finite element approximations

The truncated singular value decomposition (TSVD) is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. This method replaces the matrix A by the closest matrix Ak of low rank k, and then computes the minimal norm solution of the linear system of equations with a rank-deficient matrix so obtained. The modified TSVD (MTSVD) method improves the TSVD

Recently, numerous numerical schemes have been developed for solving single-term time-space fractional diffusion-wave equations. Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity. In this paper, we present an efficient alternating direction implicit (ADI) scheme for two-dimensional multi-term time-space fractional nonlinear

The global Krylov subspace iterative methods are an attractive class of iterative solvers for solving linear systems with several right-hand sides. In this paper, the global version of the GMRES method is applied to solve linear discrete ill-posed problems that arise from the discretization of linear ill-posed problems, pattern classification and dimensionality reduction. The regularizing properties

In this paper, we develop a new WENO weak Galerkin finite element scheme for solving the time dependent hyperbolic equations. The upwind-type stabilizer is imposed to enforce the flux direction in the scheme. For the linear convection equations, we analyze the L2-stability and error estimate for L2-norm. We also investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology

Based on a fully overlapping domain decomposition approach, a parallel stabilized finite element variational multiscale method for the incompressible Navier-Stokes equations is proposed, where the stabilizations both for the velocity and pressure are based on two local Gauss integrations at the element level. The basic idea of the method is to use a locally refined global mesh to compute a stabilized

Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the relative importance or centrality of nodes. Another popular approach to rank the nodes of a network is to compute the left Perron vector of the adjacency matrix for

In this paper, we focus on the robust partial quadratic eigenvalue assignment problem for vibrating structures by active feedback control. This problem is reformulated as an minimization problem, where the cost function can measure both the sensitivity of the closed-loop eigenvalues and the feedback norms. This can be seen as a modified version of the minimization problem proposed by Bai et al. (2016)

In this work we propose an algorithm for computing a stationary flow in a bifurcation tree. Our idea is to divide the tree into smaller basic blocks, each corresponding to one bifurcation, and solve a sequence of flow problems for the individual blocks. Numerical experiments demonstrate that the algorithm works well. We give a criteria for convergence that can be verified numerically and also an analytical

The ill-posedness of the forward-backward heat problem arises in boundary layer problems, fluid dynamics, plasma physics, astrophysics and the study of propagation of an electron beam through the solar corona. In this study, a new truly meshless method is developed for the numerical solution of the two dimensional forward-backward heat equation. We propose a novel method based on the domain decomposition

We develop a new approach to obtain derivatives of functions on arbitrary interval by extending these functions to the whole real axis based on Hermite function approximation. A modified Tikhonov regularization with a new penalty term derived from the Fourier transform is used to stabilize the extension process. A complete convergence theory is given and the process of numerical implementation is also

In this paper, we establish the unconditionally optimal error estimates of linearized second-order backward difference formula (BDF2) Galerkin finite element methods (FEMs) for the Landau-Lifshitz equation describing the magnetic behavior in ferromagnetic materials. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into

Explicit two-step Runge-Kutta (TSRK) methods offer an efficient alternative to traditional explicit Low-Storage Runge-Kutta (LSRK) schemes for solving the Navier-Stokes equations. A special class of TSRK methods that reduce requirement compared to previous TSRK schemes are derived. Schemes of fourth, fifth and sixth order are implemented and tested. The new schemes are evaluated with two common test

In this paper, a number of energy stable numerical schemes are proposed for the fractional Cahn–Hilliard equation. We prove mass conservation, unique solvability and energy stability for three time semi-discretized schemes based on the first-order semi-implicit scheme, the Crank–Nicolson scheme and the BDF2 scheme respectively. Then we present error analysis for these numerical schemes with the Fourier

Numerical schemes for nonlinear weakly singular Volterra and Fredholm integral equations of the first kind are rarely investigated in the literature. In this paper, we present numerical solutions for these types of equations including Abel equations by hybrid block-pulse functions and Legendre polynomials. Hybrid functions give us the opportunity to attend a highly accurate solution by adjusting the

Recently, there has been much interest in filter methods for solving linear and nonlinear problems. We describe extensions of such methods to nonlinear semidefinite programming problems. These extensions are based on linear subproblems that combine filter techniques and trust region methods. The results of convergence and the rate of convergence are given. Numerical examples on semidefinite programming

This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying

The histopolation with quadratic/linear rational splines of class C2 is studied. These splines keep the sign of its second derivative on the whole interval and because of that the given histogram is assumed to be strictly convex. The grid points of the histogram and spline knots between them are supposed to be placed arbitrarily. We show that there is a strictly convex histogram without the solution

It is efficient to use the Hestenes–Stiefe (HS) conjugate gradient algorithm in solving large-scale complex smooth optimization problems because of its simplicity and low calculation requirements. Additionally, this algorithm has been used to simultaneously solve large-scale nonsmooth problems, nonlinear equations, and practical application. In this paper, the authors propose some modified HS conjugate

In this paper, we propose a finite volume scheme preserving discrete maximum principle (DMP) for the imperfect interface problem. We prove the existence of the numerical solution and prove that when the interface parameter tends to 0, the numerical solutions of imperfect interface problem have a subsequence converging to the numerical solution of perfect interface problem. We apply a domain decomposition

In this work, we consider a fractional extension of the Klein–Gordon–Zakharov system which describes the propagation of strong turbulences on the Langmuir wave in a high-frequency plasma. Both components consider space-fractional derivatives of the Riesz type, and initial-boundary conditions are imposed on a closed and bounded interval of the real numbers. In a first stage, we show that the total energy

In this paper, we propose a unified and high order accurate fully-discrete one-step ADER Discontinuous Galerkin method for the simulation of linear seismic waves in the sea bottom that are generated by the propagation of free surface water waves. In particular, a hyperbolic reformulation of the Serre-Green-Naghdi model for nonlinear dispersive free surface flows is coupled with a first order velocity-stress

This work presents the development, analysis and numerical simulations of a biophysical model for 3D cell deformation and movement, which couples biochemical reactions and biomechanical forces. We propose a mechanobiochemical model which considers the actin filament network as a viscoelastic and contractile gel. The mechanical properties are modelled by a force balancing equation for the displacements

The absolute value equation appears in various fields of applied mathematics such as operational research. Here we consider its generalized versionAX+B|X|=C, where A,B,C∈Cn×n are given, |X|=(|xi,j|) and X∈Cn×n is an unknown matrix that must be determined. In this investigation, based on the Picard matrix splitting iteration method, we applied a matrix splitting method for solving it. We will see that

We show that applying any deterministic B-series method of order pd with a random step size to single integrand SDEs gives a numerical method converging in the mean-square and weak sense with order ⌊pd/2⌋. As an application, we derive high order energy-preserving methods for stochastic Poisson systems as well as further geometric numerical schemes for this wide class of Stratonovich SDEs.

In this paper, an hp-version Chebyshev collocation method is developed for the nonlinear fractional differential equations. The hp-version error bounds of the Chebyshev collocation method are derived under the L2-norm and the L∞-norm. Numerical experiments are included to illustrate the theoretical results.

In this paper, we propose an unconditionally stable numerical scheme based on finite difference for the approximation of time-fractional diffusion equation on a metric star graph. The fractional derivative is considered in Caputo sense and the so-called L1 method is used for the discrete approximation of Caputo fractional derivative. The convergence and stability of the difference scheme has been proved

In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the F12 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the complex Higgins functions, the complex Christov functions, and their sine-like and cosine-like versions. Then, after studying the asymptotic behavior of the resulting expressions, we discuss the numerical difficulties

This paper is mainly concerned with developing and establishing the reduced-order extrapolated format about the unknown coefficient vectors in numerical solutions to the finite element (FE) method for the hyperbolic type equation. To this end, the functional-form FE format, the existence, stability, and error estimates of the FE solutions, the matrix-form FE format for the hyperbolic type equation

In the present work, a novel variable-order awareness programs mathematical model with multiple time delay is presented. Two types of variable-order fractional derivatives of Atangana-Baleanu-Caputo definitions are given. The properties of the proposed model such as the basic reproduction number and the stability of the equilibrium points are studied analytically and numerically. Two numerical methods

This article is devoted to a new semi-analytical algorithm for solving time-fractional modified anomalous sub-diffusion equations (FMASDEs). In this method first, the main problem is reduced to a system of fractional-order ordinary differential equations (FODEs) under known initial value conditions by using the Chebyshev collocation procedure. After that, to solve this system, some auxiliary initial

Mixed finite element methods are applied to a Poisson problem with a singularity at a boundary point. The approximation spaces are based on quarter-point elements, the shape functions inheriting the singular behavior of their quadratic geometric maps. Two mesh scenarios are considered, by fixing some macro quarter-point elements at the coarse level, and subdividing them by mapping uniformly refined

This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally

This study aims to use the Taylor wavelet method to solve linear and nonlinear Lane-Emden equations. An advantage of the method is the orthonormality property of the polynomials which reduce the computational cost. Another advantage is that the nonlinear terms do not need to be approximated. The application of the method reduces the differential equations to a system of algebraic equations. Six differential

This paper presents an approximate solution of time variable order fractional differential equations with sub-diffusion and super-diffusion. The aim of paper is to solve and analyze this problem by a fully discrete local discontinuous Galerkin scheme. The method is based on local discontinuous Galerkin method in space and a finite difference technique in time. The numerical stability and convergence

We study the behavior of the finite element condition numbers on a class of anisotropic meshes. These newly-developed mesh algorithms can produce numerical approximations with optimal convergence to isotropic and anisotropic singular solutions of elliptic boundary value problems in two- and three-dimensions. Despite the simplicity and fewer geometric constraints in implementation, these meshes can

In this paper, a new method is proposed to solve the time-fractional reaction-diffusion equation with nonlinear variable coefficients. Based on the second-order weighted and shifted Grünwald difference (WSGD) method for time Riemann-Liouville fractional derivative, we formulate and analyze the extrapolated Crank-Nicolson (ECN) method in time and orthogonal spline collocation (OSC) method in space.

We consider in this paper numerical approximations for the hydro-dynamically coupled phase-field model for an immiscible binary mixture of passive/active nematic Liquid Crystals (LCs) and viscous fluids. The passive model is a highly coupled nonlinear system that consists of the incompressible Navier-Stokes equations, the Cahn-Hilliard equations, and the constitutive equation for the polarization field

Efficient numerical methods using Sinc quadrature formula are proposed and analyzed for second kind Fredholm integral equations over infinite intervals. These methods are obtained by combining the infinite trapezoidal rule with two concrete families of single and double exponential transformations. In such a way, the integral equation is reduced to a set of linear algebraic equations. Furthermore,

The plasmonics of two–dimensional materials, such as graphene, has become an important field over the past decade. The active tunability of graphene via electrical gating or chemical doping has generated a great deal of excitement among engineers seeking sensing devices. Consequently there is significant demand for robust and highly accurate computational capabilities which can simulate such materials

We describe a computational method for parameter estimation in linear regression, that is capable of simultaneously producing sparse estimates and dealing with outliers and heavy-tailed error distributions. The method used is based on the image restoration method proposed in Huang et al. (2017) [13]. It can be applied to problems of arbitrary size. The choice of certain parameters is discussed. Results

In this work, we discuss the stability and convergence of the Fourier pseudo-spectral scheme coupled with several linearized finite difference methods for FitzHugh-Nagumo model. The work of this article has three main features. Firstly, our full-discrete schemes are linear and easy to implement for two dimensional or three dimensional simulations. Secondly, by constructing the auxiliary interpolation

In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically

In this paper, we consider the nonstationary magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation. The Crank-Nicolson extrapolation scheme is used for time derivative terms, and the mixed finite method is used for spatial discretization. We employ the Taylor-Hood finite elements to approximate Navier-Stokes equations, Nédélec edge element for the magnetic induction

In this paper, a modified weak Galerkin (MWG) finite element method with weakly imposed boundary conditions is presented for solving convection-dominated diffusion equations. The method is shown uniformly stable for all diffusion parameters. The method converges at the optimal order for large diffusion problems in the energy norm, and at half a super-convergent order for small diffusion problems. Various

Color image restoration is an ill-posed problem, and regularization is necessary. In this paper, we first formulate the color image restoration problem into a constrained minimization problem that minimizes a quadratic functional and subject to a constraint that the total variation of the color image is less than a given parameter δ. The advantages of the constrained minimization problem over the traditional

Recently, a new stabilizer free weak Galerkin method (SFWG) is proposed, which is easier to implement and more efficient. The main idea is that by letting j≥j0 for some j0, where j is the degree of the polynomials used to compute the weak gradients, then the stabilizer term in the regular weak Galerkin method is no longer needed. Later on in [1], the optimal of such j0 for a certain type of finite

Single domain spectral/pseudospectral integration preconditioning matrices have been shown to be effective operators for solving differential equations. In this study we extend the integration precondition methodology to a multidomain computational framework, and construct global inverse matrices for the advection operator discretized by multidomain Gauss-Lobatto-Legendre pseudospectral methods. These

The main purpose of this work is to construct and analyze an efficient numerical scheme for solving the fractional neutron diffusion equation with delayed neutrons, which describes neutron transport in a nuclear reactor. The L1 approximation is used for discretization of time derivative and finite difference method is used for discretization of space derivative. The stability and convergence analysis